\(\int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx\) [276]

Optimal result

Integrand size = 29, antiderivative size = 29 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\text {Int}\left (\frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 29, 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 = {} \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx \\ \end{align*}

Mathematica [F(-1)]

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

[In]

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

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 1.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 511, normalized size of antiderivative = 17.62 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-((d*f*x + d*e - 2*f)*e^(d*x + c) - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - (a*d^2*
f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^(2*d*x + 2*c) + 2*(I*a*d^2*f^3*x^3 + 3*I*a*d^2*e*
f^2*x^2 + 3*I*a*d^2*e^2*f*x + I*a*d^2*e^3)*e^(d*x + c))*integral((-6*I*f^2 + (d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*
e^2 - 6*f^2)*e^(d*x + c))/(a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e
^4 + (a*d^2*f^4*x^4 + 4*a*d^2*e*f^3*x^3 + 6*a*d^2*e^2*f^2*x^2 + 4*a*d^2*e^3*f*x + a*d^2*e^4)*e^(2*d*x + 2*c)),
x) + 2*I*f)/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3 - (a*d^2*f^3*x^3 + 3*a*d^2*e*f^2
*x^2 + 3*a*d^2*e^2*f*x + a*d^2*e^3)*e^(2*d*x + 2*c) + 2*(I*a*d^2*f^3*x^3 + 3*I*a*d^2*e*f^2*x^2 + 3*I*a*d^2*e^2
*f*x + I*a*d^2*e^3)*e^(d*x + c))

Sympy [N/A]

Not integrable

Time = 30.13 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=- \frac {i \int \frac {\operatorname {sech}{\left (c + d x \right )}}{e^{2} \sinh {\left (c + d x \right )} - i e^{2} + 2 e f x \sinh {\left (c + d x \right )} - 2 i e f x + f^{2} x^{2} \sinh {\left (c + d x \right )} - i f^{2} x^{2}}\, dx}{a} \]

[In]

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

[Out]

-I*Integral(sech(c + d*x)/(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 [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 436, normalized size of antiderivative = 15.03 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

-((d*f*x*e^c + (d*e - 2*f)*e^c)*e^(d*x) + 2*I*f)/(a*d^2*f^3*x^3 + 3*a*d^2*e*f^2*x^2 + 3*a*d^2*e^2*f*x + a*d^2*
e^3 - (a*d^2*f^3*x^3*e^(2*c) + 3*a*d^2*e*f^2*x^2*e^(2*c) + 3*a*d^2*e^2*f*x*e^(2*c) + a*d^2*e^3*e^(2*c))*e^(2*d
*x) + 2*(I*a*d^2*f^3*x^3*e^c + 3*I*a*d^2*e*f^2*x^2*e^c + 3*I*a*d^2*e^2*f*x*e^c + I*a*d^2*e^3*e^c)*e^(d*x)) + 2
*integrate((d^2*f^2*x^2 + 2*d^2*e*f*x + d^2*e^2 - 12*f^2)/(-4*I*a*d^2*f^4*x^4 - 16*I*a*d^2*e*f^3*x^3 - 24*I*a*
d^2*e^2*f^2*x^2 - 16*I*a*d^2*e^3*f*x - 4*I*a*d^2*e^4 + 4*(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) + 2*integrate(1/(4*I*a*f^2*x^2 + 8*I*a*e*f
*x + 4*I*a*e^2 + 4*(a*f^2*x^2*e^c + 2*a*e*f*x*e^c + a*e^2*e^c)*e^(d*x)), x)

Giac [N/A]

Not integrable

Time = 32.95 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{{\left (f x + e\right )}^{2} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {\text {sech}(c+d x)}{(e+f x)^2 (a+i a \sinh (c+d x))} \, dx=\int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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