\(\int \cos ^2(e+f x) (a+b (c \tan (e+f x))^n)^p \, dx\) [495]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\text {Int}\left (\cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

Unintegrable(cos(f*x+e)^2*(a+b*(c*tan(f*x+e))^n)^p,x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, 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 = {} \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

Int[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Defer[Int][Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x]

Rubi steps \begin{align*} \text {integral}& = \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

[In]

Integrate[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Integrate[Cos[e + f*x]^2*(a + b*(c*Tan[e + f*x])^n)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.62 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

\[\int \cos \left (f x +e \right )^{2} \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]

[In]

int(cos(f*x+e)^2*(a+b*(c*tan(f*x+e))^n)^p,x)

[Out]

int(cos(f*x+e)^2*(a+b*(c*tan(f*x+e))^n)^p,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(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)^2, x)

Sympy [F(-1)]

Timed out. \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\text {Timed out} \]

[In]

integrate(cos(f*x+e)**2*(a+b*(c*tan(f*x+e))**n)**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 7.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(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)^2, x)

Giac [N/A]

Not integrable

Time = 1.51 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cos \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(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)^2, x)

Mupad [N/A]

Not integrable

Time = 13.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cos ^2(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]

[In]

int(cos(e + f*x)^2*(a + b*(c*tan(e + f*x))^n)^p,x)

[Out]

int(cos(e + f*x)^2*(a + b*(c*tan(e + f*x))^n)^p, x)