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3.324 \(\int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=279 \[ \frac {\left (8 a^2 A-2 a b B+A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}+\frac {\left (16 a^3 B+8 a^2 A b+2 a b^2 B-A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}-\frac {\sqrt {a-i b} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(6 a B+A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

[Out]

1/8*(8*A*a^2*b-A*b^3+16*B*a^3+2*B*a*b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/a^(5/2)/d-(I*A+B)*arctanh((a+
b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))*(a-I*b)^(1/2)/d+(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))*(a+I*
b)^(1/2)/d+1/8*(8*A*a^2+A*b^2-2*B*a*b)*cot(d*x+c)*(a+b*tan(d*x+c))^(1/2)/a^2/d-1/12*(A*b+6*B*a)*cot(d*x+c)^2*(
a+b*tan(d*x+c))^(1/2)/a/d-1/3*A*cot(d*x+c)^3*(a+b*tan(d*x+c))^(1/2)/d

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Rubi [A]  time = 1.17, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3608, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {\left (8 a^2 A b+16 a^3 B+2 a b^2 B-A b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}+\frac {\left (8 a^2 A-2 a b B+A b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {\sqrt {a-i b} (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {(6 a B+A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

((8*a^2*A*b - A*b^3 + 16*a^3*B + 2*a*b^2*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*a^(5/2)*d) - (Sqrt[a
 - I*b]*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (Sqrt[a + I*b]*(I*A - B)*ArcTanh[Sqrt[a
 + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + ((8*a^2*A + A*b^2 - 2*a*b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*
a^2*d) - ((A*b + 6*a*B)*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(12*a*d) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c
 + d*x]])/(3*d)

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 3539

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
 + I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] &&  !IntegerQ[m]

Rule 3608

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*b - a*B)*(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n)/(
f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(b*(m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f
*x])^(n - 1)*Simp[b*B*(b*c*(m + 1) + a*d*n) + A*b*(a*c*(m + 1) - b*d*n) - b*(A*(b*c - a*d) - B*(a*c + b*d))*(m
 + 1)*Tan[e + f*x] - b*d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B},
 x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && LtQ[0, n, 1] && (IntegerQ[
m] || IntegersQ[2*m, 2*n])

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
 /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rule 3649

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*(b*B - a*C))*(a + b*T
an[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n + 1))/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2)), x] + Dist[1/((m + 1)*(
b*c - a*d)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1)
 - b^2*d*(m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d)*(A*b - a*B - b*C)*Tan[e
+ f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C,
 n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] &&  !(ILtQ[n, -1] && ( !I
ntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rubi steps

\begin {align*} \int \cot ^4(c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx &=-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {1}{3} \int \frac {\cot ^3(c+d x) \left (\frac {1}{2} (-A b-6 a B)+3 (a A-b B) \tan (c+d x)+\frac {5}{2} A b \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {\cot ^2(c+d x) \left (-\frac {3}{4} \left (8 a^2 A+A b^2-2 a b B\right )-6 a (A b+a B) \tan (c+d x)-\frac {3}{4} b (A b+6 a B) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a}\\ &=\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\int \frac {\cot (c+d x) \left (\frac {3}{8} \left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right )-6 a^2 (a A-b B) \tan (c+d x)-\frac {3}{8} b \left (8 a^2 A+A b^2-2 a b B\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}\\ &=\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {\int \frac {-6 a^2 (a A-b B)-6 a^2 (A b+a B) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{16 a^2}\\ &=\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{2} ((a-i b) (A-i B)) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} ((a+i b) (A+i B)) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 a^2 d}\\ &=\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {(i (a-i b) (A-i B)) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac {((i a-b) (A+i B)) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{8 a^2 b d}\\ &=\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}+\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {((a-i b) (A-i B)) \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i a}{b}+\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}-\frac {((a+i b) (A+i B)) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i a}{b}-\frac {i x^2}{b}} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{b d}\\ &=\frac {\left (8 a^2 A b-A b^3+16 a^3 B+2 a b^2 B\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 a^{5/2} d}-\frac {\sqrt {a-i b} (i A+B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} (i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2 A+A b^2-2 a b B\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 a^2 d}-\frac {(A b+6 a B) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 a d}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\\ \end {align*}

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Mathematica [B]  time = 6.44, size = 564, normalized size = 2.02 \[ \frac {2 b^4 \left (-\frac {(a A-b B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{2 a^{3/2} b^3}-\frac {3 (a B+A b) \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a b}\right )}{8 a b^2}+\frac {5 A \left (\frac {3 \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\cot (c+d x) \sqrt {a+b \tan (c+d x)}}{a b}\right )}{a}+\frac {2 \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{a b^2}\right )}{48 b}+\frac {(a B+A b) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} b^4}-\frac {(a B+A b) \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{4 a b^4}+\frac {(a A-b B) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{2 a b^4}-\frac {A \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{6 b^4}-\frac {\left (a A b+a \sqrt {-b^2} B+A \sqrt {-b^2} b+b^2 (-B)\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+\sqrt {-b^2}}}\right )}{2 \left (-b^2\right )^{5/2} \sqrt {a+\sqrt {-b^2}}}+\frac {\left (a A b-a \sqrt {-b^2} B-A \sqrt {-b^2} b+b^2 (-B)\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {-b^2}}}\right )}{2 b^4 \sqrt {-b^2} \sqrt {a-\sqrt {-b^2}}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

(2*b^4*(((A*b + a*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(Sqrt[a]*b^4) - ((a*A - b*B)*ArcTanh[Sqrt[a +
b*Tan[c + d*x]]/Sqrt[a]])/(2*a^(3/2)*b^3) + ((a*A*b - A*b*Sqrt[-b^2] - b^2*B - a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a - Sqrt[-b^2]]])/(2*b^4*Sqrt[-b^2]*Sqrt[a - Sqrt[-b^2]]) - ((a*A*b + A*b*Sqrt[-b^2] -
b^2*B + a*Sqrt[-b^2]*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + Sqrt[-b^2]]])/(2*(-b^2)^(5/2)*Sqrt[a + Sqrt[
-b^2]]) + ((a*A - b*B)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(2*a*b^4) - ((A*b + a*B)*Cot[c + d*x]^2*Sqrt[a +
 b*Tan[c + d*x]])/(4*a*b^4) - (A*Cot[c + d*x]^3*Sqrt[a + b*Tan[c + d*x]])/(6*b^4) - (3*(A*b + a*B)*(ArcTanh[Sq
rt[a + b*Tan[c + d*x]]/Sqrt[a]]/a^(3/2) - (Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*b)))/(8*a*b^2) + (5*A*((2
*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d*x]])/(a*b^2) + (3*(ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]]/a^(3/2) - (C
ot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(a*b)))/a))/(48*b)))/d

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 4.65, size = 118304, normalized size = 424.03 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^4*(a+b*tan(d*x+c))^(1/2)*(A+B*tan(d*x+c)),x, algorithm="maxima")

[Out]

integrate((B*tan(d*x + c) + A)*sqrt(b*tan(d*x + c) + a)*cot(d*x + c)^4, x)

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mupad [B]  time = 8.46, size = 16796, normalized size = 60.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(A + B*tan(c + d*x))*(a + b*tan(c + d*x))^(1/2),x)

[Out]

atan(((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4
 + 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/
(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d
^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2)
- (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2
)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 -
 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2
- 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)
/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*
d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^
2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 +
2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^
14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2
) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4
*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16
- 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 +
 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B
^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^
9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A
^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((224*A*a^4*b^1
1*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a
^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2
) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1
/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B
^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d
^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 -
64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 +
2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^
2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*
B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 -
 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*
A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^
4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2
*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4
*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*
A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B
^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3
*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a
)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b
*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*1i)/((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^
4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^1
0*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 -
 B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2
*d^2))^(1/2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan
(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 10
24*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 +
4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*
A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (1
6*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96
*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528
*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^
6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b
^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))
^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a
^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a
^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b
^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d
^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^
(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + (((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*
B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(
a + b*tan(c + d*x))^(1/2)*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^
2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*
((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*
B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^
3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*
B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a
^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d
^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^
3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*
a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A
*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*
b^8*d^2)/(a^4*d^5))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
- A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*
x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 -
 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b
^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 51
2*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 -
B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*
d^2))^(1/2) + (12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5
*a^8*b^9 + 7*B^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12
+ 112*A^2*B^3*a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b
^11 - 64*A^3*B^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A
*B^4*a^8*b^9 + (3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*
a^9*b^8)/(a^4*d^5)))*((B^2*a)/(4*d^2) - (A^2*a)/(4*d^2) - (2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) + (A*B*b)/(2*d^2))^(1/2)*2i + atan(((((((224
*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b
^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B
^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d
^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2
*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A
*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*
b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b
^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^
2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*
d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*
b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15
*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 16
8*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2
*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) +
(B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b
^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^
10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12
*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4
))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1
/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i - (((((224*A*a^4*b^11*d^4 - 32*A*a
^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((20
48*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a
^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) +
(A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A
*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (
(a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^1
2*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b
^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*
d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^
(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^1
2*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13
*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 9
6*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4
 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b
)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 +
 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5
*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A
*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 -
 B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4
*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*1i)/((((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*
b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a
^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4
*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/
(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a
*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/
2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^1
0*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b
^9*d^2))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 -
4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13
*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*
d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^1
1*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 2
88*A^2*B*a^8*b^8*d^2)/(a^4*d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3
*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a +
 b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*
A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*
A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*
a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2
*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2)
+ (A*B*b)/(2*d^2))^(1/2) + (((((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4
 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c +
d*x))^(1/2)*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*
b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2))/(4*a^4*d^4))*((2*A^2*B^2*b^
2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^
2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - ((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 5
12*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^
2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(4*a^4*d^4))*((2*A
^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d
^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) - (16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2
 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 -
(A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d
^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(a^4*
d^5))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)
^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) + ((a + b*tan(c + d*x))^(1/2)*(A^4
*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^1
4 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2
*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^
11 - 256*A^3*B*a^7*b^9))/(4*a^4*d^4))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*
A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2) +
(12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5*a^8*b^9 + 7*B
^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12 + 112*A^2*B^3*
a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b^11 - 64*A^3*B
^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A*B^4*a^8*b^9 +
 (3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*a^9*b^8)/(a^4*
d^5)))*((2*A^2*B^2*b^2*d^4 - B^4*b^2*d^4 - 4*A^2*B^2*a^2*d^4 - A^4*b^2*d^4 + 4*A*B^3*a*b*d^4 - 4*A^3*B*a*b*d^4
)^(1/2)/(4*d^4) - (A^2*a)/(4*d^2) + (B^2*a)/(4*d^2) + (A*B*b)/(2*d^2))^(1/2)*2i + ((a + b*tan(c + d*x))^(1/2)*
(A*a^2*b - (A*b^3)/8 + (B*a*b^2)/4) + ((a + b*tan(c + d*x))^(5/2)*(A*b^3 + 8*A*a^2*b - 2*B*a*b^2))/(8*a^2) - (
(A*b^3 + 6*A*a^2*b)*(a + b*tan(c + d*x))^(3/2))/(3*a))/(d*(a + b*tan(c + d*x))^3 - a^3*d - 3*a*d*(a + b*tan(c
+ d*x))^2 + 3*a^2*d*(a + b*tan(c + d*x))) + (atan(-(((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 1
7*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64
*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*
a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))
/(64*a^4*d^4) + (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B
^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^
2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^
12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a
^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304
*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64
*a^4*d^4) + (((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10
*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) - ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(2
56*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
- 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*
a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16
*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^
5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B
*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*
B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(a^5*d) + ((((a + b*tan(c + d*x))^(1/
2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*
a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1
792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B
*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4) - (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9
*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2
 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2
*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) + ((((a +
b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2
 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d
^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) - (((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*
B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4
)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 6
4*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^
7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(
256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
 - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7
*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^
2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(a^5*d
))/((12*A^5*a^4*b^13 - (A^3*B^2*b^17)/4 - (A^5*a^2*b^15)/4 - (A^5*b^17)/4 + 44*A^5*a^6*b^11 + 32*A^5*a^8*b^9 +
 7*B^5*a^3*b^14 + 63*B^5*a^5*b^12 + 56*B^5*a^7*b^10 + (79*A^2*B^3*a^3*b^14)/4 + 67*A^2*B^3*a^5*b^12 + 112*A^2*
B^3*a^7*b^10 + 64*A^2*B^3*a^9*b^8 - (17*A^3*B^2*a^2*b^15)/4 + 20*A^3*B^2*a^4*b^13 - 40*A^3*B^2*a^6*b^11 - 64*A
^3*B^2*a^8*b^9 + (3*A^4*B*a*b^16)/4 - 4*A*B^4*a^2*b^15 + 8*A*B^4*a^4*b^13 - 84*A*B^4*a^6*b^11 - 96*A*B^4*a^8*b
^9 + (3*A^2*B^3*a*b^16)/4 + (51*A^4*B*a^3*b^14)/4 + 4*A^4*B*a^5*b^12 + 56*A^4*B*a^7*b^10 + 64*A^4*B*a^9*b^8)/(
a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^14 + 208*A^4*a^4*b^12 + 192*A
^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10 + 384*B^4*a^8*b^8 + 5*A^2*B
^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A*B^3*a^3*b^13 + 1280*A*B^3*a
^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4) + (((16*A^3*a^3*b^13*d^2 -
144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d^2 + 96*B^3*a^6*b^10*d^2 + 9
6*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^2 + 528*A*B^2*a^5*b^11*d^2 +
 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A^2*B*a^6*b^10*d^2 - 288*A^2*
B*a^8*b^8*d^2)/(16*a^4*d^5) - ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 - 512*A^2*a^5*b^10*d^2 - 1280
*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*d^2 + 4*A^2*a*b^14*d^2 - 16*
A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) + (((224*A*a^4*b^11*d^4 - 32*A*
a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*a^7*b^8*d^4)/(16*a^4*d^5) -
((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^
4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(1024*a^9*d^5))*(
256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b
 - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7
*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^
2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d
))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^
10*b - 4*A*B*a^6*b^5)^(1/2))/(a^5*d) + ((((a + b*tan(c + d*x))^(1/2)*(A^4*b^16 - A^2*B^2*b^16 - 17*A^4*a^2*b^1
4 + 208*A^4*a^4*b^12 + 192*A^4*a^6*b^10 + 128*A^4*a^8*b^8 - 4*B^4*a^2*b^14 + 68*B^4*a^4*b^12 + 64*B^4*a^6*b^10
 + 384*B^4*a^8*b^8 + 5*A^2*B^2*a^2*b^14 + 236*A^2*B^2*a^4*b^12 + 1792*A^2*B^2*a^6*b^10 + 4*A*B^3*a*b^15 + 12*A
*B^3*a^3*b^13 + 1280*A*B^3*a^7*b^9 - 60*A^3*B*a^3*b^13 + 512*A^3*B*a^5*b^11 - 256*A^3*B*a^7*b^9))/(64*a^4*d^4)
 - (((16*A^3*a^3*b^13*d^2 - 144*A^3*a^5*b^11*d^2 - 160*A^3*a^7*b^9*d^2 - 2*B^3*a^2*b^14*d^2 - 2*B^3*a^4*b^12*d
^2 + 96*B^3*a^6*b^10*d^2 + 96*B^3*a^8*b^8*d^2 - (A^2*B*b^16*d^2)/2 + 2*A*B^2*a*b^15*d^2 + 50*A*B^2*a^3*b^13*d^
2 + 528*A*B^2*a^5*b^11*d^2 + 480*A*B^2*a^7*b^9*d^2 - (49*A^2*B*a^2*b^14*d^2)/2 + 168*A^2*B*a^4*b^12*d^2 - 96*A
^2*B*a^6*b^10*d^2 - 288*A^2*B*a^8*b^8*d^2)/(16*a^4*d^5) + ((((a + b*tan(c + d*x))^(1/2)*(16*B^2*a^3*b^12*d^2 -
 512*A^2*a^5*b^10*d^2 - 1280*A^2*a^7*b^8*d^2 - 64*A^2*a^3*b^12*d^2 + 1024*B^2*a^5*b^10*d^2 + 2304*B^2*a^7*b^8*
d^2 + 4*A^2*a*b^14*d^2 - 16*A*B*a^2*b^13*d^2 + 2048*A*B*a^4*b^11*d^2 + 4096*A*B*a^6*b^9*d^2))/(64*a^4*d^4) - (
((224*A*a^4*b^11*d^4 - 32*A*a^2*b^13*d^4 + 256*A*a^6*b^9*d^4 + 64*B*a^3*b^12*d^4 + 448*B*a^5*b^10*d^4 + 384*B*
a^7*b^8*d^4)/(16*a^4*d^5) + ((2048*a^4*b^10*d^4 + 3072*a^6*b^8*d^4)*(a + b*tan(c + d*x))^(1/2)*(256*B^2*a^11 +
 A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b
^5)^(1/2))/(1024*a^9*d^5))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*
B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4
+ 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^
2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A
*B*a^6*b^5)^(1/2))/(16*a^5*d))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 +
 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2))/(a^5*d)))*(256*B^2*a^11 + A^2*a^5*b^6 - 16*A^2*a^7*b^
4 + 64*A^2*a^9*b^2 + 4*B^2*a^7*b^4 + 64*B^2*a^9*b^2 + 256*A*B*a^10*b - 4*A*B*a^6*b^5)^(1/2)*1i)/(8*a^5*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \cot ^{4}{\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**4*(a+b*tan(d*x+c))**(1/2)*(A+B*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))*sqrt(a + b*tan(c + d*x))*cot(c + d*x)**4, x)

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