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