\(\int (e x)^{-1+2 n} (a+b \cosh (c+d x^n))^p \, dx\) [46]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\frac {x^{-2 n} (e x)^{2 n} \text {Int}\left (x^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p,x\right )}{e} \]

[Out]

(e*x)^(2*n)*Unintegrable(x^(-1+2*n)*(a+b*cosh(c+d*x^n))^p,x)/e/(x^(2*n))

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, 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 = {} \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx \]

[In]

Int[(e*x)^(-1 + 2*n)*(a + b*Cosh[c + d*x^n])^p,x]

[Out]

((e*x)^(2*n)*Defer[Int][x^(-1 + 2*n)*(a + b*Cosh[c + d*x^n])^p, x])/(e*x^(2*n))

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx}{e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx \]

[In]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Cosh[c + d*x^n])^p,x]

[Out]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Cosh[c + d*x^n])^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00

\[\int \left (e x \right )^{2 n -1} {\left (a +b \cosh \left (c +d \,x^{n}\right )\right )}^{p}d x\]

[In]

int((e*x)^(2*n-1)*(a+b*cosh(c+d*x^n))^p,x)

[Out]

int((e*x)^(2*n-1)*(a+b*cosh(c+d*x^n))^p,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{2 \, n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((e*x)^(2*n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Sympy [N/A]

Not integrable

Time = 24.08 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int \left (e x\right )^{2 n - 1} \left (a + b \cosh {\left (c + d x^{n} \right )}\right )^{p}\, dx \]

[In]

integrate((e*x)**(-1+2*n)*(a+b*cosh(c+d*x**n))**p,x)

[Out]

Integral((e*x)**(2*n - 1)*(a + b*cosh(c + d*x**n))**p, x)

Maxima [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{2 \, n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((e*x)^(2*n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Giac [N/A]

Not integrable

Time = 7.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{2 \, n - 1} {\left (b \cosh \left (d x^{n} + c\right ) + a\right )}^{p} \,d x } \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*cosh(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^(2*n - 1)*(b*cosh(d*x^n + c) + a)^p, x)

Mupad [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int (e x)^{-1+2 n} \left (a+b \cosh \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^{2\,n-1}\,{\left (a+b\,\mathrm {cosh}\left (c+d\,x^n\right )\right )}^p \,d x \]

[In]

int((e*x)^(2*n - 1)*(a + b*cosh(c + d*x^n))^p,x)

[Out]

int((e*x)^(2*n - 1)*(a + b*cosh(c + d*x^n))^p, x)