\(\int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx\) [151]

Optimal result

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

[Out]

Unintegrable((d*x+c)^m*(a+I*a*sinh(f*x+e))^n,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.39 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.09 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

integral((d*x + c)^m*(1/2*(I*a*e^(2*f*x + 2*e) + 2*a*e^(f*x + e) - I*a)*e^(-f*x - e))^n, x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int (c+d x)^m (a+i a \sinh (e+f x))^n \, dx=\int {\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n\,{\left (c+d\,x\right )}^m \,d x \]

[In]

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

[Out]

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