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

Optimal. Leaf size=135 \[ \frac {(e x)^{m+1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (e^{2 a d} \left (c x^n\right )^{2 b d}-1\right )^p F_1\left (\frac {m+1}{2 b d n};-p,p;\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]

[Out]

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

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Rubi [F]  time = 0.10, 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 = {} \[ \int (e x)^m \tanh ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.60, size = 174, normalized size = 1.29 \[ \frac {x (e x)^m \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )^{-p} \left (\frac {e^{2 a d} \left (c x^n\right )^{2 b d}-1}{e^{2 a d} \left (c x^n\right )^{2 b d}+1}\right )^p \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )^p F_1\left (\frac {m+1}{2 b d n};-p,p;\frac {m+1}{2 b d n}+1;e^{2 a d} \left (c x^n\right )^{2 b d},-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{m+1} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((e*x)^m*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 \[ \int \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \left (\tanh ^{p}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x)^m*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 \[ \int \left (e x\right )^{m} \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((e*x)^m*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 \[ \int {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n)))^p*(e*x)^m,x)

[Out]

int(tanh(d*(a + b*log(c*x^n)))^p*(e*x)^m, x)

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

[Out]

Timed out

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