\(\int \cot ^2(e+f x) (a+b (c \sec (e+f x))^n)^p \, dx\) [470]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.60 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

Sympy [N/A]

Not integrable

Time = 73.55 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int \left (a + b \left (c \sec {\left (e + f x \right )}\right )^{n}\right )^{p} \cot ^{2}{\left (e + f x \right )}\, dx \]

[In]

integrate(cot(f*x+e)**2*(a+b*(c*sec(f*x+e))**n)**p,x)

[Out]

Integral((a + b*(c*sec(e + f*x))**n)**p*cot(e + f*x)**2, x)

Maxima [N/A]

Not integrable

Time = 6.35 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

Giac [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int { {\left (\left (c \sec \left (f x + e\right )\right )^{n} b + a\right )}^{p} \cot \left (f x + e\right )^{2} \,d x } \]

[In]

integrate(cot(f*x+e)^2*(a+b*(c*sec(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*sec(f*x + e))^n*b + a)^p*cot(f*x + e)^2, x)

Mupad [N/A]

Not integrable

Time = 21.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int \cot ^2(e+f x) \left (a+b (c \sec (e+f x))^n\right )^p \, dx=\int {\mathrm {cot}\left (e+f\,x\right )}^2\,{\left (a+b\,{\left (\frac {c}{\cos \left (e+f\,x\right )}\right )}^n\right )}^p \,d x \]

[In]

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

[Out]

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