3.3.21 \(\int \frac {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [221]
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 = 0.05, 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 {\text {csch}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \end {gather*}
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 \begin {gather*} \text {\$Aborted} \end {gather*}
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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Maple [A]
time = 180.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
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*f*x*e - 4*I*a*d*e^2 + 4*(a*d*f^2*x^2*e^c + 2*a*d*f*x*e^(c + 1) +
a*d*e^(c + 2))*e^(d*x)), x) - 8*(4*d*f*x + 4*d*e + (3*d*f*x*e^(4*c) - f*e^(4*c) + 3*d*e^(4*c + 1))*e^(4*d*x) +
(-3*I*d*f*x*e^(3*c) + I*f*e^(3*c) - 3*I*d*e^(3*c + 1))*e^(3*d*x) - (5*d*f*x*e^(2*c) - f*e^(2*c) + 5*d*e^(2*c
+ 1))*e^(2*d*x) + (I*d*f*x*e^c + I*d*e^(c + 1) - I*f*e^c)*e^(d*x))/(-8*I*a*d^2*f^2*x^2 - 16*I*a*d^2*f*x*e - 8*
I*a*d^2*e^2 + 8*(a*d^2*f^2*x^2*e^(5*c) + 2*a*d^2*f*x*e^(5*c + 1) + a*d^2*e^(5*c + 2))*e^(5*d*x) - 8*(I*a*d^2*f
^2*x^2*e^(4*c) + 2*I*a*d^2*f*x*e^(4*c + 1) + I*a*d^2*e^(4*c + 2))*e^(4*d*x) - 16*(a*d^2*f^2*x^2*e^(3*c) + 2*a*
d^2*f*x*e^(3*c + 1) + a*d^2*e^(3*c + 2))*e^(3*d*x) - 16*(-I*a*d^2*f^2*x^2*e^(2*c) - 2*I*a*d^2*f*x*e^(2*c + 1)
- I*a*d^2*e^(2*c + 2))*e^(2*d*x) + 8*(a*d^2*f^2*x^2*e^c + 2*a*d^2*f*x*e^(c + 1) + a*d^2*e^(c + 2))*e^(d*x)) -
8*integrate(1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 + 2*I*d*f*e - 2*f^2 + 2*(3*d^2*f*e + I*d*f^2)*x)/(a*d^2*f^3*x^3 +
3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3 + (a*d^2*f^3*x^3*e^c + 3*a*d^2*f^2*x^2*e^(c + 1) + 3*a*d^2*f*x
*e^(c + 2) + a*d^2*e^(c + 3))*e^(d*x)), x) - 8*integrate(-1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 2*I*d*f*e - 2*f^2
+ 2*(3*d^2*f*e - I*d*f^2)*x)/(a*d^2*f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3 - (a*d^2*f^3*x^3
*e^c + 3*a*d^2*f^2*x^2*e^(c + 1) + 3*a*d^2*f*x*e^(c + 2) + a*d^2*e^(c + 3))*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(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*f*x*e
- I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*f
*x*e - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2)*e^(3*d*x + 3*c) - 2*(-I*a*
d^2*f^2*x^2 - 2*I*a*d^2*f*x*e - I*a*d^2*e^2)*e^(2*d*x + 2*c) + (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2)*e^(
d*x + c))*integral((4*d*f^2*x + 4*d*f*e - (3*d^2*f^2*x^2 + 2*d*f^2*x + 3*d^2*e^2 - 2*f^2 + 2*(3*d^2*f*x + d*f)
*e)*e^(2*d*x + 2*c) + (3*I*d^2*f^2*x^2 + 2*I*d*f^2*x + 3*I*d^2*e^2 - 2*I*f^2 - 2*(-3*I*d^2*f*x - I*d*f)*e)*e^(
d*x + c))/(I*a*d^2*f^3*x^3 + 3*I*a*d^2*f^2*x^2*e + 3*I*a*d^2*f*x*e^2 + I*a*d^2*e^3 + (a*d^2*f^3*x^3 + 3*a*d^2*
f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3)*e^(3*d*x + 3*c) + (-I*a*d^2*f^3*x^3 - 3*I*a*d^2*f^2*x^2*e - 3*I*a*d^2
*f*x*e^2 - I*a*d^2*e^3)*e^(2*d*x + 2*c) - (a*d^2*f^3*x^3 + 3*a*d^2*f^2*x^2*e + 3*a*d^2*f*x*e^2 + a*d^2*e^3)*e^
(d*x + c)), x))/(-I*a*d^2*f^2*x^2 - 2*I*a*d^2*f*x*e - I*a*d^2*e^2 + (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2
)*e^(5*d*x + 5*c) + (-I*a*d^2*f^2*x^2 - 2*I*a*d^2*f*x*e - I*a*d^2*e^2)*e^(4*d*x + 4*c) - 2*(a*d^2*f^2*x^2 + 2*
a*d^2*f*x*e + a*d^2*e^2)*e^(3*d*x + 3*c) - 2*(-I*a*d^2*f^2*x^2 - 2*I*a*d^2*f*x*e - I*a*d^2*e^2)*e^(2*d*x + 2*c
) + (a*d^2*f^2*x^2 + 2*a*d^2*f*x*e + a*d^2*e^2)*e^(d*x + c))
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {i \int \frac {\operatorname {csch}^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \end {gather*}
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]
-I*Integral(csch(c + d*x)**3/(e*sinh(c + d*x) - I*e + f*x*sinh(c + d*x) - I*f*x), x)/a
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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(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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \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 \end {gather*}
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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