3.282 \(\int \frac{\text{sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)
Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{\text{sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]
[Out]
Unintegrable[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]
________________________________________________________________________________________
Rubi [A] time = 0.0767881, 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 = {}
\[ \int \frac{\text{sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
[In]
Int[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]
[Out]
Defer[Int][Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])), x]
Rubi steps
\begin{align*} \int \frac{\text{sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx &=\int \frac{\text{sech}^2(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\\ \end{align*}
Mathematica [F] time = 180.02, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
[In]
Integrate[Sech[c + d*x]^2/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]
[Out]
$Aborted
________________________________________________________________________________________
Maple [A] time = 1.753, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{ \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)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
[Out]
int(sech(d*x+c)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
________________________________________________________________________________________
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)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima")
[Out]
-4*I*f*integrate(1/(4*I*a*d*f^3*x^3 + 12*I*a*d*e*f^2*x^2 + 12*I*a*d*e^2*f*x + 4*I*a*d*e^3 + 4*(a*d*f^3*x^3*e^c
+ 3*a*d*e*f^2*x^2*e^c + 3*a*d*e^2*f*x*e^c + a*d*e^3*e^c)*e^(d*x)), x) - 4*(2*d^2*f^2*x^2 + 4*d^2*e*f*x + 2*d^
2*e^2 - 3*f^2*e^(2*d*x + 2*c) - 3*f^2 + (I*d*f^2*x*e^(3*c) + (I*d*e*f - 3*I*f^2)*e^(3*c))*e^(3*d*x) + (4*I*d^2
*f^2*x^2*e^c + (8*I*d^2*e*f + I*d*f^2)*x*e^c + (4*I*d^2*e^2 + I*d*e*f - 3*I*f^2)*e^c)*e^(d*x))/(6*a*d^3*f^4*x^
4 + 24*a*d^3*e*f^3*x^3 + 36*a*d^3*e^2*f^2*x^2 + 24*a*d^3*e^3*f*x + 6*a*d^3*e^4 - 6*(a*d^3*f^4*x^4*e^(4*c) + 4*
a*d^3*e*f^3*x^3*e^(4*c) + 6*a*d^3*e^2*f^2*x^2*e^(4*c) + 4*a*d^3*e^3*f*x*e^(4*c) + a*d^3*e^4*e^(4*c))*e^(4*d*x)
- (-12*I*a*d^3*f^4*x^4*e^(3*c) - 48*I*a*d^3*e*f^3*x^3*e^(3*c) - 72*I*a*d^3*e^2*f^2*x^2*e^(3*c) - 48*I*a*d^3*e
^3*f*x*e^(3*c) - 12*I*a*d^3*e^4*e^(3*c))*e^(3*d*x) - (-12*I*a*d^3*f^4*x^4*e^c - 48*I*a*d^3*e*f^3*x^3*e^c - 72*
I*a*d^3*e^2*f^2*x^2*e^c - 48*I*a*d^3*e^3*f*x*e^c - 12*I*a*d^3*e^4*e^c)*e^(d*x)) - 4*integrate((5*d^2*f^3*x^2 +
10*d^2*e*f^2*x + 5*d^2*e^2*f - 24*f^3)/(12*a*d^3*f^5*x^5 + 60*a*d^3*e*f^4*x^4 + 120*a*d^3*e^2*f^3*x^3 + 120*a
*d^3*e^3*f^2*x^2 + 60*a*d^3*e^4*f*x + 12*a*d^3*e^5 + (12*I*a*d^3*f^5*x^5*e^c + 60*I*a*d^3*e*f^4*x^4*e^c + 120*
I*a*d^3*e^2*f^3*x^3*e^c + 120*I*a*d^3*e^3*f^2*x^2*e^c + 60*I*a*d^3*e^4*f*x*e^c + 12*I*a*d^3*e^5*e^c)*e^(d*x)),
x)
________________________________________________________________________________________
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)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas")
[Out]
-(4*d^2*f^2*x^2 + 8*d^2*e*f*x + 4*d^2*e^2 - 6*f^2*e^(2*d*x + 2*c) - 6*f^2 + (2*I*d*f^2*x + 2*I*d*e*f - 6*I*f^2
)*e^(3*d*x + 3*c) + (8*I*d^2*f^2*x^2 + 8*I*d^2*e^2 + 2*I*d*e*f - 6*I*f^2 + (16*I*d^2*e*f + 2*I*d*f^2)*x)*e^(d*
x + c) - (3*a*d^3*f^4*x^4 + 12*a*d^3*e*f^3*x^3 + 18*a*d^3*e^2*f^2*x^2 + 12*a*d^3*e^3*f*x + 3*a*d^3*e^4 - 3*(a*
d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*e^(4*d*x + 4*c) - (-6*I*a
*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(3*d*x +
3*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^
4)*e^(d*x + c))*integral(-1/3*(8*d^2*f^3*x^2 + 16*d^2*e*f^2*x + 8*d^2*e^2*f - 24*f^3 - (2*I*d^2*f^3*x^2 + 4*I*
d^2*e*f^2*x + 2*I*d^2*e^2*f - 24*I*f^3)*e^(d*x + c))/(a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*x^3
+ 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5 + (a*d^3*f^5*x^5 + 5*a*d^3*e*f^4*x^4 + 10*a*d^3*e^2*f^3*
x^3 + 10*a*d^3*e^3*f^2*x^2 + 5*a*d^3*e^4*f*x + a*d^3*e^5)*e^(2*d*x + 2*c)), x))/(3*a*d^3*f^4*x^4 + 12*a*d^3*e*
f^3*x^3 + 18*a*d^3*e^2*f^2*x^2 + 12*a*d^3*e^3*f*x + 3*a*d^3*e^4 - 3*(a*d^3*f^4*x^4 + 4*a*d^3*e*f^3*x^3 + 6*a*d
^3*e^2*f^2*x^2 + 4*a*d^3*e^3*f*x + a*d^3*e^4)*e^(4*d*x + 4*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3*e*f^3*x^3 - 3
6*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(3*d*x + 3*c) - (-6*I*a*d^3*f^4*x^4 - 24*I*a*d^3
*e*f^3*x^3 - 36*I*a*d^3*e^2*f^2*x^2 - 24*I*a*d^3*e^3*f*x - 6*I*a*d^3*e^4)*e^(d*x + c))
________________________________________________________________________________________
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)**2/(f*x+e)**2/(a+I*a*sinh(d*x+c)),x)
[Out]
Timed out
________________________________________________________________________________________
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)^2/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
[Out]
Timed out