\(\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [287]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 51.85 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.78 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 1783, normalized size of antiderivative = 57.52 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-1/12*(4*I*d^2*f^3*x^2 + 8*I*d^2*e*f^2*x + 4*I*d^2*e^2*f - 6*I*f^3 + (9*d^3*f^3*x^3 + 9*d^3*e^3 - 9*d^2*e^2*f
- 2*d*e*f^2 + 6*f^3 + 9*(3*d^3*e*f^2 - d^2*f^3)*x^2 + (27*d^3*e^2*f - 18*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 - 3*I*d^2*e^2*f + I*f^3 + 3*(3*I*d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(3*I*d^3
*e^2*f - 2*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 - 4*d^2*e^2*f - 2*d*e*f^2 + 6*f^3 +
(9*d^3*e*f^2 - 4*d^2*f^3)*x^2 + (9*d^3*e^2*f - 8*d^2*e*f^2 - 2*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 - 11*I*d^2*e^2*f + 6*I*f^3 + (-27*I*d^3*e*f^2 - 11*I*d^2*f^3)*x^2 + (-27*I*d^3*e^2*f - 22*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 + d^2*e^2*f - 2*d*e*f^2 + 6*f^3 + (27*d^3*e*f^2 + d^2*
f^3)*x^2 + (27*d^3*e^2*f + 2*d^2*e*f^2 - 2*d*f^3)*x)*e^(d*x + c) - 12*(a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a
*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4 - (a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*
a*d^4*e^3*f*x + a*d^4*e^4)*e^(6*d*x + 6*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*e*f^3*x^3 + 6*I*a*d^4*e^2*f^2*x^2
+ 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(5*d*x + 5*c) - (a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2
+ 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(4*d*x + 4*c) + 4*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*e*f^3*x^3 + 6*I*a*d^4*e^2*f^2*
x^2 + 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(3*d*x + 3*c) + (a*d^4*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*
x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(2*d*x + 2*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*e*f^3*x^3 + 6*I*a*d^4*e^2*
f^2*x^2 + 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(d*x + c))*integral(1/12*(-8*I*d^2*f^4*x^2 - 16*I*d^2*e*f^3*x - 8
*I*d^2*e^2*f^2 + 24*I*f^4 + (9*d^4*f^4*x^4 + 36*d^4*e*f^3*x^3 + 9*d^4*e^4 - 20*d^2*e^2*f^2 + 24*f^4 + 2*(27*d^
4*e^2*f^2 - 10*d^2*f^4)*x^2 + 4*(9*d^4*e^3*f - 10*d^2*e*f^3)*x)*e^(d*x + 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 + (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)), x))/(a*d^4
*f^4*x^4 + 4*a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4 - (a*d^4*f^4*x^4 + 4*a*d^4*e*
f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(6*d*x + 6*c) + 2*(I*a*d^4*f^4*x^4 + 4*I*a*d^4*
e*f^3*x^3 + 6*I*a*d^4*e^2*f^2*x^2 + 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(5*d*x + 5*c) - (a*d^4*f^4*x^4 + 4*a*d^
4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(4*d*x + 4*c) + 4*(I*a*d^4*f^4*x^4 + 4*I*a*
d^4*e*f^3*x^3 + 6*I*a*d^4*e^2*f^2*x^2 + 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(3*d*x + 3*c) + (a*d^4*f^4*x^4 + 4*
a*d^4*e*f^3*x^3 + 6*a*d^4*e^2*f^2*x^2 + 4*a*d^4*e^3*f*x + a*d^4*e^4)*e^(2*d*x + 2*c) + 2*(I*a*d^4*f^4*x^4 + 4*
I*a*d^4*e*f^3*x^3 + 6*I*a*d^4*e^2*f^2*x^2 + 4*I*a*d^4*e^3*f*x + I*a*d^4*e^4)*e^(d*x + c))

Sympy [N/A]

Not integrable

Time = 8.77 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 150.91 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {\text {sech}^3(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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