### 3.411 $$\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx$$

Optimal. Leaf size=82 $\text{Unintegrable}\left (x^m \coth (a+b x),x\right )+\frac{e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \text{Gamma}(m+1,-2 b x)}{b}+\frac{e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \text{Gamma}(m+1,2 b x)}{b}$

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m) + Unintegrable[x^m*Coth[a + b*x], x]

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Rubi [A]  time = 0.131473, 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 = {} $\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m) + Defer[Int][x^m*Coth[a + b*x], x]

Rubi steps

\begin{align*} \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x^m \coth (a+b x) \, dx+\int x^m \cosh (a+b x) \sinh (a+b x) \, dx\\ &=\int x^m \coth (a+b x) \, dx+\int \frac{1}{2} x^m \sinh (2 a+2 b x) \, dx\\ &=\frac{1}{2} \int x^m \sinh (2 a+2 b x) \, dx+\int x^m \coth (a+b x) \, dx\\ &=\frac{1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac{1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx+\int x^m \coth (a+b x) \, dx\\ &=\frac{2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac{2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}+\int x^m \coth (a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 22.3679, size = 0, normalized size = 0. $\int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx$

Veriﬁcation is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.074, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cosh \left ( bx+a \right ) \right ) ^{3}{\rm csch} \left (bx+a\right )\, 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),x)

[Out]

int(x^m*cosh(b*x+a)^3*csch(b*x+a),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 )\,{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),x, algorithm="maxima")

[Out]

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

[Out]

integral(x^m*cosh(b*x + a)^3*csch(b*x + a), 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),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 )\,{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),x, algorithm="giac")

[Out]

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