Optimal. Leaf size=59 \[ \frac {x}{a^3}+\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {29 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2844, 3047,
3098, 2814, 2727} \begin {gather*} \frac {x}{a^3}+\frac {29 \cos (x)}{15 \left (a^3 \sin (x)+a^3\right )}+\frac {\sin ^2(x) \cos (x)}{5 (a \sin (x)+a)^3}-\frac {7 \cos (x)}{15 a (a \sin (x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
Rule 2844
Rule 3047
Rule 3098
Rubi steps
\begin {align*} \int \frac {\sin ^3(x)}{(a+a \sin (x))^3} \, dx &=\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {\int \frac {\sin (x) (2 a-5 a \sin (x))}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {\int \frac {2 a \sin (x)-5 a \sin ^2(x)}{(a+a \sin (x))^2} \, dx}{5 a^2}\\ &=\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {\int \frac {-14 a^2+15 a^2 \sin (x)}{a+a \sin (x)} \, dx}{15 a^4}\\ &=\frac {x}{a^3}+\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}-\frac {29 \int \frac {1}{a+a \sin (x)} \, dx}{15 a^2}\\ &=\frac {x}{a^3}+\frac {\cos (x) \sin ^2(x)}{5 (a+a \sin (x))^3}-\frac {7 \cos (x)}{15 a (a+a \sin (x))^2}+\frac {29 \cos (x)}{15 \left (a^3+a^3 \sin (x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 112, normalized size = 1.90 \begin {gather*} \frac {\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right ) \left (30 (-9+5 x) \cos \left (\frac {x}{2}\right )+(230-75 x) \cos \left (\frac {3 x}{2}\right )-15 x \cos \left (\frac {5 x}{2}\right )-370 \sin \left (\frac {x}{2}\right )+150 x \sin \left (\frac {x}{2}\right )-90 \sin \left (\frac {3 x}{2}\right )+75 x \sin \left (\frac {3 x}{2}\right )+64 \sin \left (\frac {5 x}{2}\right )-15 x \sin \left (\frac {5 x}{2}\right )\right )}{60 a^3 (1+\sin (x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 64, normalized size = 1.08
method | result | size |
risch | \(\frac {x}{a^{3}}+\frac {18 i {\mathrm e}^{3 i x}+6 \,{\mathrm e}^{4 i x}-\frac {46 i {\mathrm e}^{i x}}{3}-\frac {74 \,{\mathrm e}^{2 i x}}{3}+\frac {64}{15}}{a^{3} \left ({\mathrm e}^{i x}+i\right )^{5}}\) | \(54\) |
default | \(\frac {-\frac {4}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {8}{5 \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {4}{3 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {2}{\left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {16}{8 \tan \left (\frac {x}{2}\right )+8}+2 \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}\) | \(64\) |
norman | \(\frac {\frac {x}{a}+\frac {x \left (\tan ^{11}\left (\frac {x}{2}\right )\right )}{a}+\frac {2 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}+\frac {10 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}+\frac {48 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {44}{15 a}+\frac {5 x \tan \left (\frac {x}{2}\right )}{a}+\frac {13 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a}+\frac {25 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a}+\frac {38 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{a}+\frac {46 x \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {46 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{a}+\frac {38 x \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{a}+\frac {25 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{a}+\frac {13 x \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{a}+\frac {5 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{a}+\frac {38 \tan \left (\frac {x}{2}\right )}{3 a}+\frac {68 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{a}+\frac {128 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {76 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{3 a}+\frac {344 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5 a}+\frac {422 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15 a}+\frac {1004 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{15 a}}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{3} a^{2} \left (\tan \left (\frac {x}{2}\right )+1\right )^{5}}\) | \(271\) |
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. 144 vs.
\(2 (53) = 106\).
time = 0.51, size = 144, normalized size = 2.44 \begin {gather*} \frac {2 \, {\left (\frac {95 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {145 \, \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 22\right )}}{15 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {10 \, a^{3} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (x\right )^{5}}{{\left (\cos \left (x\right ) + 1\right )}^{5}}\right )}} + \frac {2 \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 119 vs.
\(2 (53) = 106\).
time = 0.35, size = 119, normalized size = 2.02 \begin {gather*} \frac {{\left (15 \, x + 32\right )} \cos \left (x\right )^{3} + {\left (45 \, x - 19\right )} \cos \left (x\right )^{2} - 6 \, {\left (5 \, x + 9\right )} \cos \left (x\right ) + {\left ({\left (15 \, x - 32\right )} \cos \left (x\right )^{2} - 3 \, {\left (10 \, x + 17\right )} \cos \left (x\right ) - 60 \, x + 3\right )} \sin \left (x\right ) - 60 \, x - 3}{15 \, {\left (a^{3} \cos \left (x\right )^{3} + 3 \, a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3} + {\left (a^{3} \cos \left (x\right )^{2} - 2 \, a^{3} \cos \left (x\right ) - 4 \, a^{3}\right )} \sin \left (x\right )\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. 777 vs.
\(2 (58) = 116\).
time = 4.75, size = 777, normalized size = 13.17 \begin {gather*} \frac {15 x \tan ^{5}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {75 x \tan ^{4}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {150 x \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {150 x \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {75 x \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {15 x}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {30 \tan ^{4}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {150 \tan ^{3}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {290 \tan ^{2}{\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {190 \tan {\left (\frac {x}{2} \right )}}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} + \frac {44}{15 a^{3} \tan ^{5}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan ^{4}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{3}{\left (\frac {x}{2} \right )} + 150 a^{3} \tan ^{2}{\left (\frac {x}{2} \right )} + 75 a^{3} \tan {\left (\frac {x}{2} \right )} + 15 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.50, size = 51, normalized size = 0.86 \begin {gather*} \frac {x}{a^{3}} + \frac {2 \, {\left (15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 75 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 145 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 95 \, \tan \left (\frac {1}{2} \, x\right ) + 22\right )}}{15 \, a^{3} {\left (\tan \left (\frac {1}{2} \, x\right ) + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.74, size = 50, normalized size = 0.85 \begin {gather*} \frac {x}{a^3}+\frac {2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+10\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+\frac {58\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{3}+\frac {38\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {44}{15}}{a^3\,{\left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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