3.177 \(\int \frac{\tanh (d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 \, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x}-\frac{1}{x} \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.0294655, 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 = {} \[ \int \frac{\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

Defer[Int][Tanh[d*(a + b*Log[c*x^n])]/x^2, x]

Rubi steps

\begin{align*} \int \frac{\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=\int \frac{\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx\\ \end{align*}

Mathematica [B]  time = 3.21226, size = 126, normalized size = 2.14 \[ \frac{\frac{e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac{1}{2 b d n};2-\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n-1}+\, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.796, size = 0, normalized size = 0. \begin{align*} \int{\frac{\tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(tanh(d*(a+b*ln(c*x^n)))/x^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\tanh \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")

[Out]

integral(tanh(b*d*log(c*x^n) + a*d)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

Integral(tanh(a*d + b*d*log(c*x**n))/x**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")

[Out]

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