Optimal. Leaf size=103 \[ -\frac {3 a b \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {2833, 12, 2739,
632, 210} \begin {gather*} -\frac {3 a b \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rubi steps
\begin {align*} \int \frac {\sin (x)}{(a+b \sin (x))^3} \, dx &=-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\int \frac {2 b-a \sin (x)}{(a+b \sin (x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int -\frac {3 a b}{a+b \sin (x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {(3 a b) \int \frac {1}{a+b \sin (x)} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {(3 a b) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {(6 a b) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{\left (a^2-b^2\right )^2}\\ &=-\frac {3 a b \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {a \cos (x)}{2 \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\left (a^2+2 b^2\right ) \cos (x)}{2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 94, normalized size = 0.91 \begin {gather*} -\frac {3 a b \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {\cos (x) \left (a \left (2 a^2+b^2\right )+b \left (a^2+2 b^2\right ) \sin (x)\right )}{2 (a-b)^2 (a+b)^2 (a+b \sin (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(220\) vs.
\(2(93)=186\).
time = 0.28, size = 221, normalized size = 2.15
method | result | size |
default | \(\frac {-\frac {3 a^{2} b \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {\left (2 a^{4}+5 a^{2} b^{2}+2 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{2}+4 b^{2}\right ) b \tan \left (\frac {x}{2}\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}-\frac {\left (2 a^{2}+b^{2}\right ) a}{a^{4}-2 a^{2} b^{2}+b^{4}}}{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )^{2}}-\frac {3 a b \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\) | \(221\) |
risch | \(\frac {i \left (-3 i a \,b^{3} {\mathrm e}^{3 i x}+4 i a^{3} b \,{\mathrm e}^{i x}+5 i a \,b^{3} {\mathrm e}^{i x}+2 a^{4} {\mathrm e}^{2 i x}+5 a^{2} b^{2} {\mathrm e}^{2 i x}+2 b^{4} {\mathrm e}^{2 i x}-a^{2} b^{2}-2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} b}-\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(274\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 212 vs.
\(2 (93) = 186\).
time = 0.39, size = 490, normalized size = 4.76 \begin {gather*} \left [-\frac {2 \, {\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (x\right ) \sin \left (x\right ) - 3 \, {\left (a b^{3} \cos \left (x\right )^{2} - 2 \, a^{2} b^{2} \sin \left (x\right ) - a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (x\right ) \sin \left (x\right ) + b \cos \left (x\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (x\right )^{2} - 2 \, a b \sin \left (x\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right )}{4 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (x\right )\right )}}, -\frac {{\left (a^{4} b + a^{2} b^{3} - 2 \, b^{5}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \, {\left (a b^{3} \cos \left (x\right )^{2} - 2 \, a^{2} b^{2} \sin \left (x\right ) - a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (x\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (x\right )}\right ) + {\left (2 \, a^{5} - a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right )}{2 \, {\left (a^{8} - 2 \, a^{6} b^{2} + 2 \, a^{2} b^{6} - b^{8} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (x\right )\right )}}\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs.
\(2 (93) = 186\).
time = 0.43, size = 189, normalized size = 1.83 \begin {gather*} -\frac {3 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} a b}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {3 \, a^{3} b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 5 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 5 \, a^{3} b \tan \left (\frac {1}{2} \, x\right ) + 4 \, a b^{3} \tan \left (\frac {1}{2} \, x\right ) + 2 \, a^{4} + a^{2} b^{2}}{{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.51, size = 310, normalized size = 3.01 \begin {gather*} -\frac {\frac {2\,a^3+a\,b^2}{a^4-2\,a^2\,b^2+b^4}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{a^4-2\,a^2\,b^2+b^4}+\frac {b\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (5\,a^2+4\,b^2\right )}{a^4-2\,a^2\,b^2+b^4}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2+b^2\right )\,\left (a^2+2\,b^2\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^2+4\,b^2\right )+a^2+a^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+4\,a\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+4\,a\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}-\frac {3\,a\,b\,\mathrm {atan}\left (\frac {\left (\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}}+\frac {3\,a\,b^2\,\left (2\,a^4-4\,a^2\,b^2+2\,b^4\right )}{2\,{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}\,\left (a^4-2\,a^2\,b^2+b^4\right )}\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{3\,a\,b}\right )}{{\left (a+b\right )}^{5/2}\,{\left (a-b\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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