\(\int (e x)^m (b \cos (c+d x^n))^p \, dx\) [63]

Optimal result

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

[Out]

Unintegrable((e*x)^m*(b*cos(c+d*x^n))^p,x)

Rubi [N/A]

Not integrable

Time = 0.02 (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 = {} \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx \]

[In]

Int[(e*x)^m*(b*Cos[c + d*x^n])^p,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.33 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \left (e x \right )^{m} {\left (b \cos \left (c +d \,x^{n}\right )\right )}^{p}d x\]

[In]

int((e*x)^m*(b*cos(c+d*x^n))^p,x)

[Out]

int((e*x)^m*(b*cos(c+d*x^n))^p,x)

Fricas [N/A]

Not integrable

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

[In]

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

[Out]

integral((e*x)^m*(b*cos(d*x^n + c))^p, x)

Sympy [N/A]

Not integrable

Time = 8.54 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int \left (b \cos {\left (c + d x^{n} \right )}\right )^{p} \left (e x\right )^{m}\, dx \]

[In]

integrate((e*x)**m*(b*cos(c+d*x**n))**p,x)

[Out]

Integral((b*cos(c + d*x**n))**p*(e*x)**m, x)

Maxima [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{m} \left (b \cos \left (d x^{n} + c\right )\right )^{p} \,d x } \]

[In]

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

[Out]

integrate((e*x)^m*(b*cos(d*x^n + c))^p, x)

Giac [N/A]

Not integrable

Time = 0.94 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int { \left (e x\right )^{m} \left (b \cos \left (d x^{n} + c\right )\right )^{p} \,d x } \]

[In]

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

[Out]

integrate((e*x)^m*(b*cos(d*x^n + c))^p, x)

Mupad [N/A]

Not integrable

Time = 13.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int (e x)^m \left (b \cos \left (c+d x^n\right )\right )^p \, dx=\int {\left (e\,x\right )}^m\,{\left (b\,\cos \left (c+d\,x^n\right )\right )}^p \,d x \]

[In]

int((e*x)^m*(b*cos(c + d*x^n))^p,x)

[Out]

int((e*x)^m*(b*cos(c + d*x^n))^p, x)