\(\int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [288]

Optimal result

Integrand size = 31, antiderivative size = 31 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \] Output:

Defer(Int)(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
 

Mathematica [F(-1)]

Timed out. \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {\$Aborted} \] Input:

Integrate[Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]
 

Output:

$Aborted
 

Rubi [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\)

\(\Big \downarrow \) 6111

\(\displaystyle \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))}dx\)

Input:

Int[Sech[c + d*x]^3/((e + f*x)^2*(a + I*a*Sinh[c + d*x])),x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

\[\int \frac {\operatorname {sech}\left (d x +c \right )^{3}}{\left (f x +e \right )^{2} \left (a +i a \sinh \left (d x +c \right )\right )}d x\]

Input:

int(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
 

Output:

int(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
 

Fricas [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 2026, normalized size of antiderivative = 65.35 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \] Input:

integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas 
")
 

Output:

-1/12*(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) - 6*(3*I*d^3*f^3*x^3 + 3*I*d^3*e^3 - 6*I*d^2*e^2*f + 4*I*f^3 + 3*(3* 
I*d^3*e*f^2 - 2*I*d^2*f^3)*x^2 + 3*(3*I*d^3*e^2*f - 4*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) - 2*(-9*I*d^3*f^3*x^3 - 9*I*d^3*e^3 - 22*I*d^2*e^2*f 
 + 24*I*f^3 + (-27*I*d^3*e*f^2 - 22*I*d^2*f^3)*x^2 + (-27*I*d^3*e^2*f - 44 
*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 + 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 - (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) + 2*(I*a*d^4 
*f^5*x^5 + 5*I*a*d^4*e*f^4*x^4 + 10*I*a*d^4*e^2*f^3*x^3 + 10*I*a*d^4*e^3*f 
^2*x^2 + 5*I*a*d^4*e^4*f*x + I*a*d^4*e^5)*e^(5*d*x + 5*c) - (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) + 4*(I*a*d^4*f^5*x^5 + 5*I*a*d^4*e 
*f^4*x^4 + 10*I*a*d^4*e^2*f^3*x^3 + 10*I*a*d^4*e^3*f^2*x^2 + 5*I*a*d^4*...
 

Sympy [N/A]

Not integrable

Time = 40.80 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{e^{2} \sinh {\left (c + d x \right )} - i e^{2} + 2 e f x \sinh {\left (c + d x \right )} - 2 i e f x + f^{2} x^{2} \sinh {\left (c + d x \right )} - i f^{2} x^{2}}\, dx}{a} \] Input:

integrate(sech(d*x+c)**3/(f*x+e)**2/(a+I*a*sinh(d*x+c)),x)
 

Output:

-I*Integral(sech(c + d*x)**3/(e**2*sinh(c + d*x) - I*e**2 + 2*e*f*x*sinh(c 
 + d*x) - 2*I*e*f*x + f**2*x**2*sinh(c + d*x) - I*f**2*x**2), x)/a
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima 
")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F(-1)]

Timed out. \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Timed out} \] Input:

integrate(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [N/A]

Not integrable

Time = 2.13 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:

int(1/(cosh(c + d*x)^3*(e + f*x)^2*(a + a*sinh(c + d*x)*1i)),x)
 

Output:

int(1/(cosh(c + d*x)^3*(e + f*x)^2*(a + a*sinh(c + d*x)*1i)), x)
 

Reduce [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.26 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\frac {\int \frac {\mathrm {sech}\left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) e^{2} i +2 \sinh \left (d x +c \right ) e f i x +\sinh \left (d x +c \right ) f^{2} i \,x^{2}+e^{2}+2 e f x +f^{2} x^{2}}d x}{a} \] Input:

int(sech(d*x+c)^3/(f*x+e)^2/(a+I*a*sinh(d*x+c)),x)
 

Output:

int(sech(c + d*x)**3/(sinh(c + d*x)*e**2*i + 2*sinh(c + d*x)*e*f*i*x + sin 
h(c + d*x)*f**2*i*x**2 + e**2 + 2*e*f*x + f**2*x**2),x)/a