\(\int (c+d x)^m (b \cosh (e+f x))^n \, dx\) [75]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\text {Int}\left ((c+d x)^m (b \cosh (e+f x))^n,x\right ) \]

[Out]

Unintegrable((d*x+c)^m*(b*cosh(f*x+e))^n,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 18, 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 = {} \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int (c+d x)^m (b \cosh (e+f x))^n \, dx \]

[In]

Int[(c + d*x)^m*(b*Cosh[e + f*x])^n,x]

[Out]

Defer[Int][(c + d*x)^m*(b*Cosh[e + f*x])^n, x]

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

Mathematica [N/A]

Not integrable

Time = 2.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int (c+d x)^m (b \cosh (e+f x))^n \, dx \]

[In]

Integrate[(c + d*x)^m*(b*Cosh[e + f*x])^n,x]

[Out]

Integrate[(c + d*x)^m*(b*Cosh[e + f*x])^n, x]

Maple [N/A] (verified)

Not integrable

Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \left (d x +c \right )^{m} \left (b \cosh \left (f x +e \right )\right )^{n}d x\]

[In]

int((d*x+c)^m*(b*cosh(f*x+e))^n,x)

[Out]

int((d*x+c)^m*(b*cosh(f*x+e))^n,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} \left (b \cosh \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*x+c)^m*(b*cosh(f*x+e))^n,x, algorithm="fricas")

[Out]

integral((d*x + c)^m*(b*cosh(f*x + e))^n, x)

Sympy [N/A]

Not integrable

Time = 8.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int \left (b \cosh {\left (e + f x \right )}\right )^{n} \left (c + d x\right )^{m}\, dx \]

[In]

integrate((d*x+c)**m*(b*cosh(f*x+e))**n,x)

[Out]

Integral((b*cosh(e + f*x))**n*(c + d*x)**m, x)

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} \left (b \cosh \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*x+c)^m*(b*cosh(f*x+e))^n,x, algorithm="maxima")

[Out]

integrate((d*x + c)^m*(b*cosh(f*x + e))^n, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int { {\left (d x + c\right )}^{m} \left (b \cosh \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*x+c)^m*(b*cosh(f*x+e))^n,x, algorithm="giac")

[Out]

integrate((d*x + c)^m*(b*cosh(f*x + e))^n, x)

Mupad [N/A]

Not integrable

Time = 1.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (c+d x)^m (b \cosh (e+f x))^n \, dx=\int {\left (b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^n\,{\left (c+d\,x\right )}^m \,d x \]

[In]

int((b*cosh(e + f*x))^n*(c + d*x)^m,x)

[Out]

int((b*cosh(e + f*x))^n*(c + d*x)^m, x)