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]

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

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Rubi [F]  time = 0.03, 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 = {} \[ \int \frac {\tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]

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*}

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Mathematica [B]  time = 3.31, 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*1/(b*d*n), 1 - 1/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]))])/x

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tanh \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \]

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)

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

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)

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maple [F]  time = 1.03, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]

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)

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

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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n)))/x^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

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)

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