3.185 \(\int \frac {\tanh ^2(d (a+b \log (c x^n)))}{x^3} \, dx\)

Optimal. Leaf size=136 \[ -\frac {2 \, _2F_1\left (1,-\frac {1}{b d n};1-\frac {1}{b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n x^2}+\frac {1-e^{2 a d} \left (c x^n\right )^{2 b d}}{b d n x^2 \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}+\frac {2-b d n}{2 b d n x^2} \]

[Out]

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

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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 = {} \[ \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tanh[d*(a + b*Log[c*x^n])]^2/x^3,x]

[Out]

Defer[Int][Tanh[d*(a + b*Log[c*x^n])]^2/x^3, x]

Rubi steps

\begin {align*} \int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx &=\int \frac {\tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.59, size = 159, normalized size = 1.17 \[ -\frac {2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac {1}{b d n};2-\frac {1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n-1) \left (2 \, _2F_1\left (1,-\frac {1}{b d n};1-\frac {1}{b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+2 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )}{2 b d n x^2 (b d n-1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[d*(a + b*Log[c*x^n])]^2/x^3,x]

[Out]

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

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="fricas")

[Out]

integral(tanh(b*d*log(c*x^n) + a*d)^2/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )^{2}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="giac")

[Out]

integrate(tanh((b*log(c*x^n) + a)*d)^2/x^3, x)

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maple [F]  time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a+b*ln(c*x^n)))^2/x^3,x)

[Out]

int(tanh(d*(a+b*ln(c*x^n)))^2/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n - 4}{2 \, {\left (b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n x^{2}\right )}} + 4 \, \int \frac {1}{b c^{2 \, b d} d n x^{3} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*log(c*x^n)))^2/x^3,x, algorithm="maxima")

[Out]

-1/2*(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) + b*d*n - 4)/(b*c^(2*b*d)*d*n*x^2*e^(2*b*d*log(x^n) + 2*a*d)
+ b*d*n*x^2) + 4*integrate(1/(b*c^(2*b*d)*d*n*x^3*e^(2*b*d*log(x^n) + 2*a*d) + b*d*n*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tanh}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}^2}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*(a + b*log(c*x^n)))^2/x^3,x)

[Out]

int(tanh(d*(a + b*log(c*x^n)))^2/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(d*(a+b*ln(c*x**n)))**2/x**3,x)

[Out]

Integral(tanh(a*d + b*d*log(c*x**n))**2/x**3, x)

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