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

Optimal. Leaf size=78 \[ \text {Int}\left (x^m \coth (a+b x) \text {csch}(a+b x),x\right )+\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b}-\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b} \]

[Out]

CannotIntegrate(x^m*coth(b*x+a)*csch(b*x+a),x)+1/2*exp(a)*x^m*GAMMA(1+m,-b*x)/b/((-b*x)^m)-1/2*x^m*GAMMA(1+m,b
*x)/b/exp(a)/((b*x)^m)

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Rubi [A]  time = 0.15, 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 = {} \[ \int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx \]

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*}

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Mathematica [A]  time = 38.88, size = 0, normalized size = 0.00 \[ \int x^m \cosh (a+b x) \coth ^2(a+b x) \, dx \]

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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fricas [A]  time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.44, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\cosh ^{3}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )^{2}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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