3.180 \(\int x^2 \tanh ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=137 \[ -\frac {2 x^3 \, _2F_1\left (1,\frac {3}{2 b d n};1+\frac {3}{2 b d n};-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac {x^3 \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}+\frac {1}{3} x^3 \left (\frac {3}{b d n}+1\right ) \]

[Out]

1/3*(1+3/b/d/n)*x^3+x^3*(1-exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n/(1+exp(2*a*d)*(c*x^n)^(2*b*d))-2*x^3*hypergeom([1
, 3/2/b/d/n],[1+3/2/b/d/n],-exp(2*a*d)*(c*x^n)^(2*b*d))/b/d/n

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Rubi [F]  time = 0.06, 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 = {} \[ \int x^2 \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 8.17, size = 169, normalized size = 1.23 \[ \frac {x^3 \left (9 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac {3}{2 b d n};2+\frac {3}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+3) \left (-3 \, _2F_1\left (1,\frac {3}{2 b d n};1+\frac {3}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-3 \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )\right )}{3 b d n (2 b d n+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^3*(9*E^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 + 3/(2*b*d*n), 2 + 3/(2*b*d*n), -E^(2*d*(a + b*Log[c
*x^n]))] + (3 + 2*b*d*n)*(b*d*n - 3*Hypergeometric2F1[1, 3/(2*b*d*n), 1 + 3/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^
n]))] - 3*Tanh[d*(a + b*Log[c*x^n])])))/(3*b*d*n*(3 + 2*b*d*n))

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fricas [F]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} \tanh \left (b d \log \left (c x^{n}\right ) + a d\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 1.07, size = 0, normalized size = 0.00 \[ \int x^{2} \left (\tanh ^{2}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b c^{2 \, b d} d n x^{3} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + {\left (b d n + 6\right )} x^{3}}{3 \, {\left (b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n\right )}} - 6 \, \int \frac {x^{2}}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} + b d n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(b*c^(2*b*d)*d*n*x^3*e^(2*b*d*log(x^n) + 2*a*d) + (b*d*n + 6)*x^3)/(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*
a*d) + b*d*n) - 6*integrate(x^2/(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) + b*d*n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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