\(\int \cot ^2(c+d x) (a+b \sin ^n(c+d x))^p \, dx\) [589]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\text {Int}\left (\cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.76 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 113.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \left (a + b \sin ^{n}{\left (c + d x \right )}\right )^{p} \cot ^{2}{\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 4.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.57 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^2(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]

[In]

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

[Out]

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