\(\int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\) [191]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 37.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.31 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 12.08 (sec) , antiderivative size = 425, normalized size of antiderivative = 14.66 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=- \frac {2}{- i a d e - i a d f x + \left (a d e e^{c} + a d f x e^{c}\right ) e^{d x}} - \frac {i \left (\int \left (- \frac {2 i f}{e^{2} e^{c} e^{d x} - i e^{2} + 2 e f x e^{c} e^{d x} - 2 i e f x + f^{2} x^{2} e^{c} e^{d x} - i f^{2} x^{2}}\right )\, dx + \int \left (- \frac {i d e}{e^{2} e^{c} e^{d x} - i e^{2} + 2 e f x e^{c} e^{d x} - 2 i e f x + f^{2} x^{2} e^{c} e^{d x} - i f^{2} x^{2}}\right )\, dx + \int \left (- \frac {i d f x}{e^{2} e^{c} e^{d x} - i e^{2} + 2 e f x e^{c} e^{d x} - 2 i e f x + f^{2} x^{2} e^{c} e^{d x} - i f^{2} x^{2}}\right )\, dx + \int \frac {d e e^{c} e^{d x}}{e^{2} e^{c} e^{d x} - i e^{2} + 2 e f x e^{c} e^{d x} - 2 i e f x + f^{2} x^{2} e^{c} e^{d x} - i f^{2} x^{2}}\, dx + \int \frac {d f x e^{c} e^{d x}}{e^{2} e^{c} e^{d x} - i e^{2} + 2 e f x e^{c} e^{d x} - 2 i e f x + f^{2} x^{2} e^{c} e^{d x} - i f^{2} x^{2}}\, dx\right )}{a d} \]

[In]

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

[Out]

-2/(-I*a*d*e - I*a*d*f*x + (a*d*e*exp(c) + a*d*f*x*exp(c))*exp(d*x)) - I*(Integral(-2*I*f/(e**2*exp(c)*exp(d*x
) - I*e**2 + 2*e*f*x*exp(c)*exp(d*x) - 2*I*e*f*x + f**2*x**2*exp(c)*exp(d*x) - I*f**2*x**2), x) + Integral(-I*
d*e/(e**2*exp(c)*exp(d*x) - I*e**2 + 2*e*f*x*exp(c)*exp(d*x) - 2*I*e*f*x + f**2*x**2*exp(c)*exp(d*x) - I*f**2*
x**2), x) + Integral(-I*d*f*x/(e**2*exp(c)*exp(d*x) - I*e**2 + 2*e*f*x*exp(c)*exp(d*x) - 2*I*e*f*x + f**2*x**2
*exp(c)*exp(d*x) - I*f**2*x**2), x) + Integral(d*e*exp(c)*exp(d*x)/(e**2*exp(c)*exp(d*x) - I*e**2 + 2*e*f*x*ex
p(c)*exp(d*x) - 2*I*e*f*x + f**2*x**2*exp(c)*exp(d*x) - I*f**2*x**2), x) + Integral(d*f*x*exp(c)*exp(d*x)/(e**
2*exp(c)*exp(d*x) - I*e**2 + 2*e*f*x*exp(c)*exp(d*x) - 2*I*e*f*x + f**2*x**2*exp(c)*exp(d*x) - I*f**2*x**2), x
))/(a*d)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.00 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int { \frac {\sinh \left (d x + c\right )}{{\left (f x + e\right )} {\left (i \, a \sinh \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {\sinh (c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx=\int \frac {\mathrm {sinh}\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]

[In]

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

[Out]

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