3.221 \(\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\)

Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 180.02, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]

Verification is Not applicable to the result.

[In]

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

[Out]

$Aborted

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fricas [A]  time = 0.90, size = 0, normalized size = 0.00 \[ -\frac {4 \, d f x + 4 \, d e + {\left (3 \, d f x + 3 \, d e - f\right )} e^{\left (4 \, d x + 4 \, c\right )} - {\left (3 i \, d f x + 3 i \, d e - i \, f\right )} e^{\left (3 \, d x + 3 \, c\right )} - {\left (5 \, d f x + 5 \, d e - f\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (-i \, d f x - i \, d e + i \, f\right )} e^{\left (d x + c\right )} - {\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a d^{2} e f x - i \, a d^{2} e^{2} + {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a d^{2} e f x - i \, a d^{2} e^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (2 i \, a d^{2} f^{2} x^{2} + 4 i \, a d^{2} e f x + 2 i \, a d^{2} e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (d x + c\right )}\right )} {\rm integral}\left (\frac {4 \, d f^{2} x + 4 \, d e f - {\left (3 \, d^{2} f^{2} x^{2} + 3 \, d^{2} e^{2} + 2 \, d e f - 2 \, f^{2} + 2 \, {\left (3 \, d^{2} e f + d f^{2}\right )} x\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (3 i \, d^{2} f^{2} x^{2} + 3 i \, d^{2} e^{2} + 2 i \, d e f - 2 i \, f^{2} + {\left (6 i \, d^{2} e f + 2 i \, d f^{2}\right )} x\right )} e^{\left (d x + c\right )}}{i \, a d^{2} f^{3} x^{3} + 3 i \, a d^{2} e f^{2} x^{2} + 3 i \, a d^{2} e^{2} f x + i \, a d^{2} e^{3} + {\left (a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (-i \, a d^{2} f^{3} x^{3} - 3 i \, a d^{2} e f^{2} x^{2} - 3 i \, a d^{2} e^{2} f x - i \, a d^{2} e^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )} - {\left (a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3}\right )} e^{\left (d x + c\right )}}, x\right )}{-i \, a d^{2} f^{2} x^{2} - 2 i \, a d^{2} e f x - i \, a d^{2} e^{2} + {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )} + {\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a d^{2} e f x - i \, a d^{2} e^{2}\right )} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )} + {\left (2 i \, a d^{2} f^{2} x^{2} + 4 i \, a d^{2} e f x + 2 i \, a d^{2} e^{2}\right )} e^{\left (2 \, d x + 2 \, c\right )} + {\left (a d^{2} f^{2} x^{2} + 2 \, a d^{2} e f x + a d^{2} e^{2}\right )} e^{\left (d x + c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(4*d*f*x + 4*d*e + (3*d*f*x + 3*d*e - f)*e^(4*d*x + 4*c) - (3*I*d*f*x + 3*I*d*e - I*f)*e^(3*d*x + 3*c) - (5*d
*f*x + 5*d*e - f)*e^(2*d*x + 2*c) - (-I*d*f*x - I*d*e + I*f)*e^(d*x + c) - (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x
- I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e
*f*x - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(3*d*x + 3*c) + (2*I*a*d
^2*f^2*x^2 + 4*I*a*d^2*e*f*x + 2*I*a*d^2*e^2)*e^(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^
(d*x + c))*integral((4*d*f^2*x + 4*d*e*f - (3*d^2*f^2*x^2 + 3*d^2*e^2 + 2*d*e*f - 2*f^2 + 2*(3*d^2*e*f + d*f^2
)*x)*e^(2*d*x + 2*c) + (3*I*d^2*f^2*x^2 + 3*I*d^2*e^2 + 2*I*d*e*f - 2*I*f^2 + (6*I*d^2*e*f + 2*I*d*f^2)*x)*e^(
d*x + c))/(I*a*d^2*f^3*x^3 + 3*I*a*d^2*e*f^2*x^2 + 3*I*a*d^2*e^2*f*x + I*a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*
e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^(3*d*x + 3*c) + (-I*a*d^2*f^3*x^3 - 3*I*a*d^2*e*f^2*x^2 - 3*I*a*d^2
*e^2*f*x - I*a*d^2*e^3)*e^(2*d*x + 2*c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^
(d*x + c)), x))/(-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2
)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*e*f*x - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*
a*d^2*e*f*x + a*d^2*e^2)*e^(3*d*x + 3*c) + (2*I*a*d^2*f^2*x^2 + 4*I*a*d^2*e*f*x + 2*I*a*d^2*e^2)*e^(2*d*x + 2*
c) + (a*d^2*f^2*x^2 + 2*a*d^2*e*f*x + a*d^2*e^2)*e^(d*x + c))

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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(csch(d*x+c)^3/(f*x+e)/(a+I*a*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.70, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {csch}\left (d x +c \right )^{3}}{\left (f x +e \right ) \left (a +i a \sinh \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -8 \, f \int \frac {1}{-4 i \, a d f^{2} x^{2} - 8 i \, a d e f x - 4 i \, a d e^{2} + 4 \, {\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac {8 \, {\left (4 \, d f x + 4 \, d e + {\left (3 \, d f x e^{\left (4 \, c\right )} + {\left (3 \, d e - f\right )} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} + {\left (-3 i \, d f x e^{\left (3 \, c\right )} + {\left (-3 i \, d e + i \, f\right )} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} - {\left (5 \, d f x e^{\left (2 \, c\right )} + {\left (5 \, d e - f\right )} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + {\left (i \, d f x e^{c} + {\left (i \, d e - i \, f\right )} e^{c}\right )} e^{\left (d x\right )}\right )}}{-8 i \, a d^{2} f^{2} x^{2} - 16 i \, a d^{2} e f x - 8 i \, a d^{2} e^{2} + 8 \, {\left (a d^{2} f^{2} x^{2} e^{\left (5 \, c\right )} + 2 \, a d^{2} e f x e^{\left (5 \, c\right )} + a d^{2} e^{2} e^{\left (5 \, c\right )}\right )} e^{\left (5 \, d x\right )} + {\left (-8 i \, a d^{2} f^{2} x^{2} e^{\left (4 \, c\right )} - 16 i \, a d^{2} e f x e^{\left (4 \, c\right )} - 8 i \, a d^{2} e^{2} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} - 16 \, {\left (a d^{2} f^{2} x^{2} e^{\left (3 \, c\right )} + 2 \, a d^{2} e f x e^{\left (3 \, c\right )} + a d^{2} e^{2} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} + {\left (16 i \, a d^{2} f^{2} x^{2} e^{\left (2 \, c\right )} + 32 i \, a d^{2} e f x e^{\left (2 \, c\right )} + 16 i \, a d^{2} e^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + 8 \, {\left (a d^{2} f^{2} x^{2} e^{c} + 2 \, a d^{2} e f x e^{c} + a d^{2} e^{2} e^{c}\right )} e^{\left (d x\right )}} - 8 \, \int \frac {3 \, d^{2} f^{2} x^{2} + 3 \, d^{2} e^{2} + 2 i \, d e f - 2 \, f^{2} + 2 \, {\left (3 \, d^{2} e f + i \, d f^{2}\right )} x}{16 \, {\left (a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3} + {\left (a d^{2} f^{3} x^{3} e^{c} + 3 \, a d^{2} e f^{2} x^{2} e^{c} + 3 \, a d^{2} e^{2} f x e^{c} + a d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} - 8 \, \int -\frac {3 \, d^{2} f^{2} x^{2} + 3 \, d^{2} e^{2} - 2 i \, d e f - 2 \, f^{2} + 2 \, {\left (3 \, d^{2} e f - i \, d f^{2}\right )} x}{16 \, {\left (a d^{2} f^{3} x^{3} + 3 \, a d^{2} e f^{2} x^{2} + 3 \, a d^{2} e^{2} f x + a d^{2} e^{3} - {\left (a d^{2} f^{3} x^{3} e^{c} + 3 \, a d^{2} e f^{2} x^{2} e^{c} + 3 \, a d^{2} e^{2} f x e^{c} + a d^{2} e^{3} e^{c}\right )} e^{\left (d x\right )}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*f*integrate(1/(-4*I*a*d*f^2*x^2 - 8*I*a*d*e*f*x - 4*I*a*d*e^2 + 4*(a*d*f^2*x^2*e^c + 2*a*d*e*f*x*e^c + a*d*
e^2*e^c)*e^(d*x)), x) - 8*(4*d*f*x + 4*d*e + (3*d*f*x*e^(4*c) + (3*d*e - f)*e^(4*c))*e^(4*d*x) + (-3*I*d*f*x*e
^(3*c) + (-3*I*d*e + I*f)*e^(3*c))*e^(3*d*x) - (5*d*f*x*e^(2*c) + (5*d*e - f)*e^(2*c))*e^(2*d*x) + (I*d*f*x*e^
c + (I*d*e - I*f)*e^c)*e^(d*x))/(-8*I*a*d^2*f^2*x^2 - 16*I*a*d^2*e*f*x - 8*I*a*d^2*e^2 + 8*(a*d^2*f^2*x^2*e^(5
*c) + 2*a*d^2*e*f*x*e^(5*c) + a*d^2*e^2*e^(5*c))*e^(5*d*x) + (-8*I*a*d^2*f^2*x^2*e^(4*c) - 16*I*a*d^2*e*f*x*e^
(4*c) - 8*I*a*d^2*e^2*e^(4*c))*e^(4*d*x) - 16*(a*d^2*f^2*x^2*e^(3*c) + 2*a*d^2*e*f*x*e^(3*c) + a*d^2*e^2*e^(3*
c))*e^(3*d*x) + (16*I*a*d^2*f^2*x^2*e^(2*c) + 32*I*a*d^2*e*f*x*e^(2*c) + 16*I*a*d^2*e^2*e^(2*c))*e^(2*d*x) + 8
*(a*d^2*f^2*x^2*e^c + 2*a*d^2*e*f*x*e^c + a*d^2*e^2*e^c)*e^(d*x)) - 8*integrate(1/16*(3*d^2*f^2*x^2 + 3*d^2*e^
2 + 2*I*d*e*f - 2*f^2 + 2*(3*d^2*e*f + I*d*f^2)*x)/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^
2*e^3 + (a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c + 3*a*d^2*e^2*f*x*e^c + a*d^2*e^3*e^c)*e^(d*x)), x) - 8*int
egrate(-1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 2*I*d*e*f - 2*f^2 + 2*(3*d^2*e*f - I*d*f^2)*x)/(a*d^2*f^3*x^3 + 3*a*
d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - (a*d^2*f^3*x^3*e^c + 3*a*d^2*e*f^2*x^2*e^c + 3*a*d^2*e^2*f*x*e^c
+ a*d^2*e^3*e^c)*e^(d*x)), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(c + d*x)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)),x)

[Out]

int(1/(sinh(c + d*x)^3*(e + f*x)*(a + a*sinh(c + d*x)*1i)), 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(csch(d*x+c)**3/(f*x+e)/(a+I*a*sinh(d*x+c)),x)

[Out]

Timed out

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