3.5.90 \(\int \frac {\coth ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\) [490]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\coth ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

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Rubi [A]
time = 0.05, 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 = {} \begin {gather*} \int \frac {\coth ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Coth[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][Coth[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\coth ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\coth ^3(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end {align*}

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Mathematica [F]
time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Coth[c + d*x]^3/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

$Aborted

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\coth ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-(a*f - 2*(b*d*f*x*e^(3*c) + b*d*e^(3*c + 1))*e^(3*d*x) + (2*a*d*f*x*e^(2*c) - a*f*e^(2*c) + 2*a*d*e^(2*c + 1)
)*e^(2*d*x) + 2*(b*d*f*x*e^c + b*d*e^(c + 1))*e^(d*x))/(a^2*d^2*f^2*x^2 + 2*a^2*d^2*f*x*e + a^2*d^2*e^2 + (a^2
*d^2*f^2*x^2*e^(4*c) + 2*a^2*d^2*f*x*e^(4*c + 1) + a^2*d^2*e^(4*c + 2))*e^(4*d*x) - 2*(a^2*d^2*f^2*x^2*e^(2*c)
+ 2*a^2*d^2*f*x*e^(2*c + 1) + a^2*d^2*e^(2*c + 2))*e^(2*d*x)) + integrate(-(a*b*d*f*e + a^2*f^2 + (a^2*d^2*f^
2 + b^2*d^2*f^2)*x^2 + (a*b*d*f^2 + 2*(a^2*d^2*f + b^2*d^2*f)*e)*x + (a^2*d^2 + b^2*d^2)*e^2)/(a^3*d^2*f^3*x^3
+ 3*a^3*d^2*f^2*x^2*e + 3*a^3*d^2*f*x*e^2 + a^3*d^2*e^3 - (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*f^2*x^2*e^(c + 1)
+ 3*a^3*d^2*f*x*e^(c + 2) + a^3*d^2*e^(c + 3))*e^(d*x)), x) - integrate(-(a*b*d*f*e - a^2*f^2 - (a^2*d^2*f^2 +
b^2*d^2*f^2)*x^2 + (a*b*d*f^2 - 2*(a^2*d^2*f + b^2*d^2*f)*e)*x - (a^2*d^2 + b^2*d^2)*e^2)/(a^3*d^2*f^3*x^3 +
3*a^3*d^2*f^2*x^2*e + 3*a^3*d^2*f*x*e^2 + a^3*d^2*e^3 + (a^3*d^2*f^3*x^3*e^c + 3*a^3*d^2*f^2*x^2*e^(c + 1) + 3
*a^3*d^2*f*x*e^(c + 2) + a^3*d^2*e^(c + 3))*e^(d*x)), x) + integrate(2*(a^2*b + b^3 - (a^3*e^c + a*b^2*e^c)*e^
(d*x))/(a^3*b*f*x + a^3*b*e - (a^3*b*f*x*e^(2*c) + a^3*b*e^(2*c + 1))*e^(2*d*x) - 2*(a^4*f*x*e^c + a^4*e^(c +
1))*e^(d*x)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(coth(d*x + c)^3/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)**3/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(coth(c + d*x)**3/((a + b*sinh(c + d*x))*(e + f*x)), x)

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)^3/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)^3/((e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(coth(c + d*x)^3/((e + f*x)*(a + b*sinh(c + d*x))), x)

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