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

Optimal. Leaf size=55 \[ \frac{x^2}{2}-x^2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

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

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Rubi [F]  time = 0.0240282, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int x \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Tanh[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 7.58517, size = 122, normalized size = 2.22 \[ \frac{x^2 \left (e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{1}{b d n};2+\frac{1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(b d n+1) \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b d n+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.783, size = 0, normalized size = 0. \begin{align*} \int x\tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, x^{2} - 2 \, \int \frac{x}{c^{2 \, b d} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x \tanh \left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \tanh{\left (a d + b d \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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