Optimal. Leaf size=53 \[ -\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)} \]
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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2846, 2827,
2715, 8, 2713} \begin {gather*} -\frac {3 x}{2 a}+\frac {4 \cos ^3(x)}{3 a}-\frac {4 \cos (x)}{a}+\frac {\sin ^3(x) \cos (x)}{a \sin (x)+a}+\frac {3 \sin (x) \cos (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rule 2846
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{a+a \sin (x)} \, dx &=\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {\int \sin ^2(x) (3 a-4 a \sin (x)) \, dx}{a^2}\\ &=\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {3 \int \sin ^2(x) \, dx}{a}+\frac {4 \int \sin ^3(x) \, dx}{a}\\ &=\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}-\frac {3 \int 1 \, dx}{2 a}-\frac {4 \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (x)\right )}{a}\\ &=-\frac {3 x}{2 a}-\frac {4 \cos (x)}{a}+\frac {4 \cos ^3(x)}{3 a}+\frac {3 \cos (x) \sin (x)}{2 a}+\frac {\cos (x) \sin ^3(x)}{a+a \sin (x)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 101, normalized size = 1.91 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (-3 (7+12 x) \cos \left (\frac {x}{2}\right )-18 \cos \left (\frac {3 x}{2}\right )-2 \cos \left (\frac {5 x}{2}\right )+\cos \left (\frac {7 x}{2}\right )+69 \sin \left (\frac {x}{2}\right )-36 x \sin \left (\frac {x}{2}\right )-18 \sin \left (\frac {3 x}{2}\right )+2 \sin \left (\frac {5 x}{2}\right )+\sin \left (\frac {7 x}{2}\right )\right )}{24 a (1+\sin (x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 66, normalized size = 1.25
method | result | size |
risch | \(-\frac {3 x}{2 a}-\frac {7 \,{\mathrm e}^{i x}}{8 a}-\frac {7 \,{\mathrm e}^{-i x}}{8 a}-\frac {2}{\left ({\mathrm e}^{i x}+i\right ) a}+\frac {\cos \left (3 x \right )}{12 a}+\frac {\sin \left (2 x \right )}{4 a}\) | \(61\) |
default | \(\frac {-\frac {2 \left (\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2}+\tan ^{4}\left (\frac {x}{2}\right )+4 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-\frac {\tan \left (\frac {x}{2}\right )}{2}+\frac {5}{3}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3}}-3 \arctan \left (\tan \left (\frac {x}{2}\right )\right )-\frac {2}{\tan \left (\frac {x}{2}\right )+1}}{a}\) | \(66\) |
norman | \(\frac {-\frac {3 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}-\frac {11 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {11 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x}{2 a}-\frac {16}{3 a}-\frac {3 x \tan \left (\frac {x}{2}\right )}{2 a}-\frac {6 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}-\frac {9 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {9 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}-\frac {6 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}-\frac {3 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {3 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{2 a}-\frac {21 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}-\frac {7 \tan \left (\frac {x}{2}\right )}{3 a}-\frac {55 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3 a}-\frac {31 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{3 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{4} \left (\tan \left (\frac {x}{2}\right )+1\right )}\) | \(224\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 180 vs.
\(2 (47) = 94\).
time = 0.54, size = 180, normalized size = 3.40 \begin {gather*} -\frac {\frac {7 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {39 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {24 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {24 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {9 \, \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {9 \, \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + 16}{3 \, {\left (a + \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {3 \, a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {3 \, a \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {3 \, a \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}} + \frac {a \sin \left (x\right )^{6}}{{\left (\cos \left (x\right ) + 1\right )}^{6}} + \frac {a \sin \left (x\right )^{7}}{{\left (\cos \left (x\right ) + 1\right )}^{7}}\right )}} - \frac {3 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 70, normalized size = 1.32 \begin {gather*} \frac {2 \, \cos \left (x\right )^{4} - \cos \left (x\right )^{3} - 3 \, {\left (3 \, x + 5\right )} \cos \left (x\right ) - 12 \, \cos \left (x\right )^{2} + {\left (2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - 9 \, x - 9 \, \cos \left (x\right ) + 6\right )} \sin \left (x\right ) - 9 \, x - 6}{6 \, {\left (a \cos \left (x\right ) + a \sin \left (x\right ) + a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1221 vs.
\(2 (49) = 98\).
time = 2.39, size = 1221, normalized size = 23.04 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 67, normalized size = 1.26 \begin {gather*} -\frac {3 \, x}{2 \, a} - \frac {2}{a {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}} - \frac {3 \, \tan \left (\frac {1}{2} \, x\right )^{5} + 6 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 3 \, \tan \left (\frac {1}{2} \, x\right ) + 10}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.82, size = 78, normalized size = 1.47 \begin {gather*} -\frac {3\,x}{2\,a}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+13\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+\frac {7\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {16}{3}}{a\,{\left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )}^3\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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