3.526 \(\int \cot ^4(c+d x) (a+b \tan (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=237 \[ \frac {5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d} \]

[Out]

-I*(a-I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d+I*(a+I*b)^(5/2)*arctanh((a+b*tan(d*x+c))^(1/2
)/(a+I*b)^(1/2))/d+5/8*b*(8*a^2-b^2)*arctanh((a+b*tan(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+1/8*(8*a^2-11*b^2)*cot(
d*x+c)*(a+b*tan(d*x+c))^(1/2)/d-13/12*a*b*cot(d*x+c)^2*(a+b*tan(d*x+c))^(1/2)/d-1/3*a^2*cot(d*x+c)^3*(a+b*tan(
d*x+c))^(1/2)/d

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Rubi [A]  time = 1.07, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3565, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac {5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

(5*b*(8*a^2 - b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/(8*Sqrt[a]*d) - (I*(a - I*b)^(5/2)*ArcTanh[Sqrt[
a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (I*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d
+ ((8*a^2 - 11*b^2)*Cot[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(8*d) - (13*a*b*Cot[c + d*x]^2*Sqrt[a + b*Tan[c + d
*x]])/(12*d) - (a^2*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 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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) (a+b \tan (c+d x))^{5/2} \, dx &=-\frac {a^2 \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 {13 a^2 b}{2}-3 a \left (a^2-3 b^2\right ) \tan (c+d x)-\frac {1}{2} b \left (5 a^2-6 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \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} a^2 \left (8 a^2-11 b^2\right )+6 a b \left (3 a^2-b^2\right ) \tan (c+d x)+\frac {39}{4} a^2 b^2 \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a}\\ &=\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {\cot (c+d x) \left (-\frac {15}{8} a^2 b \left (8 a^2-b^2\right )+6 a^3 \left (a^2-3 b^2\right ) \tan (c+d x)+\frac {3}{8} a^2 b \left (8 a^2-11 b^2\right ) \tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}\\ &=\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {\int \frac {6 a^3 \left (a^2-3 b^2\right )+6 a^2 b \left (3 a^2-b^2\right ) \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{6 a^2}-\frac {1}{16} \left (5 b \left (8 a^2-b^2\right )\right ) \int \frac {\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {1}{2} (a-i b)^3 \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (a+i b)^3 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx-\frac {\left (5 b \left (8 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan (c+d x)\right )}{16 d}\\ &=\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}+\frac {(i a-b)^3 \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)^3 \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 (5 \left (8 a^2-b^2\right )\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 d}\\ &=\frac {5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}+\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}-\frac {(a-i b)^3 \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)^3 \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 {5 b \left (8 a^2-b^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{8 \sqrt {a} d}-\frac {i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {\left (8 a^2-11 b^2\right ) \cot (c+d x) \sqrt {a+b \tan (c+d x)}}{8 d}-\frac {13 a b \cot ^2(c+d x) \sqrt {a+b \tan (c+d x)}}{12 d}-\frac {a^2 \cot ^3(c+d x) \sqrt {a+b \tan (c+d x)}}{3 d}\\ \end {align*}

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Mathematica [A]  time = 3.63, size = 185, normalized size = 0.78 \[ -\frac {\frac {15 b \left (b^2-8 a^2\right ) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\cot (c+d x) \sqrt {a+b \tan (c+d x)} \left (8 a^2 \cot ^2(c+d x)-24 a^2+26 a b \cot (c+d x)+33 b^2\right )+24 i (a-i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-24 i (a+i b)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{24 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^4*(a + b*Tan[c + d*x])^(5/2),x]

[Out]

-1/24*((15*b*(-8*a^2 + b^2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a]])/Sqrt[a] + (24*I)*(a - I*b)^(5/2)*ArcTan
h[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (24*I)*(a + I*b)^(5/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I
*b]] + Cot[c + d*x]*(-24*a^2 + 33*b^2 + 26*a*b*Cot[c + d*x] + 8*a^2*Cot[c + d*x]^2)*Sqrt[a + b*Tan[c + d*x]])/
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))^(5/2),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))^(5/2),x, algorithm="giac")

[Out]

Timed out

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maple [C]  time = 3.44, size = 88284, normalized size = 372.51 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^4*(a+b*tan(d*x+c))^(5/2),x)

[Out]

result too large to display

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maxima [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))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 10.91, size = 4455, normalized size = 18.80 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^4*(a + b*tan(c + d*x))^(5/2),x)

[Out]

(((5*a*b^3)/3 - 2*a^3*b)*(a + b*tan(c + d*x))^(3/2) + (a^4*b - (5*a^2*b^3)/8)*(a + b*tan(c + d*x))^(1/2) + (a^
2*b - (11*b^3)/8)*(a + b*tan(c + d*x))^(5/2))/(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))) - log((5*b^9*(a^2 + b^2)^3*(11*b^8 - 128*a^8 + 15*a^2*b^6 - 896*a^4*b^4 + 592
*a^6*b^2))/(8*d^5) - ((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^
2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4
)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2
)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*((32
*b^9*(32*a^4 - 5*b^4 + 27*a^2*b^2))/d - 128*b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5
*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (a*b^8*(a + b*tan(c +
d*x))^(1/2)*(320*a^6 + 1191*b^6 + 80*a^2*b^4 - 4864*a^4*b^2))/d^2))/2 - (a*b^9*(407*b^8 - 736*a^8 - 3225*a^2*
b^6 + 1088*a^4*b^4 + 3984*a^6*b^2))/d^3))/2 - (b^8*(a + b*tan(c + d*x))^(1/2)*(128*a^12 + 153*b^12 - 7*a^2*b^1
0 + 9895*a^4*b^8 - 27465*a^6*b^6 + 26320*a^8*b^4 - 832*a^10*b^2))/(4*d^4)))/2)*(-((20*a^2*b^8*d^4 - b^10*d^4 -
110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/(4*d^4))^
(1/2) - log((5*b^9*(a^2 + b^2)^3*(11*b^8 - 128*a^8 + 15*a^2*b^6 - 896*a^4*b^4 + 592*a^6*b^2))/(8*d^5) - ((((-b
^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*
(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4
+ b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 -
10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((32*b^9*(32*a^4 - 5*b^4 + 27*a^2*b
^2))/d - 128*b^8*(((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4
)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/2)))/2 - (a*b^8*(a + b*tan(c + d*x))^(1/2)*(320*a^6 + 1191*b^6
+ 80*a^2*b^4 - 4864*a^4*b^2))/d^2))/2 - (a*b^9*(407*b^8 - 736*a^8 - 3225*a^2*b^6 + 1088*a^4*b^4 + 3984*a^6*b^
2))/d^3))/2 - (b^8*(a + b*tan(c + d*x))^(1/2)*(128*a^12 + 153*b^12 - 7*a^2*b^10 + 9895*a^4*b^8 - 27465*a^6*b^6
+ 26320*a^8*b^4 - 832*a^10*b^2))/(4*d^4)))/2)*(((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^
4 - 25*a^8*b^2*d^4)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/(4*d^4))^(1/2) + log((5*b^9*(a^2 + b^2)^3*
(11*b^8 - 128*a^8 + 15*a^2*b^6 - 896*a^4*b^4 + 592*a^6*b^2))/(8*d^5) - ((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)
^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/
2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^
5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(((((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 -
5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*((32*b^9*(32*a^4 - 5*b^4 + 27*a^2*b^2))/d + 128*b^8*(((-b^2*d^4*(5*a
^4 + b^4 - 10*a^2*b^2)^2)^(1/2) - a^5*d^2 - 5*a*b^4*d^2 + 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*ta
n(c + d*x))^(1/2)))/2 + (a*b^8*(a + b*tan(c + d*x))^(1/2)*(320*a^6 + 1191*b^6 + 80*a^2*b^4 - 4864*a^4*b^2))/d^
2))/2 - (a*b^9*(407*b^8 - 736*a^8 - 3225*a^2*b^6 + 1088*a^4*b^4 + 3984*a^6*b^2))/d^3))/2 + (b^8*(a + b*tan(c +
d*x))^(1/2)*(128*a^12 + 153*b^12 - 7*a^2*b^10 + 9895*a^4*b^8 - 27465*a^6*b^6 + 26320*a^8*b^4 - 832*a^10*b^2))
/(4*d^4)))/2)*((20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4)
- a^5/(4*d^2) - (5*a*b^4)/(4*d^2) + (5*a^3*b^2)/(2*d^2))^(1/2) + log((5*b^9*(a^2 + b^2)^3*(11*b^8 - 128*a^8 +
15*a^2*b^6 - 896*a^4*b^4 + 592*a^6*b^2))/(8*d^5) - ((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2
+ 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a
*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d
^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(((-((-b^2*d^4*(5*a^4 + b^4 - 10*a^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 1
0*a^3*b^2*d^2)/d^4)^(1/2)*((32*b^9*(32*a^4 - 5*b^4 + 27*a^2*b^2))/d + 128*b^8*(-((-b^2*d^4*(5*a^4 + b^4 - 10*a
^2*b^2)^2)^(1/2) + a^5*d^2 + 5*a*b^4*d^2 - 10*a^3*b^2*d^2)/d^4)^(1/2)*(3*a^2 + 2*b^2)*(a + b*tan(c + d*x))^(1/
2)))/2 + (a*b^8*(a + b*tan(c + d*x))^(1/2)*(320*a^6 + 1191*b^6 + 80*a^2*b^4 - 4864*a^4*b^2))/d^2))/2 - (a*b^9*
(407*b^8 - 736*a^8 - 3225*a^2*b^6 + 1088*a^4*b^4 + 3984*a^6*b^2))/d^3))/2 + (b^8*(a + b*tan(c + d*x))^(1/2)*(1
28*a^12 + 153*b^12 - 7*a^2*b^10 + 9895*a^4*b^8 - 27465*a^6*b^6 + 26320*a^8*b^4 - 832*a^10*b^2))/(4*d^4)))/2)*(
(5*a^3*b^2)/(2*d^2) - a^5/(4*d^2) - (5*a*b^4)/(4*d^2) - (20*a^2*b^8*d^4 - b^10*d^4 - 110*a^4*b^6*d^4 + 100*a^6
*b^4*d^4 - 25*a^8*b^2*d^4)^(1/2)/(4*d^4))^(1/2) + (b*atan(((b*(8*a^2 - b^2)*(((a + b*tan(c + d*x))^(1/2)*(153*
b^20 - 7*a^2*b^18 + 9895*a^4*b^16 - 27465*a^6*b^14 + 26320*a^8*b^12 - 832*a^10*b^10 + 128*a^12*b^8))/(4*d^4) +
(5*b*((407*a*b^17*d^2 - 3225*a^3*b^15*d^2 + 1088*a^5*b^13*d^2 + 3984*a^7*b^11*d^2 - 736*a^9*b^9*d^2)/d^5 + (5
*b*(((a + b*tan(c + d*x))^(1/2)*(4764*a*b^14*d^2 + 320*a^3*b^12*d^2 - 19456*a^5*b^10*d^2 + 1280*a^7*b^8*d^2))/
(4*d^4) - (5*b*((864*a^2*b^11*d^4 - 160*b^13*d^4 + 1024*a^4*b^9*d^4)/d^5 - (5*b*(8*a^2 - b^2)*(2048*b^10*d^4 +
3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(64*a^(1/2)*d^5))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))
/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d))*5i)/(16*a^(1/2)*d) + (b*(8*a^2 - b^2)*(((a + b*tan(c + d*x))^(
1/2)*(153*b^20 - 7*a^2*b^18 + 9895*a^4*b^16 - 27465*a^6*b^14 + 26320*a^8*b^12 - 832*a^10*b^10 + 128*a^12*b^8))
/(4*d^4) - (5*b*((407*a*b^17*d^2 - 3225*a^3*b^15*d^2 + 1088*a^5*b^13*d^2 + 3984*a^7*b^11*d^2 - 736*a^9*b^9*d^2
)/d^5 - (5*b*(((a + b*tan(c + d*x))^(1/2)*(4764*a*b^14*d^2 + 320*a^3*b^12*d^2 - 19456*a^5*b^10*d^2 + 1280*a^7*
b^8*d^2))/(4*d^4) + (5*b*((864*a^2*b^11*d^4 - 160*b^13*d^4 + 1024*a^4*b^9*d^4)/d^5 + (5*b*(8*a^2 - b^2)*(2048*
b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(64*a^(1/2)*d^5))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a
^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d))*5i)/(16*a^(1/2)*d))/(((55*b^23)/4 + 60*a^2*b^21 - (2
045*a^4*b^19)/2 - 2550*a^6*b^17 - (5125*a^8*b^15)/4 + 620*a^10*b^13 + 260*a^12*b^11 - 160*a^14*b^9)/d^5 + (5*b
*(8*a^2 - b^2)*(((a + b*tan(c + d*x))^(1/2)*(153*b^20 - 7*a^2*b^18 + 9895*a^4*b^16 - 27465*a^6*b^14 + 26320*a^
8*b^12 - 832*a^10*b^10 + 128*a^12*b^8))/(4*d^4) + (5*b*((407*a*b^17*d^2 - 3225*a^3*b^15*d^2 + 1088*a^5*b^13*d^
2 + 3984*a^7*b^11*d^2 - 736*a^9*b^9*d^2)/d^5 + (5*b*(((a + b*tan(c + d*x))^(1/2)*(4764*a*b^14*d^2 + 320*a^3*b^
12*d^2 - 19456*a^5*b^10*d^2 + 1280*a^7*b^8*d^2))/(4*d^4) - (5*b*((864*a^2*b^11*d^4 - 160*b^13*d^4 + 1024*a^4*b
^9*d^4)/d^5 - (5*b*(8*a^2 - b^2)*(2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(64*a^(1/2)*d^
5))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d)))/(16*a^(1/2)*
d) - (5*b*(8*a^2 - b^2)*(((a + b*tan(c + d*x))^(1/2)*(153*b^20 - 7*a^2*b^18 + 9895*a^4*b^16 - 27465*a^6*b^14 +
26320*a^8*b^12 - 832*a^10*b^10 + 128*a^12*b^8))/(4*d^4) - (5*b*((407*a*b^17*d^2 - 3225*a^3*b^15*d^2 + 1088*a^
5*b^13*d^2 + 3984*a^7*b^11*d^2 - 736*a^9*b^9*d^2)/d^5 - (5*b*(((a + b*tan(c + d*x))^(1/2)*(4764*a*b^14*d^2 + 3
20*a^3*b^12*d^2 - 19456*a^5*b^10*d^2 + 1280*a^7*b^8*d^2))/(4*d^4) + (5*b*((864*a^2*b^11*d^4 - 160*b^13*d^4 + 1
024*a^4*b^9*d^4)/d^5 + (5*b*(8*a^2 - b^2)*(2048*b^10*d^4 + 3072*a^2*b^8*d^4)*(a + b*tan(c + d*x))^(1/2))/(64*a
^(1/2)*d^5))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d))*(8*a^2 - b^2))/(16*a^(1/2)*d)))/(16
*a^(1/2)*d)))*(8*a^2 - b^2)*5i)/(8*a^(1/2)*d)

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sympy [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))**(5/2),x)

[Out]

Timed out

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