3.402 \(\int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx\)

Optimal. Leaf size=297 \[ \frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )}-\frac {2 A}{a d \sqrt {\tan (c+d x)}} \]

[Out]

-2*b^(3/2)*(A*b-B*a)*arctan(b^(1/2)*tan(d*x+c)^(1/2)/a^(1/2))/a^(3/2)/(a^2+b^2)/d-1/2*(a*(A-B)+b*(A+B))*arctan
(-1+2^(1/2)*tan(d*x+c)^(1/2))/(a^2+b^2)/d*2^(1/2)-1/2*(a*(A-B)+b*(A+B))*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/(a^
2+b^2)/d*2^(1/2)+1/4*(b*(A-B)-a*(A+B))*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-1/4*(b*(A
-B)-a*(A+B))*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(a^2+b^2)/d*2^(1/2)-2*A/a/d/tan(d*x+c)^(1/2)

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Rubi [A]  time = 0.64, antiderivative size = 297, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3609, 3653, 3534, 1168, 1162, 617, 204, 1165, 628, 3634, 63, 205} \[ -\frac {2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d \left (a^2+b^2\right )}+\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \left (a^2+b^2\right )}-\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \left (a^2+b^2\right )}+\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {(b (A-B)-a (A+B)) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \left (a^2+b^2\right )}-\frac {2 A}{a d \sqrt {\tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]

[Out]

((a*(A - B) + b*(A + B))*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - ((a*(A - B) + b*(A
+ B))*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*(a^2 + b^2)*d) - (2*b^(3/2)*(A*b - a*B)*ArcTan[(Sqrt[b]
*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(a^(3/2)*(a^2 + b^2)*d) + ((b*(A - B) - a*(A + B))*Log[1 - Sqrt[2]*Sqrt[Tan[c +
d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - ((b*(A - B) - a*(A + B))*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]
] + Tan[c + d*x]])/(2*Sqrt[2]*(a^2 + b^2)*d) - (2*A)/(a*d*Sqrt[Tan[c + d*x]])

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 205

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

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1168

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[-(a*c)]

Rule 3534

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3609

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

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 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 \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x))} \, dx &=-\frac {2 A}{a d \sqrt {\tan (c+d x)}}-\frac {2 \int \frac {\frac {1}{2} (A b-a B)+\frac {1}{2} a A \tan (c+d x)+\frac {1}{2} A b \tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a}\\ &=-\frac {2 A}{a d \sqrt {\tan (c+d x)}}-\frac {2 \int \frac {\frac {1}{2} a (A b-a B)+\frac {1}{2} a (a A+b B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{a \left (a^2+b^2\right )}-\frac {\left (b^2 (A b-a B)\right ) \int \frac {1+\tan ^2(c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))} \, dx}{a \left (a^2+b^2\right )}\\ &=-\frac {2 A}{a d \sqrt {\tan (c+d x)}}-\frac {4 \operatorname {Subst}\left (\int \frac {\frac {1}{2} a (A b-a B)+\frac {1}{2} a (a A+b B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right ) d}-\frac {\left (b^2 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} (a+b x)} \, dx,x,\tan (c+d x)\right )}{a \left (a^2+b^2\right ) d}\\ &=-\frac {2 A}{a d \sqrt {\tan (c+d x)}}-\frac {\left (2 b^2 (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac {2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}-\frac {2 A}{a d \sqrt {\tan (c+d x)}}+\frac {(b (A-B)-a (A+B)) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (A+B)) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \left (a^2+b^2\right ) d}\\ &=-\frac {2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 A}{a d \sqrt {\tan (c+d x)}}-\frac {(a (A-B)+b (A+B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}+\frac {(a (A-B)+b (A+B)) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}\\ &=\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {(a (A-B)+b (A+B)) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 b^{3/2} (A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{a^{3/2} \left (a^2+b^2\right ) d}+\frac {(b (A-B)-a (A+B)) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {(b (A-B)-a (A+B)) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} \left (a^2+b^2\right ) d}-\frac {2 A}{a d \sqrt {\tan (c+d x)}}\\ \end {align*}

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Mathematica [C]  time = 0.60, size = 153, normalized size = 0.52 \[ \frac {\frac {\sqrt [4]{-1} a \left ((b-i a) (A-i B) \tan ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+(b+i a) (A+i B) \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )\right )}{a^2+b^2}+\frac {2 b^{3/2} (a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {\tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} \left (a^2+b^2\right )}-\frac {2 A}{\sqrt {\tan (c+d x)}}}{a d} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + b*Tan[c + d*x])),x]

[Out]

((2*b^(3/2)*(-(A*b) + a*B)*ArcTan[(Sqrt[b]*Sqrt[Tan[c + d*x]])/Sqrt[a]])/(Sqrt[a]*(a^2 + b^2)) + ((-1)^(1/4)*a
*(((-I)*a + b)*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + (I*a + b)*(A + I*B)*ArcTanh[(-1)^(3/4)*Sqrt[T
an[c + d*x]]]))/(a^2 + b^2) - (2*A)/Sqrt[Tan[c + d*x]])/(a*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((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (b \tan \left (d x + c\right ) + a\right )} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

integrate((B*tan(d*x + c) + A)/((b*tan(d*x + c) + a)*tan(d*x + c)^(3/2)), x)

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maple [B]  time = 0.34, size = 628, normalized size = 2.11 \[ -\frac {2 b^{3} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) A}{d a \left (a^{2}+b^{2}\right ) \sqrt {a b}}+\frac {2 b^{2} \arctan \left (\frac {\left (\sqrt {\tan }\left (d x +c \right )\right ) b}{\sqrt {a b}}\right ) B}{d \left (a^{2}+b^{2}\right ) \sqrt {a b}}-\frac {2 A}{a d \sqrt {\tan \left (d x +c \right )}}-\frac {A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}-\frac {A \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d \left (a^{2}+b^{2}\right )}-\frac {A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a}{4 d \left (a^{2}+b^{2}\right )}+\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {A \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) a}{4 d \left (a^{2}+b^{2}\right )}-\frac {A \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {A \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) a}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right ) b}{4 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \sqrt {2}\, \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right ) b}{2 d \left (a^{2}+b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x)

[Out]

-2/d/a*b^3/(a^2+b^2)/(a*b)^(1/2)*arctan(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*A+2/d*b^2/(a^2+b^2)/(a*b)^(1/2)*arctan
(tan(d*x+c)^(1/2)*b/(a*b)^(1/2))*B-2*A/a/d/tan(d*x+c)^(1/2)-1/2/d/(a^2+b^2)*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*
x+c)^(1/2))*b-1/4/d/(a^2+b^2)*A*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)
+tan(d*x+c)))*b-1/2/d/(a^2+b^2)*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*b+1/2/d/(a^2+b^2)*B*2^(1/2)*arcta
n(-1+2^(1/2)*tan(d*x+c)^(1/2))*a+1/4/d/(a^2+b^2)*B*2^(1/2)*ln((1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/
2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a+1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*a-1/4/d/(a^2+b
^2)*A*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))*a-1/2/d/(a^2
+b^2)*A*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*a-1/2/d/(a^2+b^2)*A*2^(1/2)*arctan(1+2^(1/2)*tan(d*x+c)^(1
/2))*a-1/4/d/(a^2+b^2)*B*2^(1/2)*ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*
x+c)))*b-1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*b-1/2/d/(a^2+b^2)*B*2^(1/2)*arctan(1+2^
(1/2)*tan(d*x+c)^(1/2))*b

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maxima [A]  time = 0.63, size = 235, normalized size = 0.79 \[ \frac {\frac {8 \, {\left (B a b^{2} - A b^{3}\right )} \arctan \left (\frac {b \sqrt {\tan \left (d x + c\right )}}{\sqrt {a b}}\right )}{{\left (a^{3} + a b^{2}\right )} \sqrt {a b}} - \frac {2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left ({\left (A - B\right )} a + {\left (A + B\right )} b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left ({\left (A + B\right )} a - {\left (A - B\right )} b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )}{a^{2} + b^{2}} - \frac {8 \, A}{a \sqrt {\tan \left (d x + c\right )}}}{4 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/4*(8*(B*a*b^2 - A*b^3)*arctan(b*sqrt(tan(d*x + c))/sqrt(a*b))/((a^3 + a*b^2)*sqrt(a*b)) - (2*sqrt(2)*((A - B
)*a + (A + B)*b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*((A - B)*a + (A + B)*b)*arct
an(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt(2)*((A + B)*a - (A - B)*b)*log(sqrt(2)*sqrt(tan(d*x +
c)) + tan(d*x + c) + 1) + sqrt(2)*((A + B)*a - (A - B)*b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1)
)/(a^2 + b^2) - 8*A/(a*sqrt(tan(d*x + c))))/d

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mupad [B]  time = 11.60, size = 15318, normalized size = 51.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(c + d*x))/(tan(c + d*x)^(3/2)*(a + b*tan(c + d*x))),x)

[Out]

atan(((tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c
+ d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (((6
4*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2
*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*
b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*A*a^8*b^9*d^8 - 640*A*a^10*b^7*d^8 + 256*A*a^12*b^5*d^8 +
384*A*a^14*b^3*d^8))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*
b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*a^
13*b^2*d^6))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(1
6*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i + (tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^
5) + (((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d
^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A
^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4
))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^
2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*
b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*A*a^8*b^9*d^
8 + 640*A*a^10*b^7*d^8 - 256*A*a^12*b^5*d^8 - 384*A*a^14*b^3*d^8))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 1
6*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 128*A^3*
a^7*b^8*d^6 + 32*A^3*a^11*b^4*d^6 + 32*A^3*a^13*b^2*d^6))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4
+ 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((tan(c + d*x)^
(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^
2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c + d*x)^(1/2)*(512*A
^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (((64*A^4*a^2*b^2*d^4 -
A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)
))^(1/2)*(tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8
*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b
^5*d^9 - 512*a^15*b^3*d^9) - 512*A*a^8*b^9*d^8 - 640*A*a^10*b^7*d^8 + 256*A*a^12*b^5*d^8 + 384*A*a^14*b^3*d^8)
)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4
+ b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*a^13*b^2*d^6))*(((64*A
^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2) - (tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (((64*A^4*a^2*b^2*
d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2)*((tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^
2*a^14*b^2*d^7) - (((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^
2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(512*a
^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*A*a^8*b^9*d^8 + 640*A*a^10*b^7*d^8
- 256*A*a^12*b^5*d^8 - 384*A*a^14*b^3*d^8))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*
d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 128*A^3*a^7*b^8*d^6 + 32*A^3*a^
11*b^4*d^6 + 32*A^3*a^13*b^2*d^6))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/
2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)))*(((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i + at
an(((tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c
+ d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (-((6
4*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^
2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11
*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*A*a^8*b^9*d^8 - 640*A*a^10*b^7*d^8 + 256*A*a^12*b^5*d^8
+ 384*A*a^14*b^3*d^8))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*
a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*
a^13*b^2*d^6))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)
/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i + (tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5
*d^5) + (-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a
^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 1
28*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^
2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(-((64*A^4*
a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 +
2*a^2*b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*A*a^8*
b^9*d^8 + 640*A*a^10*b^7*d^8 - 256*A*a^12*b^5*d^8 - 384*A*a^14*b^3*d^8))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*
d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 1
28*A^3*a^7*b^8*d^6 + 32*A^3*a^11*b^4*d^6 + 32*A^3*a^13*b^2*d^6))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16
*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((tan(c
+ d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^
4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c + d*x)^(1/
2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (-((64*A^4*a^2*
b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^
2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4)
)^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 -
512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) - 512*A*a^8*b^9*d^8 - 640*A*a^10*b^7*d^8 + 256*A*a^12*b^5*d^8 + 384*A*a^
14*b^3*d^8))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(
16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + 128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*a^13*b^2*d
^6))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*
d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (-((6
4*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*
d^4 + 2*a^2*b^2*d^4)))^(1/2)*((tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b
^4*d^7 + 64*A^2*a^14*b^2*d^7) - (-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
+ 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(tan(c + d*x)^(1/2)*(-((64*A^4*a^2*b^2*d^4 -
A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4
)))^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9) + 512*A*a^8*b^9*d^8 + 640
*A*a^10*b^7*d^8 - 256*A*a^12*b^5*d^8 - 384*A*a^14*b^3*d^8))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*
d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - 128*A^3*a^7*b^
8*d^6 + 32*A^3*a^11*b^4*d^6 + 32*A^3*a^13*b^2*d^6))*(-((64*A^4*a^2*b^2*d^4 - A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32
*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)))*(-((64*A^4*a^2*b^2*d^4
- A^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*A^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d
^4)))^(1/2)*2i - atan(((((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a
^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*
d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16
*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 +
32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/
2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 +
16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*
B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2) + (96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 1
6*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i - (((32
*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2
))/d^3 + (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
- 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^
4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B
^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^
2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4
))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^
4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) -
(96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))
^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*(5*B^3*a*b^5 + B^3*a^3*b^3)
)/d^3 - (((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 - (32*tan(c + d*x)^(1
/2)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d
^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-
((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b
^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6
*d^2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(1
6*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*
b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*B^4*b^5*tan(c + d*x)^(1
/2))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*
(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16*B*b^8*d^2 + 2
8*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(-((64*B^4*a^2*b^2*d^4 - B^
4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))
^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*
a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)
+ (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(-((64*B^4*a^2*b^2*
d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^
2*d^4)))^(1/2))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2
)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(-((64*B^4*a^2*b^2*d^
4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*
d^4)))^(1/2)))*(-((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) - 8*B^2*a*b*d^2)
/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - atan(((((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16
*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2
*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a
^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^
4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*
d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(((64*B
^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4
+ 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B
^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*B^4*a
^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2
*a^2*b^2*d^4)))^(1/2)*1i - (((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8
*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b
^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 +
16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4
+ 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(
1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*
B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^
4 + 2*a^2*b^2*d^4)))^(1/2) - (96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16
*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*1i)/((((32*
(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2)
)/d^3 - (32*tan(c + d*x)^(1/2)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) +
8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4
- 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*
a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a
^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(
1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4
+ 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (96*
B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
+ 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) + (((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (
((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 + (32*tan(c + d*x)^(1/2)*(((64
*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d
^4 + 2*a^2*b^2*d^4)))^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/d^4)*(((64*B^4*a^
2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*
a^2*b^2*d^4)))^(1/2) + (32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4)
*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 +
b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/
2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2) - (96*B^4*b^5*tan(c + d*x)^(1/2))/d^4)*(((
64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2) + 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4
*d^4 + 2*a^2*b^2*d^4)))^(1/2)))*(((64*B^4*a^2*b^2*d^4 - B^4*(16*a^4*d^4 + 16*b^4*d^4 + 32*a^2*b^2*d^4))^(1/2)
+ 8*B^2*a*b*d^2)/(16*(a^4*d^4 + b^4*d^4 + 2*a^2*b^2*d^4)))^(1/2)*2i - (A*b^5*atan(((A*b^5*(tan(c + d*x)^(1/2)*
(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (A*b^5*(32*A^3*a^11*b^4*d^6 - 128*A^3*a^7*b^8*d^6 + 32*A^3*a^13*b^
2*d^6 + (A*b^5*(tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2
*a^14*b^2*d^7) - (A*b^5*(512*A*a^8*b^9*d^8 + 640*A*a^10*b^7*d^8 - 256*A*a^12*b^5*d^8 - 384*A*a^14*b^3*d^8 + (A
*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9
*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d
^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2))*1i)/(- a^3*b^9*
d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2) + (A*b^5*(tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5
) + (A*b^5*(128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*a^13*b^2*d^6 + (A*b^5*(tan(c + d*x)^(1/2)*(512*
A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (A*b^5*(256*A*a^12*b^5*
d^8 - 640*A*a^10*b^7*d^8 - 512*A*a^8*b^9*d^8 + 384*A*a^14*b^3*d^8 + (A*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*d^9
+ 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2
)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2))
)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2))*1i)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)
)/((A*b^5*(tan(c + d*x)^(1/2)*(64*A^4*a^7*b^7*d^5 - 32*A^4*a^9*b^5*d^5) + (A*b^5*(32*A^3*a^11*b^4*d^6 - 128*A^
3*a^7*b^8*d^6 + 32*A^3*a^13*b^2*d^6 + (A*b^5*(tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 +
128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7) - (A*b^5*(512*A*a^8*b^9*d^8 + 640*A*a^10*b^7*d^8 - 256*A*a^12*b^5
*d^8 - 384*A*a^14*b^3*d^8 + (A*b^5*tan(c + d*x)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 -
512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7
*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b
^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2) - (A*b^5*(tan(c + d*x)^(1/2)*(64*A^4*a^7*b
^7*d^5 - 32*A^4*a^9*b^5*d^5) + (A*b^5*(128*A^3*a^7*b^8*d^6 - 32*A^3*a^11*b^4*d^6 - 32*A^3*a^13*b^2*d^6 + (A*b^
5*(tan(c + d*x)^(1/2)*(512*A^2*a^8*b^8*d^7 - 448*A^2*a^10*b^6*d^7 + 128*A^2*a^12*b^4*d^7 + 64*A^2*a^14*b^2*d^7
) - (A*b^5*(256*A*a^12*b^5*d^8 - 640*A*a^10*b^7*d^8 - 512*A*a^8*b^9*d^8 + 384*A*a^14*b^3*d^8 + (A*b^5*tan(c +
d*x)^(1/2)*(512*a^9*b^9*d^9 + 512*a^11*b^7*d^9 - 512*a^13*b^5*d^9 - 512*a^15*b^3*d^9))/(- a^3*b^9*d^2 - 2*a^5*
b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^
7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2)))/(- a^3*b^9*d^2 - 2*a^5*b^7*
d^2 - a^7*b^5*d^2)^(1/2)))*2i)/(- a^3*b^9*d^2 - 2*a^5*b^7*d^2 - a^7*b^5*d^2)^(1/2) - (2*A)/(a*d*tan(c + d*x)^(
1/2)) - (B*b^3*atan(((B*b^3*((96*B^4*b^5*tan(c + d*x)^(1/2))/d^4 + (B*b^3*((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^
3 + (B*b^3*((32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4 - (B*b^3*((
32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 - (32*B*b^3*tan(c + d*x)^(1/2)*(
16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^
2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1
/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))*1i)/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)
+ (B*b^3*((96*B^4*b^5*tan(c + d*x)^(1/2))/d^4 - (B*b^3*((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (B*b^3*((32*ta
n(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4 + (B*b^3*((32*(16*B*b^8*d^2 +
28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 + (32*B*b^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^
2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*b
^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2
- 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))*1i)/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))/((B*b^3*((96*B^4
*b^5*tan(c + d*x)^(1/2))/d^4 + (B*b^3*((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 + (B*b^3*((32*tan(c + d*x)^(1/2)*(
4*B^2*a^3*b^4*d^2 + 2*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4 - (B*b^3*((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2
+ 8*B*a^4*b^4*d^2 - 4*B*a^6*b^2*d^2))/d^3 - (32*B*b^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4
*b^5*d^4 - 16*a^6*b^3*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5
*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 -
a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2) - (B*b^3*((96*B^4*b^5*tan(c + d*x)^(1/
2))/d^4 - (B*b^3*((32*(5*B^3*a*b^5 + B^3*a^3*b^3))/d^3 - (B*b^3*((32*tan(c + d*x)^(1/2)*(4*B^2*a^3*b^4*d^2 + 2
*B^2*a^5*b^2*d^2 - 30*B^2*a*b^6*d^2))/d^4 + (B*b^3*((32*(16*B*b^8*d^2 + 28*B*a^2*b^6*d^2 + 8*B*a^4*b^4*d^2 - 4
*B*a^6*b^2*d^2))/d^3 + (32*B*b^3*tan(c + d*x)^(1/2)*(16*b^9*d^4 + 16*a^2*b^7*d^4 - 16*a^4*b^5*d^4 - 16*a^6*b^3
*d^4))/(d^4*(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2))))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(
1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))
/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)))*2i)/(- a*b^7*d^2 - 2*a^3*b^5*d^2 - a^5*b^3*d^2)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {A + B \tan {\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)**(3/2)/(a+b*tan(d*x+c)),x)

[Out]

Integral((A + B*tan(c + d*x))/((a + b*tan(c + d*x))*tan(c + d*x)**(3/2)), x)

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