3.174 \(\int \frac {\tan ^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=67 \[ \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

-ln(cos(a+b*ln(c*x^n)))/b/n-1/2*tan(a+b*ln(c*x^n))^2/b/n+1/4*tan(a+b*ln(c*x^n))^4/b/n

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Rubi [A]  time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3473, 3475} \[ \frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Tan[a + b*Log[c*x^n]]^5/x,x]

[Out]

-(Log[Cos[a + b*Log[c*x^n]]]/(b*n)) - Tan[a + b*Log[c*x^n]]^2/(2*b*n) + Tan[a + b*Log[c*x^n]]^4/(4*b*n)

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \tan ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\operatorname {Subst}\left (\int \tan ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\operatorname {Subst}\left (\int \tan (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 55, normalized size = 0.82 \[ -\frac {-\tan ^4\left (a+b \log \left (c x^n\right )\right )+2 \tan ^2\left (a+b \log \left (c x^n\right )\right )+4 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[a + b*Log[c*x^n]]^5/x,x]

[Out]

-1/4*(4*Log[Cos[a + b*Log[c*x^n]]] + 2*Tan[a + b*Log[c*x^n]]^2 - Tan[a + b*Log[c*x^n]]^4)/(b*n)

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fricas [B]  time = 0.45, size = 129, normalized size = 1.93 \[ -\frac {{\left (\cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + \frac {1}{2}\right ) + 4 \, \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right )^{2} + 2 \, b n \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + b n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

-1/2*((cos(2*b*n*log(x) + 2*b*log(c) + 2*a)^2 + 2*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1)*log(1/2*cos(2*b*n*
log(x) + 2*b*log(c) + 2*a) + 1/2) + 4*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 2)/(b*n*cos(2*b*n*log(x) + 2*b*lo
g(c) + 2*a)^2 + 2*b*n*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n)

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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(tan(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.00, size = 68, normalized size = 1.01 \[ \frac {\tan ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}-\frac {\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\ln \left (1+\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 n b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a+b*ln(c*x^n))^5/x,x)

[Out]

1/4*tan(a+b*ln(c*x^n))^4/b/n-1/2*tan(a+b*ln(c*x^n))^2/b/n+1/2/n/b*ln(1+tan(a+b*ln(c*x^n))^2)

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maxima [B]  time = 0.49, size = 4466, normalized size = 66.66 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

-1/2*(32*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) + 6*a)^2 + 48*(cos(4*b*log(c))^2 + sin(4*b*l
og(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 32*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + 3
2*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 48*(cos(4*b*log(c))^2 + sin(4*b*log(c))^
2)*sin(4*b*log(x^n) + 4*a)^2 + 32*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 + 8*((cos(
8*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + (cos(8*b*log(c))*cos(
4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + (cos(8*b*log(c))*cos(2*b*log(c)) + si
n(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + (cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*si
n(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + (cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*s
in(4*b*log(x^n) + 4*a) + (cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n)
+ 2*a))*cos(8*b*log(x^n) + 8*a) + 8*(10*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*co
s(4*b*log(x^n) + 4*a) + 8*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n)
+ 2*a) + 10*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*(
cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(6*b*log(c)))*
cos(6*b*log(x^n) + 6*a) + 8*(10*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*lo
g(x^n) + 2*a) + 10*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a)
+ cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 8*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + ((cos(8*b*log(c))^2
+ sin(8*b*log(c))^2)*cos(8*b*log(x^n) + 8*a)^2 + 16*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) +
6*a)^2 + 36*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 16*(cos(2*b*log(c))^2 + sin(2
*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + (cos(8*b*log(c))^2 + sin(8*b*log(c))^2)*sin(8*b*log(x^n) + 8*a)^2 +
16*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 36*(cos(4*b*log(c))^2 + sin(4*b*log(c))
^2)*sin(4*b*log(x^n) + 4*a)^2 + 16*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 + 2*(4*(c
os(8*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + 6*(cos(8*b*log(c))
*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(8*b*log(c))*cos(2*b*log(c
)) + sin(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 4*(cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*l
og(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 6*(cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b
*log(c)))*sin(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*sin(
2*b*log(x^n) + 2*a) + cos(8*b*log(c)))*cos(8*b*log(x^n) + 8*a) + 8*(6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6
*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin
(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 6*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*
sin(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^
n) + 2*a) + cos(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + 12*(4*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c)
)*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 4*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(
c)))*sin(2*b*log(x^n) + 2*a) + cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 8*cos(2*b*log(c))*cos(2*b*log(x^n) +
2*a) - 2*(4*(cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + 6*(
cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c)
)*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(8*b*log(c))*cos(6*b*log(
c)) + sin(8*b*log(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) - 6*(cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*
log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 4*(cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b*log(c))*sin(2*
b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(8*b*log(c)))*sin(8*b*log(x^n) + 8*a) - 8*(6*(cos(4*b*log(c))*sin(6*b*
log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(
6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*si
n(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))
*sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) - 12*(4*(cos(2*b*log(c))*sin(4*b*log(c)) -
cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c
))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 8*sin(2*b*log(c))*sin
(2*b*log(x^n) + 2*a) + 1)*log((cos(2*a)^2 + sin(2*a)^2)*cos(2*b*log(c))^2 + (cos(2*a)^2 + sin(2*a)^2)*sin(2*b*
log(c))^2 + 2*(cos(2*b*log(c))*cos(2*a) - sin(2*b*log(c))*sin(2*a))*cos(2*b*log(x^n)) + cos(2*b*log(x^n))^2 -
2*(cos(2*a)*sin(2*b*log(c)) + cos(2*b*log(c))*sin(2*a))*sin(2*b*log(x^n)) + sin(2*b*log(x^n))^2) - 8*((cos(6*b
*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + (cos(4*b*log(c))*sin(8*b
*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + (cos(2*b*log(c))*sin(8*b*log(c)) - cos(8
*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - (cos(8*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6
*b*log(c)))*sin(6*b*log(x^n) + 6*a) - (cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*sin(
4*b*log(x^n) + 4*a) - (cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2
*a))*sin(8*b*log(x^n) + 8*a) - 8*(10*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*cos(4
*b*log(x^n) + 4*a) + 8*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) +
2*a) - 10*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 8*(cos
(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*sin
(6*b*log(x^n) + 6*a) - 8*(10*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x
^n) + 2*a) - 10*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) +
sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 8*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/((b*cos(8*b*log(c))^2 +
b*sin(8*b*log(c))^2)*n*cos(8*b*log(x^n) + 8*a)^2 + 16*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*lo
g(x^n) + 6*a)^2 + 36*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 + 8*b*n*cos(2*b*l
og(c))*cos(2*b*log(x^n) + 2*a) + 16*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 +
(b*cos(8*b*log(c))^2 + b*sin(8*b*log(c))^2)*n*sin(8*b*log(x^n) + 8*a)^2 + 16*(b*cos(6*b*log(c))^2 + b*sin(6*b*
log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 36*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4
*a)^2 - 8*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 16*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(2
*b*log(x^n) + 2*a)^2 + b*n + 2*(b*n*cos(8*b*log(c)) + 4*(b*cos(8*b*log(c))*cos(6*b*log(c)) + b*sin(8*b*log(c))
*sin(6*b*log(c)))*n*cos(6*b*log(x^n) + 6*a) + 6*(b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b
*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 4*(b*cos(8*b*log(c))*cos(2*b*log(c)) + b*sin(8*b*log(c))*sin(2*b*log(c))
)*n*cos(2*b*log(x^n) + 2*a) + 4*(b*cos(6*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(6*b*log(c)))*n*sin(
6*b*log(x^n) + 6*a) + 6*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(
x^n) + 4*a) + 4*(b*cos(2*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2
*a))*cos(8*b*log(x^n) + 8*a) + 8*(b*n*cos(6*b*log(c)) + 6*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c
))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 4*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2
*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 6*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c
)))*n*sin(4*b*log(x^n) + 4*a) + 4*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*si
n(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 12*(b*n*cos(4*b*log(c)) + 4*(b*cos(4*b*log(c))*cos(2*b*log(c)
) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 4*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*co
s(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*b*log(x^n) + 4*a) - 2*(4*(b*cos(6*b*log(c))*si
n(8*b*log(c)) - b*cos(8*b*log(c))*sin(6*b*log(c)))*n*cos(6*b*log(x^n) + 6*a) + 6*(b*cos(4*b*log(c))*sin(8*b*lo
g(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 4*(b*cos(2*b*log(c))*sin(8*b*log(c)) -
b*cos(8*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(8*b*log(c)) - 4*(b*cos(8*b*log(c))*cos(
6*b*log(c)) + b*sin(8*b*log(c))*sin(6*b*log(c)))*n*sin(6*b*log(x^n) + 6*a) - 6*(b*cos(8*b*log(c))*cos(4*b*log(
c)) + b*sin(8*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 4*(b*cos(8*b*log(c))*cos(2*b*log(c)) + b*
sin(8*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(8*b*log(x^n) + 8*a) - 8*(6*(b*cos(4*b*log(c))*
sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) + 4*(b*cos(2*b*log(c))*sin(6*b*
log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(6*b*log(c)) - 6*(b*cos(6*b*lo
g(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 4*(b*cos(6*b*log(c))*co
s(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) - 12*(4*
(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + b*n*sin(4*
b*log(c)) - 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a
))*sin(4*b*log(x^n) + 4*a))

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mupad [B]  time = 6.59, size = 247, normalized size = 3.69 \[ \ln \relax (x)\,1{}\mathrm {i}+\frac {8}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {4}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}+\frac {4}{b\,n\,\left (4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+4\,{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,8{}\mathrm {i}}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}{b\,n}-\frac {8}{b\,n\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + b*log(c*x^n))^5/x,x)

[Out]

log(x)*1i + 8/(b*n*(2*exp(a*2i)*(c*x^n)^(b*2i) + exp(a*4i)*(c*x^n)^(b*4i) + 1)) - 4/(b*n*(exp(a*2i)*(c*x^n)^(b
*2i) + 1)) + 4/(b*n*(4*exp(a*2i)*(c*x^n)^(b*2i) + 6*exp(a*4i)*(c*x^n)^(b*4i) + 4*exp(a*6i)*(c*x^n)^(b*6i) + ex
p(a*8i)*(c*x^n)^(b*8i) + 1)) - log(exp(a*2i)*(c*x^n)^(b*2i) + 1)/(b*n) - 8/(b*n*(3*exp(a*2i)*(c*x^n)^(b*2i) +
3*exp(a*4i)*(c*x^n)^(b*4i) + exp(a*6i)*(c*x^n)^(b*6i) + 1))

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sympy [A]  time = 21.87, size = 92, normalized size = 1.37 \[ \begin {cases} \log {\relax (x )} \tan ^{5}{\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \tan ^{5}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\tan ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} + 1 \right )}}{2 b n} + \frac {\tan ^{4}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b n} - \frac {\tan ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+b*ln(c*x**n))**5/x,x)

[Out]

Piecewise((log(x)*tan(a)**5, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*tan(a + b*log(c))**5, Eq(n, 0)), (log(
tan(a + b*n*log(x) + b*log(c))**2 + 1)/(2*b*n) + tan(a + b*n*log(x) + b*log(c))**4/(4*b*n) - tan(a + b*n*log(x
) + b*log(c))**2/(2*b*n), True))

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