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

Optimal. Leaf size=169 \[ -\frac{2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d e n}+\frac{(e x)^{m+1} \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d e n \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}+\frac{(e x)^{m+1} (b d n+m+1)}{b d e (m+1) n} \]

[Out]

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

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Rubi [F]  time = 0.0769626, 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 = {} \[ \int (e x)^m \tanh ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [B]  time = 16.5395, size = 516, normalized size = 3.05 \[ \frac{(m+1) x^{-m} (e x)^m \text{sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \left (\frac{x^{m+1} \sinh (b d n \log (x)) \text{sech}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{m+1}-\frac{\cosh \left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \exp \left (-\frac{(2 m+1) \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{b n}\right ) \left ((2 b d n+m+1) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right ) \, _2F_1\left (1,\frac{m+1}{2 b d n};\frac{m+1}{2 b d n}+1;-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-(m+1) \exp \left (\frac{a (2 b d n+2 m+1)}{b n}+\frac{(2 b d n+2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )}{n}+\log (x) (2 b d n+m+1)\right ) \, _2F_1\left (1,\frac{m+2 b d n+1}{2 b d n};\frac{m+4 b d n+1}{2 b d n};-e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(2 b d n+m+1) \tanh \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \exp \left (\frac{2 a m+a+b (2 m+1) \left (\log \left (c x^n\right )-n \log (x)\right )+b (m+1) n \log (x)}{b n}\right )\right )}{(m+1) (2 b d n+m+1)}\right )}{b d n}-\frac{x (e x)^m \sinh (b d n \log (x)) \text{sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) \text{sech}\left (d \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+b d n \log (x)\right )}{b d n}+\frac{x (e x)^m}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(e*x)^m)/(1 + m) - (x*(e*x)^m*Sech[d*(a + b*(-(n*Log[x]) + Log[c*x^n]))]*Sech[b*d*n*Log[x] + d*(a + b*(-(n*
Log[x]) + Log[c*x^n]))]*Sinh[b*d*n*Log[x]])/(b*d*n) + ((1 + m)*(e*x)^m*Sech[d*(a + b*(-(n*Log[x]) + Log[c*x^n]
))]*((x^(1 + m)*Sech[d*(a + b*Log[c*x^n])]*Sinh[b*d*n*Log[x]])/(1 + m) - (Cosh[d*(a + b*(-(n*Log[x]) + Log[c*x
^n]))]*(E^((a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + 2*b*d*n)*
Hypergeometric2F1[1, (1 + m)/(2*b*d*n), 1 + (1 + m)/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]))] - E^((a*(1 + 2*m +
2*b*d*n))/(b*n) + (1 + m + 2*b*d*n)*Log[x] + ((1 + 2*m + 2*b*d*n)*(-(n*Log[x]) + Log[c*x^n]))/n)*(1 + m)*Hype
rgeometric2F1[1, (1 + m + 2*b*d*n)/(2*b*d*n), (1 + m + 4*b*d*n)/(2*b*d*n), -E^(2*d*(a + b*Log[c*x^n]))] + E^((
a + 2*a*m + b*(1 + m)*n*Log[x] + b*(1 + 2*m)*(-(n*Log[x]) + Log[c*x^n]))/(b*n))*(1 + m + 2*b*d*n)*Tanh[d*(a +
b*Log[c*x^n])]))/(E^(((1 + 2*m)*(a + b*(-(n*Log[x]) + Log[c*x^n])))/(b*n))*(1 + m)*(1 + m + 2*b*d*n))))/(b*d*n
*x^m)

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Maple [F]  time = 0.218, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( \tanh \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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