\(\int \cos (e+f x) (a+b (c \tan (e+f x))^n)^p \, dx\) [489]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\text {Int}\left (\cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

Unintegrable(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x)

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

Int[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Defer[Int][Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]

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

Mathematica [N/A]

Not integrable

Time = 3.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

Integrate[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Integrate[Cos[e + f*x]*(a + b*(c*Tan[e + f*x])^n)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

int(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x)

[Out]

int(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right ) \,d x } \]

[In]

integrate(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e), x)

Sympy [N/A]

Not integrable

Time = 139.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \left (a + b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \cos {\left (e + f x \right )}\, dx \]

[In]

integrate(cos(f*x+e)*(a+b*(c*tan(f*x+e))**n)**p,x)

[Out]

Integral((a + b*(c*tan(e + f*x))**n)**p*cos(e + f*x), x)

Maxima [N/A]

Not integrable

Time = 7.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right ) \,d x } \]

[In]

integrate(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e), x)

Giac [N/A]

Not integrable

Time = 69.53 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right ) \,d x } \]

[In]

integrate(cos(f*x+e)*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*cos(f*x + e), x)

Mupad [N/A]

Not integrable

Time = 12.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cos (e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \cos \left (e+f\,x\right )\,{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]

[In]

int(cos(e + f*x)*(a + b*(c*tan(e + f*x))^n)^p,x)

[Out]

int(cos(e + f*x)*(a + b*(c*tan(e + f*x))^n)^p, x)