Optimal. Leaf size=36 \[ -\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)}+\frac {1}{2} x \csc ^2(x) \sqrt {a \sin ^4(x)} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3286, 2715, 8}
\begin {gather*} \frac {1}{2} x \csc ^2(x) \sqrt {a \sin ^4(x)}-\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 3286
Rubi steps
\begin {align*} \int \sqrt {a \sin ^4(x)} \, dx &=\left (\csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int \sin ^2(x) \, dx\\ &=-\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)}+\frac {1}{2} \left (\csc ^2(x) \sqrt {a \sin ^4(x)}\right ) \int 1 \, dx\\ &=-\frac {1}{2} \cot (x) \sqrt {a \sin ^4(x)}+\frac {1}{2} x \csc ^2(x) \sqrt {a \sin ^4(x)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.69 \begin {gather*} \frac {1}{2} \csc (x) (-\cos (x)+x \csc (x)) \sqrt {a \sin ^4(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 33, normalized size = 0.92
method | result | size |
default | \(-\frac {\sqrt {a \left (1-\left (\cos ^{2}\left (x \right )\right )\right )^{2}}\, \left (\sin \left (x \right ) \cos \left (x \right )-x \right ) \sqrt {16}}{8 \sin \left (x \right )^{2}}\) | \(33\) |
risch | \(-\frac {\sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{2 i x} x}{2 \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}\, {\mathrm e}^{4 i x}}{8 \left ({\mathrm e}^{2 i x}-1\right )^{2}}+\frac {i \sqrt {a \left ({\mathrm e}^{2 i x}-1\right )^{4} {\mathrm e}^{-4 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )^{2}}\) | \(102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 22, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, \sqrt {a} x - \frac {\sqrt {a} \tan \left (x\right )}{2 \, {\left (\tan \left (x\right )^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 36, normalized size = 1.00 \begin {gather*} \frac {\sqrt {a \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a} {\left (\cos \left (x\right ) \sin \left (x\right ) - x\right )}}{2 \, {\left (\cos \left (x\right )^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a \sin ^{4}{\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 15, normalized size = 0.42 \begin {gather*} \frac {1}{4} \, \sqrt {a} {\left (2 \, x - \sin \left (2 \, x\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \sqrt {a\,{\sin \left (x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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