Optimal. Leaf size=145 \[ -\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}-\frac {\tanh ^{-1}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))} \]
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Rubi [A]
time = 0.25, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {2881, 3134,
3080, 3855, 2739, 632, 210} \begin {gather*} -\frac {\tanh ^{-1}(\cos (x))}{a^3}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {x}{2}\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 \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 2881
Rule 3080
Rule 3134
Rule 3855
Rubi steps
\begin {align*} \int \frac {\csc (x)}{(a+b \sin (x))^3} \, dx &=-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}+\frac {\int \frac {\csc (x) \left (2 \left (a^2-b^2\right )-2 a b \sin (x)+b^2 \sin ^2(x)\right )}{(a+b \sin (x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \frac {\csc (x) \left (2 \left (a^2-b^2\right )^2-a b \left (4 a^2-b^2\right ) \sin (x)\right )}{a+b \sin (x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\int \csc (x) \, dx}{a^3}-\frac {\left (b \left (6 a^4-5 a^2 b^2+2 b^4\right )\right ) \int \frac {1}{a+b \sin (x)} \, dx}{2 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}-\frac {\left (b \left (6 a^4-5 a^2 b^2+2 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 )}{a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\tanh ^{-1}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}+\frac {\left (2 b \left (6 a^4-5 a^2 b^2+2 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 )}{a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (6 a^4-5 a^2 b^2+2 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \left (a^2-b^2\right )^{5/2}}-\frac {\tanh ^{-1}(\cos (x))}{a^3}-\frac {b^2 \cos (x)}{2 a \left (a^2-b^2\right ) (a+b \sin (x))^2}-\frac {b^2 \left (5 a^2-2 b^2\right ) \cos (x)}{2 a^2 \left (a^2-b^2\right )^2 (a+b \sin (x))}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 140, normalized size = 0.97 \begin {gather*} -\frac {\frac {2 b \left (6 a^4-5 a^2 b^2+2 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}}+2 \log \left (\cos \left (\frac {x}{2}\right )\right )-2 \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {a b^2 \cos (x) \left (6 a^3-3 a b^2+b \left (5 a^2-2 b^2\right ) \sin (x)\right )}{(a-b)^2 (a+b)^2 (a+b \sin (x))^2}}{2 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.45, size = 270, normalized size = 1.86
method | result | size |
default | \(\frac {\ln \left (\tan \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {2 b \left (\frac {\frac {a \,b^{2} \left (7 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 b \left (2 a^{4}+3 a^{2} b^{2}-2 b^{4}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {a \,b^{2} \left (17 a^{2}-8 b^{2}\right ) \tan \left (\frac {x}{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}+\frac {3 a^{2} b \left (2 a^{2}-b^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}}{\left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )^{2}}+\frac {\left (6 a^{4}-5 a^{2} b^{2}+2 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{a^{3}}\) | \(270\) |
risch | \(\frac {i b \left (-4 i a^{3} b \,{\mathrm e}^{3 i x}+i a \,b^{3} {\mathrm e}^{3 i x}+16 i a^{3} b \,{\mathrm e}^{i x}-7 i a \,b^{3} {\mathrm e}^{i x}+10 a^{4} {\mathrm e}^{2 i x}+a^{2} b^{2} {\mathrm e}^{2 i x}-2 b^{4} {\mathrm e}^{2 i x}-5 a^{2} b^{2}+2 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+i b +2 a \,{\mathrm e}^{i x}\right )^{2} a^{2} \left (a^{2}-b^{2}\right )^{2}}+\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}+\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {3 i a b \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a}-\frac {i b^{5} \ln \left ({\mathrm e}^{i x}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{\sqrt {a^{2}-b^{2}}\, b}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} a^{3}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{a^{3}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{a^{3}}\) | \(623\) |
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. 481 vs.
\(2 (135) = 270\).
time = 0.86, size = 1027, normalized size = 7.08 \begin {gather*} \left [-\frac {2 \, {\left (5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (x\right ) \sin \left (x\right ) + {\left (6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left (6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (x\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\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 ) + 6 \, {\left (2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\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 )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\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 )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{4 \, {\left (a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8} - {\left (a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} b^{7}\right )} \sin \left (x\right )\right )}}, -\frac {{\left (5 \, a^{5} b^{3} - 7 \, a^{3} b^{5} + 2 \, a b^{7}\right )} \cos \left (x\right ) \sin \left (x\right ) - {\left (6 \, a^{6} b + a^{4} b^{3} - 3 \, a^{2} b^{5} + 2 \, b^{7} - {\left (6 \, a^{4} b^{3} - 5 \, a^{2} b^{5} + 2 \, b^{7}\right )} \cos \left (x\right )^{2} + 2 \, {\left (6 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 2 \, a b^{6}\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 ) + 3 \, {\left (2 \, a^{6} b^{2} - 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} \cos \left (x\right ) + {\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 )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - {\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 )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right )}{2 \, {\left (a^{11} - 2 \, a^{9} b^{2} + 2 \, a^{5} b^{6} - a^{3} b^{8} - {\left (a^{9} b^{2} - 3 \, a^{7} b^{4} + 3 \, a^{5} b^{6} - a^{3} b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{10} b - 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} - a^{4} 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (x \right )}}{\left (a + b \sin {\left (x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 246, normalized size = 1.70 \begin {gather*} -\frac {{\left (6 \, a^{4} b - 5 \, a^{2} b^{3} + 2 \, b^{5}\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^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {7 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, a b^{5} \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 9 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, x\right )^{2} - 6 \, b^{6} \tan \left (\frac {1}{2} \, x\right )^{2} + 17 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, x\right ) - 8 \, a b^{5} \tan \left (\frac {1}{2} \, x\right ) + 6 \, a^{4} b^{2} - 3 \, a^{2} b^{4}}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} 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}} + \frac {\log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.39, size = 2191, normalized size = 15.11 \begin {gather*} \frac {\frac {3\,\left (b^4-2\,a^2\,b^2\right )}{a\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {3\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2\,\left (2\,a^4\,b^2+3\,a^2\,b^4-2\,b^6\right )}{a^3\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (8\,b^5-17\,a^2\,b^3\right )}{a^2\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3\,\left (4\,b^4-7\,a^2\,b^2\right )}{a^2\,\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 {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{a^3}+\frac {b\,\mathrm {atan}\left (\frac {\frac {b\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (\frac {8\,a^7\,b-9\,a^5\,b^3+4\,a^3\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-2\,a^{11}+24\,a^9\,b^2-62\,a^7\,b^4+68\,a^5\,b^6-36\,a^3\,b^8+8\,a\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}-\frac {b\,\left (\frac {2\,a^{10}\,b-4\,a^8\,b^3+2\,a^6\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (6\,a^{14}-32\,a^{12}\,b^2+68\,a^{10}\,b^4-72\,a^8\,b^6+38\,a^6\,b^8-8\,a^4\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}\right )\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}\right )\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )\,1{}\mathrm {i}}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}+\frac {b\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (\frac {8\,a^7\,b-9\,a^5\,b^3+4\,a^3\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-2\,a^{11}+24\,a^9\,b^2-62\,a^7\,b^4+68\,a^5\,b^6-36\,a^3\,b^8+8\,a\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {b\,\left (\frac {2\,a^{10}\,b-4\,a^8\,b^3+2\,a^6\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (6\,a^{14}-32\,a^{12}\,b^2+68\,a^{10}\,b^4-72\,a^8\,b^6+38\,a^6\,b^8-8\,a^4\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}\right )\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}\right )\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )\,1{}\mathrm {i}}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}}{\frac {2\,\left (6\,a^4\,b-5\,a^2\,b^3+2\,b^5\right )}{a^8-2\,a^6\,b^2+a^4\,b^4}+\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-24\,a^6\,b^2+26\,a^4\,b^4-13\,a^2\,b^6+2\,b^8\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}-\frac {b\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (\frac {8\,a^7\,b-9\,a^5\,b^3+4\,a^3\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-2\,a^{11}+24\,a^9\,b^2-62\,a^7\,b^4+68\,a^5\,b^6-36\,a^3\,b^8+8\,a\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}-\frac {b\,\left (\frac {2\,a^{10}\,b-4\,a^8\,b^3+2\,a^6\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (6\,a^{14}-32\,a^{12}\,b^2+68\,a^{10}\,b^4-72\,a^8\,b^6+38\,a^6\,b^8-8\,a^4\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}\right )\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}\right )\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}+\frac {b\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (\frac {8\,a^7\,b-9\,a^5\,b^3+4\,a^3\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}+\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (-2\,a^{11}+24\,a^9\,b^2-62\,a^7\,b^4+68\,a^5\,b^6-36\,a^3\,b^8+8\,a\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}+\frac {b\,\left (\frac {2\,a^{10}\,b-4\,a^8\,b^3+2\,a^6\,b^5}{a^8-2\,a^6\,b^2+a^4\,b^4}-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (6\,a^{14}-32\,a^{12}\,b^2+68\,a^{10}\,b^4-72\,a^8\,b^6+38\,a^6\,b^8-8\,a^4\,b^{10}\right )}{a^{11}-4\,a^9\,b^2+6\,a^7\,b^4-4\,a^5\,b^6+a^3\,b^8}\right )\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}\right )\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )}{2\,\left (a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}\right )}}\right )\,\sqrt {-{\left (a+b\right )}^5\,{\left (a-b\right )}^5}\,\left (6\,a^4-5\,a^2\,b^2+2\,b^4\right )\,1{}\mathrm {i}}{a^{13}-5\,a^{11}\,b^2+10\,a^9\,b^4-10\,a^7\,b^6+5\,a^5\,b^8-a^3\,b^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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