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

Optimal. Leaf size=83 \[ \text {Int}\left (x^m \coth (a+b x),x\right )+\frac {e^{2 a} 2^{-m-3} x^m (-b x)^{-m} \Gamma (m+1,-2 b x)}{b}+\frac {e^{-2 a} 2^{-m-3} x^m (b x)^{-m} \Gamma (m+1,2 b x)}{b} \]

[Out]

2^(-3-m)*exp(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/exp(2*a)/((b*x)^m)+Uninte
grable(x^m*coth(b*x+a),x)

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Rubi [A]  time = 0.13, 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 = {} \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]

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*}

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Mathematica [A]  time = 23.54, size = 0, normalized size = 0.00 \[ \int x^m \cosh ^2(a+b x) \coth (a+b x) \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right ), x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )\,{d x} \]

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)

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maple [A]  time = 0.50, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*cosh(a + b*x)^3)/sinh(a + b*x),x)

[Out]

int((x^m*cosh(a + b*x)^3)/sinh(a + b*x), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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