\(\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx\) [360]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\text {Int}\left (\cot ^2(c+d x) (a+b \sec (c+d x))^n,x\right ) \]

[Out]

Unintegrable(cot(d*x+c)^2*(a+b*sec(d*x+c))^n,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 21, 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 = {} \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 8.65 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

int(cot(d*x+c)^2*(a+b*sec(d*x+c))^n,x)

[Out]

int(cot(d*x+c)^2*(a+b*sec(d*x+c))^n,x)

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

integral((b*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)

Sympy [N/A]

Not integrable

Time = 19.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right )^{n} \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

integrate(cot(d*x+c)**2*(a+b*sec(d*x+c))**n,x)

[Out]

Integral((a + b*sec(c + d*x))**n*cot(c + d*x)**2, x)

Maxima [N/A]

Not integrable

Time = 5.85 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

integrate((b*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)

Giac [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

integrate((b*sec(d*x + c) + a)^n*cot(d*x + c)^2, x)

Mupad [N/A]

Not integrable

Time = 18.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \cot ^2(c+d x) (a+b \sec (c+d x))^n \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n \,d x \]

[In]

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

[Out]

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