3.480 \(\int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\)

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {\coth (c+d x) \text {csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]

[Out]

Unintegrable(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

Defer[Int][(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\coth (c+d x) \text {csch}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac {\coth (c+d x) \text {csch}^2(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 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

Integrate[(Coth[c + d*x]*Csch[c + d*x]^2)/((e + f*x)*(a + b*Sinh[c + d*x])),x]

[Out]

$Aborted

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fricas [A]  time = 2.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\coth \left (d x + c\right ) \operatorname {csch}\left (d x + c\right )^{2}}{a f x + a e + {\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

integral(coth(d*x + c)*csch(d*x + c)^2/(a*f*x + a*e + (b*f*x + b*e)*sinh(d*x + c)), x)

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giac [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(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 2.46, size = 0, normalized size = 0.00 \[ \int \frac {\coth \left (d x +c \right ) \mathrm {csch}\left (d x +c \right )^{2}}{\left (f x +e \right ) \left (a +b \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*x+c)*csch(d*x+c)^2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

int(coth(d*x+c)*csch(d*x+c)^2/(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 \[ -\frac {a f - 2 \, {\left (b d f x e^{\left (3 \, c\right )} + b d e e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (2 \, a d f x e^{\left (2 \, c\right )} + {\left (2 \, d e - f\right )} a e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 2 \, {\left (b d f x e^{c} + b d e e^{c}\right )} e^{\left (d x\right )}}{a^{2} d^{2} f^{2} x^{2} + 2 \, a^{2} d^{2} e f x + a^{2} d^{2} e^{2} + {\left (a^{2} d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (4 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 2 \, {\left (a^{2} d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 2 \, a^{2} d^{2} e f x e^{\left (2 \, c\right )} + a^{2} d^{2} e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}} + 4 \, \int -\frac {b^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} e^{2} + a b d e f + a^{2} f^{2} + {\left (2 \, b^{2} d^{2} e f + a b d f^{2}\right )} x}{4 \, {\left (a^{3} d^{2} f^{3} x^{3} + 3 \, a^{3} d^{2} e f^{2} x^{2} + 3 \, a^{3} d^{2} e^{2} f x + a^{3} d^{2} e^{3} - {\left (a^{3} d^{2} f^{3} x^{3} e^{c} + 3 \, a^{3} d^{2} e f^{2} x^{2} e^{c} + 3 \, a^{3} d^{2} e^{2} f x e^{c} + a^{3} d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 4 \, \int \frac {b^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} e^{2} - a b d e f + a^{2} f^{2} + {\left (2 \, b^{2} d^{2} e f - a b d f^{2}\right )} x}{4 \, {\left (a^{3} d^{2} f^{3} x^{3} + 3 \, a^{3} d^{2} e f^{2} x^{2} + 3 \, a^{3} d^{2} e^{2} f x + a^{3} d^{2} e^{3} + {\left (a^{3} d^{2} f^{3} x^{3} e^{c} + 3 \, a^{3} d^{2} e f^{2} x^{2} e^{c} + 3 \, a^{3} d^{2} e^{2} f x e^{c} + a^{3} d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} + 4 \, \int -\frac {a b^{2} e^{\left (d x + c\right )} - b^{3}}{2 \, {\left (a^{3} b f x + a^{3} b e - {\left (a^{3} b f x e^{\left (2 \, c\right )} + a^{3} b e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{4} f x e^{c} + a^{4} e e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {coth}\left (c+d\,x\right )}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c + d*x)/(sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(coth(c + d*x)/(sinh(c + d*x)^2*(e + f*x)*(a + b*sinh(c + d*x))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (c + d x \right )} \operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \sinh {\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*x+c)*csch(d*x+c)**2/(f*x+e)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(coth(c + d*x)*csch(c + d*x)**2/((a + b*sinh(c + d*x))*(e + f*x)), x)

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