Optimal. Leaf size=243 \[ \frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
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Rubi [A]
time = 0.46, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2871, 3126,
3128, 3102, 2814, 2739, 632, 210} \begin {gather*} \frac {a^2 \sin ^3(x) \cos (x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \sin ^2(x) \cos (x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {x \left (12 a^2+b^2\right )}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \sin (x) \cos (x)}{2 b^3 \left (a^2-b^2\right )^2}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2871
Rule 3102
Rule 3126
Rule 3128
Rubi steps
\begin {align*} \int \frac {\sin ^5(x)}{(a+b \sin (x))^3} \, dx &=\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {\int \frac {\sin ^2(x) \left (3 a^2-2 a b \sin (x)-2 \left (2 a^2-b^2\right ) \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\sin (x) \left (-2 a^2 \left (4 a^2-7 b^2\right )+a b \left (a^2-4 b^2\right ) \sin (x)+2 \left (6 a^4-10 a^2 b^2+b^4\right ) \sin ^2(x)\right )}{a+b \sin (x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {2 a \left (6 a^4-10 a^2 b^2+b^4\right )-2 b \left (2 a^4-4 a^2 b^2-b^4\right ) \sin (x)-6 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \sin ^2(x)}{a+b \sin (x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {2 a b \left (6 a^4-10 a^2 b^2+b^4\right )+2 \left (a^2-b^2\right )^2 \left (12 a^2+b^2\right ) \sin (x)}{a+b \sin (x)} \, dx}{4 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (2 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right )\right ) \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 )}{b^5 \left (a^2-b^2\right )^2}\\ &=\frac {\left (12 a^2+b^2\right ) x}{2 b^5}-\frac {a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{5/2}}+\frac {3 a \left (4 a^4-7 a^2 b^2+2 b^4\right ) \cos (x)}{2 b^4 \left (a^2-b^2\right )^2}-\frac {\left (6 a^4-10 a^2 b^2+b^4\right ) \cos (x) \sin (x)}{2 b^3 \left (a^2-b^2\right )^2}+\frac {a^2 \cos (x) \sin ^3(x)}{2 b \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {a^2 \left (4 a^2-7 b^2\right ) \cos (x) \sin ^2(x)}{2 b^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 164, normalized size = 0.67 \begin {gather*} \frac {2 \left (12 a^2+b^2\right ) x-\frac {4 a^3 \left (12 a^4-29 a^2 b^2+20 b^4\right ) \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}}+12 a b \cos (x)-\frac {2 a^5 b \cos (x)}{(a-b) (a+b) (a+b \sin (x))^2}+\frac {2 a^4 b \left (7 a^2-10 b^2\right ) \cos (x)}{(a-b)^2 (a+b)^2 (a+b \sin (x))}-b^2 \sin (2 x)}{4 b^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 330, normalized size = 1.36
method | result | size |
default | \(\frac {\frac {4 \left (\frac {b^{2} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4}+\frac {3 a b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\frac {b^{2} \tan \left (\frac {x}{2}\right )}{4}+\frac {3 a b}{2}\right )}{\left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\left (12 a^{2}+b^{2}\right ) \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{b^{5}}-\frac {4 a^{3} \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-8 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 b \left (2 a^{4}+a^{2} b^{2}-6 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {a \,b^{2} \left (19 a^{2}-28 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 a^{2} b \left (2 a^{2}-3 b^{2}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )^{2}}+\frac {\left (12 a^{4}-29 a^{2} b^{2}+20 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{4 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{b^{5}}\) | \(330\) |
risch | \(\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}+\frac {i {\mathrm e}^{2 i x}}{8 b^{3}}+\frac {3 a \,{\mathrm e}^{i x}}{2 b^{4}}+\frac {3 a \,{\mathrm e}^{-i x}}{2 b^{4}}-\frac {i {\mathrm e}^{-2 i x}}{8 b^{3}}-\frac {i a^{4} \left (-8 i a^{3} b \,{\mathrm e}^{3 i x}+11 i a \,b^{3} {\mathrm e}^{3 i x}+20 i a^{3} b \,{\mathrm e}^{i x}-29 i a \,b^{3} {\mathrm e}^{i x}+14 a^{4} {\mathrm e}^{2 i x}-13 a^{2} b^{2} {\mathrm e}^{2 i x}-10 b^{4} {\mathrm e}^{2 i x}-7 a^{2} b^{2}+10 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} \left (a^{2}-b^{2}\right )^{2} b^{5}}-\frac {6 i a^{7} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{5}}+\frac {29 i a^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{3}}-\frac {10 i a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b}+\frac {6 i a^{7} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{5}}-\frac {29 i a^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b^{3}}+\frac {10 i a^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} b}\) | \(668\) |
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. 513 vs.
\(2 (227) = 454\).
time = 0.44, size = 1090, normalized size = 4.49 \begin {gather*} \left [-\frac {2 \, {\left (12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right )} x \cos \left (x\right )^{2} + 8 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \cos \left (x\right )^{3} + {\left (12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - {\left (12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right )} \cos \left (x\right )^{2} + 2 \, {\left (12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right )} \sin \left (x\right )\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 (12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right )} x - 2 \, {\left (12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right )} \cos \left (x\right ) - 2 \, {\left ({\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \cos \left (x\right )^{3} + 2 \, {\left (12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right )} x + {\left (18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{4 \, {\left (a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13} - {\left (a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\right )} \sin \left (x\right )\right )}}, -\frac {{\left (12 \, a^{8} b^{2} - 35 \, a^{6} b^{4} + 33 \, a^{4} b^{6} - 9 \, a^{2} b^{8} - b^{10}\right )} x \cos \left (x\right )^{2} + 4 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} \cos \left (x\right )^{3} - {\left (12 \, a^{9} - 17 \, a^{7} b^{2} - 9 \, a^{5} b^{4} + 20 \, a^{3} b^{6} - {\left (12 \, a^{7} b^{2} - 29 \, a^{5} b^{4} + 20 \, a^{3} b^{6}\right )} \cos \left (x\right )^{2} + 2 \, {\left (12 \, a^{8} b - 29 \, a^{6} b^{3} + 20 \, a^{4} b^{5}\right )} \sin \left (x\right )\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 (12 \, a^{10} - 23 \, a^{8} b^{2} - 2 \, a^{6} b^{4} + 24 \, a^{4} b^{6} - 10 \, a^{2} b^{8} - b^{10}\right )} x - {\left (12 \, a^{9} b - 29 \, a^{7} b^{3} + 15 \, a^{5} b^{5} + 6 \, a^{3} b^{7} - 4 \, a b^{9}\right )} \cos \left (x\right ) - {\left ({\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} \cos \left (x\right )^{3} + 2 \, {\left (12 \, a^{9} b - 35 \, a^{7} b^{3} + 33 \, a^{5} b^{5} - 9 \, a^{3} b^{7} - a b^{9}\right )} x + {\left (18 \, a^{8} b^{2} - 51 \, a^{6} b^{4} + 46 \, a^{4} b^{6} - 14 \, a^{2} b^{8} + b^{10}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \, {\left (a^{8} b^{5} - 2 \, a^{6} b^{7} + 2 \, a^{2} b^{11} - b^{13} - {\left (a^{6} b^{7} - 3 \, a^{4} b^{9} + 3 \, a^{2} b^{11} - b^{13}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b^{6} - 3 \, a^{5} b^{8} + 3 \, a^{3} b^{10} - a b^{12}\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. 516 vs.
\(2 (227) = 454\).
time = 0.44, size = 516, normalized size = 2.12 \begin {gather*} -\frac {{\left (12 \, a^{7} - 29 \, a^{5} b^{2} + 20 \, a^{3} b^{4}\right )} {\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 )}}{{\left (a^{4} b^{5} - 2 \, a^{2} b^{7} + b^{9}\right )} \sqrt {a^{2} - b^{2}}} + \frac {6 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{7} - 10 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{7} + a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{7} + 12 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{6} - 5 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{6} - 20 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{6} + 4 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{6} + 54 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{5} - 90 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{5} + 17 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{5} + 4 \, b^{7} \tan \left (\frac {1}{2} \, x\right )^{5} + 36 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{4} - 15 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 66 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{4} + 24 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{4} + 90 \, a^{6} b \tan \left (\frac {1}{2} \, x\right )^{3} - 162 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} + 55 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, b^{7} \tan \left (\frac {1}{2} \, x\right )^{3} + 36 \, a^{7} \tan \left (\frac {1}{2} \, x\right )^{2} - 31 \, a^{5} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 40 \, a^{3} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} + 20 \, a b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 42 \, a^{6} b \tan \left (\frac {1}{2} \, x\right ) - 74 \, a^{4} b^{3} \tan \left (\frac {1}{2} \, x\right ) + 23 \, a^{2} b^{5} \tan \left (\frac {1}{2} \, x\right ) + 12 \, a^{7} - 21 \, a^{5} b^{2} + 6 \, a^{3} b^{4}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} {\left (a \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, x\right ) + a\right )}^{2}} + \frac {{\left (12 \, a^{2} + b^{2}\right )} x}{2 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 15.81, size = 2500, normalized size = 10.29 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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