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

Optimal. Leaf size=66 \[ -\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 66, 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} \[ -\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[a + b*Log[c*x^n]]^5/x,x]

[Out]

Log[Cosh[a + b*Log[c*x^n]]]/(b*n) - Tanh[a + b*Log[c*x^n]]^2/(2*b*n) - Tanh[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 {\tanh ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \tanh ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\operatorname {Subst}\left (\int \tanh ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\operatorname {Subst}\left (\int \tanh (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tanh ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\tanh ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 55, normalized size = 0.83 \[ \frac {-\tanh ^4\left (a+b \log \left (c x^n\right )\right )-2 \tanh ^2\left (a+b \log \left (c x^n\right )\right )+4 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.63, size = 1568, normalized size = 23.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*n*cosh(b*n*log(x) + b*log(c) + a)^8*log(x) + 8*b*n*cosh(b*n*log(x) + b*log(c) + a)*log(x)*sinh(b*n*log(x)
+ b*log(c) + a)^7 + b*n*log(x)*sinh(b*n*log(x) + b*log(c) + a)^8 + 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(
c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2*log(x) + b*n*log(x) - 1)*sinh(b*n*log(x) + b*log(c) + a
)^6 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3*log(x) + 3*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a))*
sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a)^4 + 2*(35*b*n*cosh(b*
n*log(x) + b*log(c) + a)^4*log(x) + 30*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n*log(x) - 2)*
sinh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5*log(x) + 10*(b*n*log(x) - 1)*co
sh(b*n*log(x) + b*log(c) + a)^3 + (3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(
c) + a)^3 + 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n*log(x) + 4*(7*b*n*cosh(b*n*log(x) + b*l
og(c) + a)^6*log(x) + 15*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^4 + 3*(3*b*n*log(x) - 2)*cosh(b*n*lo
g(x) + b*log(c) + a)^2 + b*n*log(x) - 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b*n*log(x) + b*log(c) + a)^
8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + sinh(b*n*log(x) + b*log(c) + a)^8 +
4*(7*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 4*cosh(b*n*log(x) + b*log(c) +
a)^6 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c)
+ a)^5 + 2*(35*cosh(b*n*log(x) + b*log(c) + a)^4 + 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x)
+ b*log(c) + a)^4 + 6*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*cosh(b*n
*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b
*n*log(x) + b*log(c) + a)^6 + 15*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*
sinh(b*n*log(x) + b*log(c) + a)^2 + 4*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7
+ 3*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*log(c) + a)
)*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(2*cosh(b*n*log(x) + b*log(c) + a)/(cosh(b*n*log(x) + b*log(c) + a)
- sinh(b*n*log(x) + b*log(c) + a))) + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7*log(x) + 3*(b*n*log(x) - 1)*cos
h(b*n*log(x) + b*log(c) + a)^5 + (3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a)^3 + (b*n*log(x) - 1)*cosh(
b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^8 + 8*b*n*co
sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sinh(b*n*log(x) + b*log(c) + a)^8 + 4*b*
n*cosh(b*n*log(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log(x) + b*lo
g(c) + a)^6 + 6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*cos
h(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4
+ 30*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^4 + 4*b*n*cosh(b*n*log(x)
+ b*log(c) + a)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*
b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) +
a)^6 + 15*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n)*sinh(b*n*log
(x) + b*log(c) + a)^2 + b*n + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)
^5 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c)
+ a))

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giac [B]  time = 0.40, size = 161, normalized size = 2.44 \[ \frac {\log \left (\sqrt {2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\relax (c) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} + 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} + 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1\right )}^{4} b n} - \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sqrt(2*x^(2*b*n)*abs(c)^(2*b)*cos(pi*b*sgn(c) - pi*b)*e^(2*a) + x^(4*b*n)*abs(c)^(4*b)*e^(4*a) + 1))/(b*n)
- 1/12*(25*c^(8*b)*x^(8*b*n)*e^(8*a) + 52*c^(6*b)*x^(6*b*n)*e^(6*a) + 102*c^(4*b)*x^(4*b*n)*e^(4*a) + 52*c^(2
*b)*x^(2*b*n)*e^(2*a) + 25)/((c^(2*b)*x^(2*b*n)*e^(2*a) + 1)^4*b*n) - log(x)

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maple [A]  time = 0.02, size = 88, normalized size = 1.33 \[ -\frac {\tanh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{4 b n}-\frac {\tanh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2 n b}-\frac {\ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2 n b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.71, size = 829, normalized size = 12.56 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(48*c^(6*b)*e^(6*b*log(x^n) + 6*a) + 108*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 88*c^(2*b)*e^(2*b*log(x^n) + 2*
a) + 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log
(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 1/24*(12*c^(6*b)*e^(6*b*log(x^n) + 6*a) + 42*c^(4
*b)*e^(4*b*log(x^n) + 4*a) + 52*c^(2*b)*e^(2*b*log(x^n) + 2*a) + 25)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b
*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a
) + b*n) + 5/8*(4*c^(6*b)*e^(6*b*log(x^n) + 6*a) + 6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c^(2*b)*e^(2*b*log(x^n
) + 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*
b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 5/12*(6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c
^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a)
+ 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) + 5/12*(4*c^(2*b)*e^(2*b*
log(x^n) + 2*a) + 1)/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*
n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n) - 5/8/(b*c^(8*b)*n*e^(8*b*log(x^n) + 8*
a) + 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 4*b*c^(2*b)*n*e^(2*b*log(x^
n) + 2*a) + b*n) + log((c^(2*b)*e^(2*b*log(x^n) + 2*a) + 1)*e^(-2*a)/c^(2*b))/(b*n) - log(x)

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mupad [B]  time = 1.06, size = 227, normalized size = 3.44 \[ \frac {8}{b\,n+3\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+3\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}}-\ln \relax (x)+\frac {4}{b\,n+b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\frac {4}{b\,n+4\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+6\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}+4\,b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}+b\,n\,{\mathrm {e}}^{8\,a}\,{\left (c\,x^n\right )}^{8\,b}}-\frac {8}{b\,n+2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+1\right )}{b\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*log(c*x^n))^5/x,x)

[Out]

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

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sympy [A]  time = 19.60, size = 97, normalized size = 1.47 \[ \begin {cases} \log {\relax (x )} \tanh ^{5}{\relax (a )} & \text {for}\: b = 0 \wedge n = 0 \\\log {\relax (x )} \tanh ^{5}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\\log {\relax (x )} \tanh ^{5}{\relax (a )} & \text {for}\: b = 0 \\\log {\relax (x )} - \frac {\log {\left (\tanh {\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} + 1 \right )}}{b n} - \frac {\tanh ^{4}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{4 b n} - \frac {\tanh ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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