3.192 \(\int \tanh ^p(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=115 \[ x \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p F_1\left (\frac {1}{2 b d n};-p,p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

x*(-1+exp(2*a*d)*(c*x^n)^(2*b*d))^p*AppellF1(1/2/b/d/n,-p,p,1+1/2/b/d/n,exp(2*a*d)*(c*x^n)^(2*b*d),-exp(2*a*d)
*(c*x^n)^(2*b*d))/((1-exp(2*a*d)*(c*x^n)^(2*b*d))^p)

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Rubi [F]  time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[d*(a + b*Log[c*x^n])]^p,x]

[Out]

Defer[Int][Tanh[d*(a + b*Log[c*x^n])]^p, x]

Rubi steps

\begin {align*} \int \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\int \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx\\ \end {align*}

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Mathematica [B]  time = 4.13, size = 387, normalized size = 3.37 \[ \frac {x (2 b d n+1) \left (\frac {e^{2 a d} \left (c x^n\right )^{2 b d}-1}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}\right )^p F_1\left (\frac {1}{2 b d n};-p,p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{-2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};1-p,p;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )-2 b d n p e^{2 a d} \left (c x^n\right )^{2 b d} F_1\left (1+\frac {1}{2 b d n};-p,p+1;2+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )+(2 b d n+1) F_1\left (\frac {1}{2 b d n};-p,p;1+\frac {1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tanh[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((1 + 2*b*d*n)*x*((-1 + E^(2*a*d)*(c*x^n)^(2*b*d))/(1 + E^(2*a*d)*(c*x^n)^(2*b*d)))^p*AppellF1[1/(2*b*d*n), -p
, p, 1 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])/(-2*b*d*E^(2*a*d)*n*p*(c*x^n)^
(2*b*d)*AppellF1[1 + 1/(2*b*d*n), 1 - p, p, 2 + 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2
*b*d))] - 2*b*d*E^(2*a*d)*n*p*(c*x^n)^(2*b*d)*AppellF1[1 + 1/(2*b*d*n), -p, 1 + p, 2 + 1/(2*b*d*n), E^(2*a*d)*
(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))] + (1 + 2*b*d*n)*AppellF1[1/(2*b*d*n), -p, p, 1 + 1/(2*b*d*n), E
^(2*a*d)*(c*x^n)^(2*b*d), -(E^(2*a*d)*(c*x^n)^(2*b*d))])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(tanh(b*d*log(c*x^n) + a*d)^p, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh((b*log(c*x^n) + a)*d)^p, x)

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maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int(tanh(d*(a+b*ln(c*x^n)))^p,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh((b*log(c*x^n) + a)*d)^p, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n)))^p,x)

[Out]

int(tanh(d*(a + b*log(c*x^n)))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tanh ^{p}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*ln(c*x**n)))**p,x)

[Out]

Integral(tanh(d*(a + b*log(c*x**n)))**p, x)

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