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

Optimal. Leaf size=88 \[ \frac {(e x)^{m+1}}{e (m+1)}-\frac {2 (e x)^{m+1} \, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{e (m+1)} \]

[Out]

(e*x)^(1+m)/e/(1+m)-2*(e*x)^(1+m)*hypergeom([1, 1/2*(1+m)/b/d/n],[1+1/2*(1+m)/b/d/n],-exp(2*a*d)*(c*x^n)^(2*b*
d))/e/(1+m)

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Rubi [F]  time = 0.05, 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 = {} \[ \int (e x)^m \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 14.33, size = 160, normalized size = 1.82 \[ \frac {x (e x)^m \left (\frac {(m+1) e^{2 a d} \left (c x^n\right )^{2 b d} \, _2F_1\left (1,\frac {m+2 b d n+1}{2 b d n};\frac {m+4 b d n+1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{2 b d n+m+1}-\, _2F_1\left (1,\frac {m+1}{2 b d n};\frac {m+1}{2 b d n}+1;-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m*(-Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]))] + (E^
(2*a*d)*(1 + m)*(c*x^n)^(2*b*d)*Hypergeometric2F1[1, (1 + m + 2*b*d*n)/(2*b*d*n), (1 + m + 4*b*d*n)/(2*b*d*n),
-(E^(2*a*d)*(c*x^n)^(2*b*d))])/(1 + m + 2*b*d*n)))/(1 + m)

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fricas [F]  time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {e^{m} x x^{m}}{m + 1} - 2 \, e^{m} \int \frac {x^{m}}{c^{2 \, b d} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^m*x*x^m/(m + 1) - 2*e^m*integrate(x^m/(c^(2*b*d)*e^(2*b*d*log(x^n) + 2*a*d) + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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