3.288 \(\int \frac{\text{sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)
Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{\text{sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable[Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]
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Rubi [A] time = 0.0811407, antiderivative size = 0, normalized size of antiderivative = 0.,
number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {}
\[ \int \frac{\text{sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]
Rubi steps
\begin{align*} \int \frac{\text{sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=\int \frac{\text{sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\\ \end{align*}
Mathematica [F] time = 180.034, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
Integrate[Sech[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 = 2.361, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{3}}{ \left ( fx+e \right ) ^{2} \left ( a+ia\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
[Out]
int(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-8*(8*I*d^2*f^3*x^2 + 16*I*d^2*e*f^2*x + 8*I*d^2*e^2*f - 24*I*f^3 + 3*(3*d^3*f^3*x^3*e^(5*c) + 3*(3*d^3*e*f^2
- 2*d^2*f^3)*x^2*e^(5*c) + (9*d^3*e^2*f - 12*d^2*e*f^2 - 2*d*f^3)*x*e^(5*c) + (3*d^3*e^3 - 6*d^2*e^2*f - 2*d*e
*f^2 + 8*f^3)*e^(5*c))*e^(5*d*x) + (-18*I*d^3*f^3*x^3*e^(4*c) + (-54*I*d^3*e*f^2 + 36*I*d^2*f^3)*x^2*e^(4*c) +
(-54*I*d^3*e^2*f + 72*I*d^2*e*f^2)*x*e^(4*c) + (-18*I*d^3*e^3 + 36*I*d^2*e^2*f - 24*I*f^3)*e^(4*c))*e^(4*d*x)
+ 2*(3*d^3*f^3*x^3*e^(3*c) + (9*d^3*e*f^2 - 8*d^2*f^3)*x^2*e^(3*c) + (9*d^3*e^2*f - 16*d^2*e*f^2 - 6*d*f^3)*x
*e^(3*c) + (3*d^3*e^3 - 8*d^2*e^2*f - 6*d*e*f^2 + 24*f^3)*e^(3*c))*e^(3*d*x) + (18*I*d^3*f^3*x^3*e^(2*c) + (54
*I*d^3*e*f^2 + 44*I*d^2*f^3)*x^2*e^(2*c) + (54*I*d^3*e^2*f + 88*I*d^2*e*f^2)*x*e^(2*c) + (18*I*d^3*e^3 + 44*I*
d^2*e^2*f - 48*I*f^3)*e^(2*c))*e^(2*d*x) + (9*d^3*f^3*x^3*e^c + (27*d^3*e*f^2 + 2*d^2*f^3)*x^2*e^c + (27*d^3*e
^2*f + 4*d^2*e*f^2 - 6*d*f^3)*x*e^c + (9*d^3*e^3 + 2*d^2*e^2*f - 6*d*e*f^2 + 24*f^3)*e^c)*e^(d*x))/(96*a*d^4*f
^5*x^5 + 480*a*d^4*e*f^4*x^4 + 960*a*d^4*e^2*f^3*x^3 + 960*a*d^4*e^3*f^2*x^2 + 480*a*d^4*e^4*f*x + 96*a*d^4*e^
5 - 96*(a*d^4*f^5*x^5*e^(6*c) + 5*a*d^4*e*f^4*x^4*e^(6*c) + 10*a*d^4*e^2*f^3*x^3*e^(6*c) + 10*a*d^4*e^3*f^2*x^
2*e^(6*c) + 5*a*d^4*e^4*f*x*e^(6*c) + a*d^4*e^5*e^(6*c))*e^(6*d*x) - (-192*I*a*d^4*f^5*x^5*e^(5*c) - 960*I*a*d
^4*e*f^4*x^4*e^(5*c) - 1920*I*a*d^4*e^2*f^3*x^3*e^(5*c) - 1920*I*a*d^4*e^3*f^2*x^2*e^(5*c) - 960*I*a*d^4*e^4*f
*x*e^(5*c) - 192*I*a*d^4*e^5*e^(5*c))*e^(5*d*x) - 96*(a*d^4*f^5*x^5*e^(4*c) + 5*a*d^4*e*f^4*x^4*e^(4*c) + 10*a
*d^4*e^2*f^3*x^3*e^(4*c) + 10*a*d^4*e^3*f^2*x^2*e^(4*c) + 5*a*d^4*e^4*f*x*e^(4*c) + a*d^4*e^5*e^(4*c))*e^(4*d*
x) - (-384*I*a*d^4*f^5*x^5*e^(3*c) - 1920*I*a*d^4*e*f^4*x^4*e^(3*c) - 3840*I*a*d^4*e^2*f^3*x^3*e^(3*c) - 3840*
I*a*d^4*e^3*f^2*x^2*e^(3*c) - 1920*I*a*d^4*e^4*f*x*e^(3*c) - 384*I*a*d^4*e^5*e^(3*c))*e^(3*d*x) + 96*(a*d^4*f^
5*x^5*e^(2*c) + 5*a*d^4*e*f^4*x^4*e^(2*c) + 10*a*d^4*e^2*f^3*x^3*e^(2*c) + 10*a*d^4*e^3*f^2*x^2*e^(2*c) + 5*a*
d^4*e^4*f*x*e^(2*c) + a*d^4*e^5*e^(2*c))*e^(2*d*x) - (-192*I*a*d^4*f^5*x^5*e^c - 960*I*a*d^4*e*f^4*x^4*e^c - 1
920*I*a*d^4*e^2*f^3*x^3*e^c - 1920*I*a*d^4*e^3*f^2*x^2*e^c - 960*I*a*d^4*e^4*f*x*e^c - 192*I*a*d^4*e^5*e^c)*e^
(d*x)) + 8*integrate((3*d^4*f^4*x^4 + 12*d^4*e*f^3*x^3 + 3*d^4*e^4 - 28*d^2*e^2*f^2 + 80*f^4 + 2*(9*d^4*e^2*f^
2 - 14*d^2*f^4)*x^2 + 4*(3*d^4*e^3*f - 14*d^2*e*f^3)*x)/(-64*I*a*d^4*f^6*x^6 - 384*I*a*d^4*e*f^5*x^5 - 960*I*a
*d^4*e^2*f^4*x^4 - 1280*I*a*d^4*e^3*f^3*x^3 - 960*I*a*d^4*e^4*f^2*x^2 - 384*I*a*d^4*e^5*f*x - 64*I*a*d^4*e^6 +
64*(a*d^4*f^6*x^6*e^c + 6*a*d^4*e*f^5*x^5*e^c + 15*a*d^4*e^2*f^4*x^4*e^c + 20*a*d^4*e^3*f^3*x^3*e^c + 15*a*d^
4*e^4*f^2*x^2*e^c + 6*a*d^4*e^5*f*x*e^c + a*d^4*e^6*e^c)*e^(d*x)), x) + 8*integrate(3*(d^2*f^2*x^2 + 2*d^2*e*f
*x + d^2*e^2 - 4*f^2)/(64*I*a*d^2*f^4*x^4 + 256*I*a*d^2*e*f^3*x^3 + 384*I*a*d^2*e^2*f^2*x^2 + 256*I*a*d^2*e^3*
f*x + 64*I*a*d^2*e^4 + 64*(a*d^2*f^4*x^4*e^c + 4*a*d^2*e*f^3*x^3*e^c + 6*a*d^2*e^2*f^2*x^2*e^c + 4*a*d^2*e^3*f
*x*e^c + a*d^2*e^4*e^c)*e^(d*x)), x)
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-(8*I*d^2*f^3*x^2 + 16*I*d^2*e*f^2*x + 8*I*d^2*e^2*f - 24*I*f^3 + 3*(3*d^3*f^3*x^3 + 3*d^3*e^3 - 6*d^2*e^2*f -
2*d*e*f^2 + 8*f^3 + 3*(3*d^3*e*f^2 - 2*d^2*f^3)*x^2 + (9*d^3*e^2*f - 12*d^2*e*f^2 - 2*d*f^3)*x)*e^(5*d*x + 5*
c) + (-18*I*d^3*f^3*x^3 - 18*I*d^3*e^3 + 36*I*d^2*e^2*f - 24*I*f^3 + (-54*I*d^3*e*f^2 + 36*I*d^2*f^3)*x^2 + (-
54*I*d^3*e^2*f + 72*I*d^2*e*f^2)*x)*e^(4*d*x + 4*c) + 2*(3*d^3*f^3*x^3 + 3*d^3*e^3 - 8*d^2*e^2*f - 6*d*e*f^2 +
24*f^3 + (9*d^3*e*f^2 - 8*d^2*f^3)*x^2 + (9*d^3*e^2*f - 16*d^2*e*f^2 - 6*d*f^3)*x)*e^(3*d*x + 3*c) + (18*I*d^
3*f^3*x^3 + 18*I*d^3*e^3 + 44*I*d^2*e^2*f - 48*I*f^3 + (54*I*d^3*e*f^2 + 44*I*d^2*f^3)*x^2 + (54*I*d^3*e^2*f +
88*I*d^2*e*f^2)*x)*e^(2*d*x + 2*c) + (9*d^3*f^3*x^3 + 9*d^3*e^3 + 2*d^2*e^2*f - 6*d*e*f^2 + 24*f^3 + (27*d^3*
e*f^2 + 2*d^2*f^3)*x^2 + (27*d^3*e^2*f + 4*d^2*e*f^2 - 6*d*f^3)*x)*e^(d*x + c) - (12*a*d^4*f^5*x^5 + 60*a*d^4*
e*f^4*x^4 + 120*a*d^4*e^2*f^3*x^3 + 120*a*d^4*e^3*f^2*x^2 + 60*a*d^4*e^4*f*x + 12*a*d^4*e^5 - 12*(a*d^4*f^5*x^
5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*x + a*d^4*e^5)*e^(6*d*x +
6*c) - (-24*I*a*d^4*f^5*x^5 - 120*I*a*d^4*e*f^4*x^4 - 240*I*a*d^4*e^2*f^3*x^3 - 240*I*a*d^4*e^3*f^2*x^2 - 120*
I*a*d^4*e^4*f*x - 24*I*a*d^4*e^5)*e^(5*d*x + 5*c) - 12*(a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x
^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*x + a*d^4*e^5)*e^(4*d*x + 4*c) - (-48*I*a*d^4*f^5*x^5 - 240*I*a*d^4*
e*f^4*x^4 - 480*I*a*d^4*e^2*f^3*x^3 - 480*I*a*d^4*e^3*f^2*x^2 - 240*I*a*d^4*e^4*f*x - 48*I*a*d^4*e^5)*e^(3*d*x
+ 3*c) + 12*(a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*
x + a*d^4*e^5)*e^(2*d*x + 2*c) - (-24*I*a*d^4*f^5*x^5 - 120*I*a*d^4*e*f^4*x^4 - 240*I*a*d^4*e^2*f^3*x^3 - 240*
I*a*d^4*e^3*f^2*x^2 - 120*I*a*d^4*e^4*f*x - 24*I*a*d^4*e^5)*e^(d*x + c))*integral(1/4*(-8*I*d^2*f^4*x^2 - 16*I
*d^2*e*f^3*x - 8*I*d^2*e^2*f^2 + 40*I*f^4 + (3*d^4*f^4*x^4 + 12*d^4*e*f^3*x^3 + 3*d^4*e^4 - 20*d^2*e^2*f^2 + 4
0*f^4 + 2*(9*d^4*e^2*f^2 - 10*d^2*f^4)*x^2 + 4*(3*d^4*e^3*f - 10*d^2*e*f^3)*x)*e^(d*x + c))/(a*d^4*f^6*x^6 + 6
*a*d^4*e*f^5*x^5 + 15*a*d^4*e^2*f^4*x^4 + 20*a*d^4*e^3*f^3*x^3 + 15*a*d^4*e^4*f^2*x^2 + 6*a*d^4*e^5*f*x + a*d^
4*e^6 + (a*d^4*f^6*x^6 + 6*a*d^4*e*f^5*x^5 + 15*a*d^4*e^2*f^4*x^4 + 20*a*d^4*e^3*f^3*x^3 + 15*a*d^4*e^4*f^2*x^
2 + 6*a*d^4*e^5*f*x + a*d^4*e^6)*e^(2*d*x + 2*c)), x))/(12*a*d^4*f^5*x^5 + 60*a*d^4*e*f^4*x^4 + 120*a*d^4*e^2*
f^3*x^3 + 120*a*d^4*e^3*f^2*x^2 + 60*a*d^4*e^4*f*x + 12*a*d^4*e^5 - 12*(a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10
*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*x + a*d^4*e^5)*e^(6*d*x + 6*c) - (-24*I*a*d^4*f^5*x^
5 - 120*I*a*d^4*e*f^4*x^4 - 240*I*a*d^4*e^2*f^3*x^3 - 240*I*a*d^4*e^3*f^2*x^2 - 120*I*a*d^4*e^4*f*x - 24*I*a*d
^4*e^5)*e^(5*d*x + 5*c) - 12*(a*d^4*f^5*x^5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2
+ 5*a*d^4*e^4*f*x + a*d^4*e^5)*e^(4*d*x + 4*c) - (-48*I*a*d^4*f^5*x^5 - 240*I*a*d^4*e*f^4*x^4 - 480*I*a*d^4*e^
2*f^3*x^3 - 480*I*a*d^4*e^3*f^2*x^2 - 240*I*a*d^4*e^4*f*x - 48*I*a*d^4*e^5)*e^(3*d*x + 3*c) + 12*(a*d^4*f^5*x^
5 + 5*a*d^4*e*f^4*x^4 + 10*a*d^4*e^2*f^3*x^3 + 10*a*d^4*e^3*f^2*x^2 + 5*a*d^4*e^4*f*x + a*d^4*e^5)*e^(2*d*x +
2*c) - (-24*I*a*d^4*f^5*x^5 - 120*I*a*d^4*e*f^4*x^4 - 240*I*a*d^4*e^2*f^3*x^3 - 240*I*a*d^4*e^3*f^2*x^2 - 120*
I*a*d^4*e^4*f*x - 24*I*a*d^4*e^5)*e^(d*x + c))
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(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)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out