\(\int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx\) [156]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 13.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx=\int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.56 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 696, normalized size of antiderivative = 30.26 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(2*(I*d^2*f^2*x^2 + 2*I*c*d*f^2*x + I*c^2*f^2 - I*d^2*m^2 + I*d^2*m + (I*d^2*f*m*x + I*d^2*m^2 + (I*c*d*f - I
*d^2)*m)*e^(2*f*x + 2*e) - (3*d^2*f^2*x^2 + 3*c^2*f^2 - 2*d^2*m^2 - (c*d*f - 2*d^2)*m + (6*c*d*f^2 - d^2*f*m)*
x)*e^(f*x + e))*(d*x + c)^m + 3*(-I*a^2*d^2*f^3*x^2 - 2*I*a^2*c*d*f^3*x - I*a^2*c^2*f^3 - (a^2*d^2*f^3*x^2 + 2
*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(3*f*x + 3*e) + 3*(I*a^2*d^2*f^3*x^2 + 2*I*a^2*c*d*f^3*x + I*a^2*c^2*f^3)*e^(2
*f*x + 2*e) + 3*(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(f*x + e))*integral(-2*(I*d^3*f^2*m*x^2 +
2*I*c*d^2*f^2*m*x - I*d^3*m^3 + 3*I*d^3*m^2 + (I*c^2*d*f^2 - 2*I*d^3)*m)*(d*x + c)^m/(-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^(f*x + e)), x))/(3*I*a^2*d^2*f^3*x^2 + 6*I*a^2*c*d*f^3*x + 3*I*a^2*c^2*f^3 + 3*
(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(3*f*x + 3*e) - 9*(I*a^2*d^2*f^3*x^2 + 2*I*a^2*c*d*f^3*x +
I*a^2*c^2*f^3)*e^(2*f*x + 2*e) - 9*(a^2*d^2*f^3*x^2 + 2*a^2*c*d*f^3*x + a^2*c^2*f^3)*e^(f*x + e))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]

[In]

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

[Out]

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