3.383 \(\int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx\)

Optimal. Leaf size=79 \[ -\text{CannotIntegrate}\left (x^m \tanh (a+b x) \text{sech}(a+b x),x\right )+\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b}+\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b} \]

[Out]

-CannotIntegrate[x^m*Sech[a + b*x]*Tanh[a + b*x], x] + (E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) + (x^m*
Gamma[1 + m, b*x])/(2*b*E^a*(b*x)^m)

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Rubi [A]  time = 0.167236, 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 = {} \[ \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^m*Sinh[a + b*x]*Tanh[a + b*x]^2,x]

[Out]

(E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) + (x^m*Gamma[1 + m, b*x])/(2*b*E^a*(b*x)^m) - Defer[Int][x^m*S
ech[a + b*x]*Tanh[a + b*x], x]

Rubi steps

\begin{align*} \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x^m \sinh (a+b x) \, dx-\int x^m \text{sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac{1}{2} \int e^{-i (i a+i b x)} x^m \, dx-\frac{1}{2} \int e^{i (i a+i b x)} x^m \, dx-\int x^m \text{sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac{e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}+\frac{e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}-\int x^m \text{sech}(a+b x) \tanh (a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 16.8345, size = 0, normalized size = 0. \[ \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^m*Sinh[a + b*x]*Tanh[a + b*x]^2,x]

[Out]

Integrate[x^m*Sinh[a + b*x]*Tanh[a + b*x]^2, x]

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Maple [A]  time = 0.08, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ({\rm sech} \left (bx+a\right ) \right ) ^{2} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sech(b*x+a)^2*sinh(b*x+a)^3,x)

[Out]

int(x^m*sech(b*x+a)^2*sinh(b*x+a)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(x^m*sech(b*x + a)^2*sinh(b*x + a)^3, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(x^m*sech(b*x + a)^2*sinh(b*x + a)^3, 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(x**m*sech(b*x+a)**2*sinh(b*x+a)**3,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^m*sech(b*x + a)^2*sinh(b*x + a)^3, x)