\(\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx\) [112]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+I*a*sinh(f*x+e)),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 = {} \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]

[In]

Int[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 26.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]

[In]

Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])),x]

[Out]

Integrate[1/((c + d*x)^2*(a + I*a*Sinh[e + f*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +i a \sinh \left (f x +e \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 8.87 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+I*a*sinh(f*x+e)),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 8.13 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\frac {2 i}{- i a c^{2} f - 2 i a c d f x - i a d^{2} f x^{2} + \left (a c^{2} f e^{e} + 2 a c d f x e^{e} + a d^{2} f x^{2} e^{e}\right ) e^{f x}} + \frac {4 i d \int \frac {1}{c^{3} e^{e} e^{f x} - i c^{3} + 3 c^{2} d x e^{e} e^{f x} - 3 i c^{2} d x + 3 c d^{2} x^{2} e^{e} e^{f x} - 3 i c d^{2} x^{2} + d^{3} x^{3} e^{e} e^{f x} - i d^{3} x^{3}}\, dx}{a f} \]

[In]

integrate(1/(d*x+c)**2/(a+I*a*sinh(f*x+e)),x)

[Out]

2*I/(-I*a*c**2*f - 2*I*a*c*d*f*x - I*a*d**2*f*x**2 + (a*c**2*f*exp(e) + 2*a*c*d*f*x*exp(e) + a*d**2*f*x**2*exp
(e))*exp(f*x)) + 4*I*d*Integral(1/(c**3*exp(e)*exp(f*x) - I*c**3 + 3*c**2*d*x*exp(e)*exp(f*x) - 3*I*c**2*d*x +
3*c*d**2*x**2*exp(e)*exp(f*x) - 3*I*c*d**2*x**2 + d**3*x**3*exp(e)*exp(f*x) - I*d**3*x**3), x)/(a*f)

Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 6.87 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+I*a*sinh(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(d*x+c)^2/(a+I*a*sinh(f*x+e)),x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(I*a*sinh(f*x + e) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^2} \,d x \]

[In]

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

[Out]

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