3.6.72 \(\int \cot ^2(c+d x) (a+b \sin ^4(c+d x))^p \, dx\) [572]

Optimal. Leaf size=26 \[ \text {Int}\left (\cot ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p,x\right ) \]

[Out]

Unintegrable(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \cot ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p,x]

[Out]

Defer[Int][Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x]

Rubi steps

\begin {align*} \int \cot ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \, dx &=\int \cot ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.02, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^2(c+d x) \left (a+b \sin ^4(c+d x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p,x]

[Out]

Integrate[Cot[c + d*x]^2*(a + b*Sin[c + d*x]^4)^p, x]

________________________________________________________________________________________

Maple [A]
time = 0.80, size = 0, normalized size = 0.00 \[\int \left (\cot ^{2}\left (d x +c \right )\right ) \left (a +b \left (\sin ^{4}\left (d x +c \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x)

[Out]

int(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c)^2, x)

________________________________________________________________________________________

Fricas [A]
time = 0.44, size = 37, normalized size = 1.42 \begin {gather*} {\rm integral}\left ({\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + a + b\right )}^{p} \cot \left (d x + c\right )^{2}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p*cot(d*x + c)^2, x)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(a+b*sin(d*x+c)**4)**p,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*sin(d*x+c)^4)^p,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p*cot(d*x + c)^2, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\mathrm {cot}\left (c+d\,x\right )}^2\,{\left (b\,{\sin \left (c+d\,x\right )}^4+a\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^2*(a + b*sin(c + d*x)^4)^p,x)

[Out]

int(cot(c + d*x)^2*(a + b*sin(c + d*x)^4)^p, x)

________________________________________________________________________________________