\(\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx\) [116]
Optimal result
Integrand size = 23, antiderivative size = 23 \[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\text {Int}\left (\frac {1}{(c+d x) (a+i a \sinh (e+f x))^2},x\right )
\]
[Out]
Unintegrable(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x)
Rubi [N/A]
Not integrable
Time = 0.04 (sec) , antiderivative size = 23, 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 = {} \[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx
\]
[In]
Int[1/((c + d*x)*(a + I*a*Sinh[e + f*x])^2),x]
[Out]
Defer[Int][1/((c + d*x)*(a + I*a*Sinh[e + f*x])^2), x]
Rubi steps \begin{align*}
\text {integral}& = \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \\
\end{align*}
Mathematica [N/A]
Not integrable
Time = 25.92 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
\[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx
\]
[In]
Integrate[1/((c + d*x)*(a + I*a*Sinh[e + f*x])^2),x]
[Out]
Integrate[1/((c + d*x)*(a + I*a*Sinh[e + f*x])^2), x]
Maple [N/A] (verified)
Not integrable
Time = 0.51 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {1}{\left (d x +c \right ) \left (a +i a \sinh \left (f x +e \right )\right )^{2}}d x\]
[In]
int(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x)
[Out]
int(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x)
Fricas [N/A]
Not integrable
Time = 0.26 (sec) , antiderivative size = 793, normalized size of antiderivative = 34.48
\[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x, algorithm="fricas")
[Out]
(-2*I*d^2*f^2*x^2 - 4*I*c*d*f^2*x - 2*I*c^2*f^2 + 4*I*d^2 - 2*(-I*d^2*f*x - I*c*d*f + 2*I*d^2)*e^(2*f*x + 2*e)
+ 2*(3*d^2*f^2*x^2 + 3*c^2*f^2 + c*d*f - 4*d^2 + (6*c*d*f^2 + d^2*f)*x)*e^(f*x + e) - 3*(-I*a^2*d^3*f^3*x^3 -
3*I*a^2*c*d^2*f^3*x^2 - 3*I*a^2*c^2*d*f^3*x - I*a^2*c^3*f^3 - (a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*
c^2*d*f^3*x + a^2*c^3*f^3)*e^(3*f*x + 3*e) + 3*(I*a^2*d^3*f^3*x^3 + 3*I*a^2*c*d^2*f^3*x^2 + 3*I*a^2*c^2*d*f^3*
x + I*a^2*c^3*f^3)*e^(2*f*x + 2*e) + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^
3)*e^(f*x + e))*integral(-2*(-I*d^3*f^2*x^2 - 2*I*c*d^2*f^2*x - I*c^2*d*f^2 + 6*I*d^3)/(-3*I*a^2*d^4*f^3*x^4 -
12*I*a^2*c*d^3*f^3*x^3 - 18*I*a^2*c^2*d^2*f^3*x^2 - 12*I*a^2*c^3*d*f^3*x - 3*I*a^2*c^4*f^3 + 3*(a^2*d^4*f^3*x
^4 + 4*a^2*c*d^3*f^3*x^3 + 6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3)*e^(f*x + e)), x))/(3*I*a^2
*d^3*f^3*x^3 + 9*I*a^2*c*d^2*f^3*x^2 + 9*I*a^2*c^2*d*f^3*x + 3*I*a^2*c^3*f^3 + 3*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^
2*f^3*x^2 + 3*a^2*c^2*d*f^3*x + a^2*c^3*f^3)*e^(3*f*x + 3*e) - 9*(I*a^2*d^3*f^3*x^3 + 3*I*a^2*c*d^2*f^3*x^2 +
3*I*a^2*c^2*d*f^3*x + I*a^2*c^3*f^3)*e^(2*f*x + 2*e) - 9*(a^2*d^3*f^3*x^3 + 3*a^2*c*d^2*f^3*x^2 + 3*a^2*c^2*d*
f^3*x + a^2*c^3*f^3)*e^(f*x + e))
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\text {Timed out}
\]
[In]
integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e))**2,x)
[Out]
Timed out
Maxima [N/A]
Not integrable
Time = 0.53 (sec) , antiderivative size = 606, normalized size of antiderivative = 26.35
\[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x, algorithm="maxima")
[Out]
-2*(I*d^2*f^2*x^2 + 2*I*c*d*f^2*x + I*c^2*f^2 - 2*I*d^2 + (-I*d^2*f*x*e^(2*e) - I*c*d*f*e^(2*e) + 2*I*d^2*e^(2
*e))*e^(2*f*x) - (3*d^2*f^2*x^2*e^e + 3*c^2*f^2*e^e + c*d*f*e^e - 4*d^2*e^e + (6*c*d*f^2*e^e + d^2*f*e^e)*x)*e
^(f*x))/(3*I*a^2*d^3*f^3*x^3 + 9*I*a^2*c*d^2*f^3*x^2 + 9*I*a^2*c^2*d*f^3*x + 3*I*a^2*c^3*f^3 + 3*(a^2*d^3*f^3*
x^3*e^(3*e) + 3*a^2*c*d^2*f^3*x^2*e^(3*e) + 3*a^2*c^2*d*f^3*x*e^(3*e) + a^2*c^3*f^3*e^(3*e))*e^(3*f*x) - 9*(I*
a^2*d^3*f^3*x^3*e^(2*e) + 3*I*a^2*c*d^2*f^3*x^2*e^(2*e) + 3*I*a^2*c^2*d*f^3*x*e^(2*e) + I*a^2*c^3*f^3*e^(2*e))
*e^(2*f*x) - 9*(a^2*d^3*f^3*x^3*e^e + 3*a^2*c*d^2*f^3*x^2*e^e + 3*a^2*c^2*d*f^3*x*e^e + a^2*c^3*f^3*e^e)*e^(f*
x)) - integrate(2/3*(d^3*f^2*x^2 + 2*c*d^2*f^2*x + c^2*d*f^2 - 6*d^3)/(a^2*d^4*f^3*x^4 + 4*a^2*c*d^3*f^3*x^3 +
6*a^2*c^2*d^2*f^3*x^2 + 4*a^2*c^3*d*f^3*x + a^2*c^4*f^3 - (-I*a^2*d^4*f^3*x^4*e^e - 4*I*a^2*c*d^3*f^3*x^3*e^e
- 6*I*a^2*c^2*d^2*f^3*x^2*e^e - 4*I*a^2*c^3*d*f^3*x*e^e - I*a^2*c^4*f^3*e^e)*e^(f*x)), x)
Giac [N/A]
Not integrable
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
\[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int { \frac {1}{{\left (d x + c\right )} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x }
\]
[In]
integrate(1/(d*x+c)/(a+I*a*sinh(f*x+e))^2,x, algorithm="giac")
[Out]
integrate(1/((d*x + c)*(I*a*sinh(f*x + e) + a)^2), x)
Mupad [N/A]
Not integrable
Time = 1.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04
\[
\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,\left (c+d\,x\right )} \,d x
\]
[In]
int(1/((a + a*sinh(e + f*x)*1i)^2*(c + d*x)),x)
[Out]
int(1/((a + a*sinh(e + f*x)*1i)^2*(c + d*x)), x)