3.3.22 \(\int \frac {\text {csch}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [222]
Optimal. Leaf size=34 \[ \text {Int}\left (\frac {\text {csch}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable(csch(d*x+c)^3/(f*x+e)^2/(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)^2 (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)^2*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Csch[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]
Rubi steps
\begin {align*} \int \frac {\text {csch}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=\int \frac {\text {csch}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\\ \end {align*}
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Mathematica [F]
time = 180.03, 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)^2*(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 \[\int \frac {\mathrm {csch}\left (d x +c \right )^{3}}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
[Out]
int(csch(d*x+c)^3/(f*x+e)^2/(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)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-8*f*integrate(1/(-2*I*a*d*f^3*x^3 - 6*I*a*d*f^2*x^2*e - 6*I*a*d*f*x*e^2 - 2*I*a*d*e^3 + 2*(a*d*f^3*x^3*e^c +
3*a*d*f^2*x^2*e^(c + 1) + 3*a*d*f*x*e^(c + 2) + a*d*e^(c + 3))*e^(d*x)), x) - 8*(4*d*f*x + 4*d*e + (3*d*f*x*e^
(4*c) - 2*f*e^(4*c) + 3*d*e^(4*c + 1))*e^(4*d*x) + (-3*I*d*f*x*e^(3*c) + 2*I*f*e^(3*c) - 3*I*d*e^(3*c + 1))*e^
(3*d*x) - (5*d*f*x*e^(2*c) - 2*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) - 2*I*f*e
^c)*e^(d*x))/(-8*I*a*d^2*f^3*x^3 - 24*I*a*d^2*f^2*x^2*e - 24*I*a*d^2*f*x*e^2 - 8*I*a*d^2*e^3 + 8*(a*d^2*f^3*x^
3*e^(5*c) + 3*a*d^2*f^2*x^2*e^(5*c + 1) + 3*a*d^2*f*x*e^(5*c + 2) + a*d^2*e^(5*c + 3))*e^(5*d*x) - 8*(I*a*d^2*
f^3*x^3*e^(4*c) + 3*I*a*d^2*f^2*x^2*e^(4*c + 1) + 3*I*a*d^2*f*x*e^(4*c + 2) + I*a*d^2*e^(4*c + 3))*e^(4*d*x) -
16*(a*d^2*f^3*x^3*e^(3*c) + 3*a*d^2*f^2*x^2*e^(3*c + 1) + 3*a*d^2*f*x*e^(3*c + 2) + a*d^2*e^(3*c + 3))*e^(3*d
*x) - 16*(-I*a*d^2*f^3*x^3*e^(2*c) - 3*I*a*d^2*f^2*x^2*e^(2*c + 1) - 3*I*a*d^2*f*x*e^(2*c + 2) - I*a*d^2*e^(2*
c + 3))*e^(2*d*x) + 8*(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)) - 8*integrate(1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 + 4*I*d*f*e - 6*f^2 + 2*(3*d^2*f*e + 2*I*d*f^2)*x)/(a
*d^2*f^4*x^4 + 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f*x*e^3 + a*d^2*e^4 + (a*d^2*f^4*x^4*e^c + 4*
a*d^2*f^3*x^3*e^(c + 1) + 6*a*d^2*f^2*x^2*e^(c + 2) + 4*a*d^2*f*x*e^(c + 3) + a*d^2*e^(c + 4))*e^(d*x)), x) -
8*integrate(-1/16*(3*d^2*f^2*x^2 + 3*d^2*e^2 - 4*I*d*f*e - 6*f^2 + 2*(3*d^2*f*e - 2*I*d*f^2)*x)/(a*d^2*f^4*x^4
+ 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f*x*e^3 + a*d^2*e^4 - (a*d^2*f^4*x^4*e^c + 4*a*d^2*f^3*x^
3*e^(c + 1) + 6*a*d^2*f^2*x^2*e^(c + 2) + 4*a*d^2*f*x*e^(c + 3) + a*d^2*e^(c + 4))*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)^2/(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 - 2*f)*e^(4*d*x + 4*c) - (3*I*d*f*x + 3*I*d*e - 2*I*f)*e^(3*d*x + 3*c) -
(5*d*f*x + 5*d*e - 2*f)*e^(2*d*x + 2*c) - (-I*d*f*x - I*d*e + 2*I*f)*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^(5*d*x + 5*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^(4*d*x + 4
*c) - 2*(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) - 2*(-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))*integral((8*d*f^2*x + 8*d*f*e - (3*d^2*f^2*x^2 + 4*d*f^2*x + 3*d
^2*e^2 - 6*f^2 + 2*(3*d^2*f*x + 2*d*f)*e)*e^(2*d*x + 2*c) + (3*I*d^2*f^2*x^2 + 4*I*d*f^2*x + 3*I*d^2*e^2 - 6*I
*f^2 - 2*(-3*I*d^2*f*x - 2*I*d*f)*e)*e^(d*x + c))/(I*a*d^2*f^4*x^4 + 4*I*a*d^2*f^3*x^3*e + 6*I*a*d^2*f^2*x^2*e
^2 + 4*I*a*d^2*f*x*e^3 + I*a*d^2*e^4 + (a*d^2*f^4*x^4 + 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f*x*
e^3 + a*d^2*e^4)*e^(3*d*x + 3*c) + (-I*a*d^2*f^4*x^4 - 4*I*a*d^2*f^3*x^3*e - 6*I*a*d^2*f^2*x^2*e^2 - 4*I*a*d^2
*f*x*e^3 - I*a*d^2*e^4)*e^(2*d*x + 2*c) - (a*d^2*f^4*x^4 + 4*a*d^2*f^3*x^3*e + 6*a*d^2*f^2*x^2*e^2 + 4*a*d^2*f
*x*e^3 + a*d^2*e^4)*e^(d*x + c)), x))/(-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^(5*d*x + 5*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^(4*d*x + 4*c) - 2*(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) - 2*(-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))
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Sympy [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)**2/(a+I*a*sinh(d*x+c)),x)
[Out]
Timed out
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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)^2/(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 )}^2\,\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)^2*(a + a*sinh(c + d*x)*1i)),x)
[Out]
int(1/(sinh(c + d*x)^3*(e + f*x)^2*(a + a*sinh(c + d*x)*1i)), x)
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