### 3.438 $$\int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx$$

Optimal. Leaf size=77 $\text{CannotIntegrate}\left (x^m \coth (a+b x) \text{csch}(a+b x),x\right )+\frac{e^a x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b}-\frac{e^{-a} x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b}$

[Out]

CannotIntegrate[x^m*Coth[a + b*x]*Csch[a + b*x], x] + (E^a*x^m*Gamma[1 + m, -(b*x)])/(2*b*(-(b*x))^m) - (x^m*G
amma[1 + m, b*x])/(2*b*E^a*(b*x)^m)

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Rubi [A]  time = 0.153311, 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 = {} $\int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Int[x^m*Cosh[a + b*x]*Coth[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*C
oth[a + b*x]*Csch[a + b*x], x]

Rubi steps

\begin{align*} \int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx &=\int x^m \cosh (a+b x) \, dx+\int x^m \coth (a+b x) \text{csch}(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 \coth (a+b x) \text{csch}(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 \coth (a+b x) \text{csch}(a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 15.9238, size = 0, normalized size = 0. $\int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[x^m*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

Integrate[x^m*Cosh[a + b*x]*Coth[a + b*x]^2, x]

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Maple [A]  time = 0.082, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosh(b*x+a)^3*csch(b*x+a)^2,x)

[Out]

int(x^m*cosh(b*x+a)^3*csch(b*x+a)^2,x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x + a)^3*csch(b*x + a)^2, x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*cosh(b*x + a)^3*csch(b*x + a)^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(x**m*cosh(b*x+a)**3*csch(b*x+a)**2,x)

[Out]

Timed out

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*cosh(b*x + a)^3*csch(b*x + a)^2, x)