\(\int \cot ^4(c+d x) (a+b \sin ^n(c+d x))^p \, dx\) [590]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\text {Int}\left (\cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p,x\right ) \]

[Out]

Unintegrable(cot(d*x+c)^4*(a+b*sin(d*x+c)^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 = {} \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx \]

[In]

Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p,x]

[Out]

Defer[Int][Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x]

Rubi steps \begin{align*} \text {integral}& = \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 32.47 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx \]

[In]

Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p,x]

[Out]

Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x]^n)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.86 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \left (\cot ^{4}\left (d x +c \right )\right ) {\left (a +b \left (\sin ^{n}\left (d x +c \right )\right )\right )}^{p}d x\]

[In]

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^n)^p,x)

[Out]

int(cot(d*x+c)^4*(a+b*sin(d*x+c)^n)^p,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

Sympy [F(-1)]

Timed out. \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\text {Timed out} \]

[In]

integrate(cot(d*x+c)**4*(a+b*sin(d*x+c)**n)**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 6.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

Giac [N/A]

Not integrable

Time = 4.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{n} + a\right )}^{p} \cot \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(cot(d*x+c)^4*(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)^4, x)

Mupad [N/A]

Not integrable

Time = 14.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) \left (a+b \sin ^n(c+d x)\right )^p \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,{\sin \left (c+d\,x\right )}^n\right )}^p \,d x \]

[In]

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^n)^p,x)

[Out]

int(cot(c + d*x)^4*(a + b*sin(c + d*x)^n)^p, x)