Optimal. Leaf size=266 \[ -\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{3/2} d}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))} \]
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Rubi [A]
time = 0.56, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2968, 3127,
3128, 3102, 2814, 2739, 632, 210} \begin {gather*} -\frac {\left (4 a^2-3 b^2\right ) \sin ^2(c+d x) \cos (c+d x)}{2 b^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {x \left (12 a^2-b^2\right )}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 d \left (a^2-b^2\right )}+\frac {\left (6 a^2-5 b^2\right ) \sin (c+d x) \cos (c+d x)}{2 b^3 d \left (a^2-b^2\right )}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{b^5 d \left (a^2-b^2\right )^{3/2}}-\frac {\sin ^3(c+d x) \cos (c+d x)}{2 b d (a+b \sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2739
Rule 2814
Rule 2968
Rule 3102
Rule 3127
Rule 3128
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \sin ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \frac {\sin ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx\\ &=-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\int \frac {\sin ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\sin (c+d x) \left (2 \left (4 a^4-7 a^2 b^2+3 b^4\right )-a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (6 a^2-5 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 b^2 \left (a^2-b^2\right )^2}\\ &=\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {-2 a \left (6 a^4-11 a^2 b^2+5 b^4\right )+2 b \left (2 a^4-3 a^2 b^2+b^4\right ) \sin (c+d x)+2 a \left (12 a^2-11 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)}{a+b \sin (c+d x)} \, dx}{4 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {-2 a b \left (6 a^4-11 a^2 b^2+5 b^4\right )-2 \left (a^2-b^2\right )^2 \left (12 a^2-b^2\right ) \sin (c+d x)}{a+b \sin (c+d 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 {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 b^5 \left (a^2-b^2\right )}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (a \left (12 a^4-19 a^2 b^2+6 b^4\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right ) d}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (2 a \left (12 a^4-19 a^2 b^2+6 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 {1}{2} (c+d x)\right )\right )}{b^5 \left (a^2-b^2\right ) d}\\ &=-\frac {\left (12 a^2-b^2\right ) x}{2 b^5}+\frac {a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{b^5 \left (a^2-b^2\right )^{3/2} d}-\frac {a \left (12 a^2-11 b^2\right ) \cos (c+d x)}{2 b^4 \left (a^2-b^2\right ) d}+\frac {\left (6 a^2-5 b^2\right ) \cos (c+d x) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}-\frac {\cos (c+d x) \sin ^3(c+d x)}{2 b d (a+b \sin (c+d x))^2}-\frac {\left (4 a^2-3 b^2\right ) \cos (c+d x) \sin ^2(c+d x)}{2 b^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 4.19, size = 288, normalized size = 1.08 \begin {gather*} \frac {\frac {4 a \left (12 a^4-19 a^2 b^2+6 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {-4 a b \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin (c+d x)-2 b^2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x) \sin ^2(c+d x)+\cos (c+d x) \left (-24 a^5 b+22 a^3 b^3-8 a b^3 \left (a^2-b^2\right ) \sin ^2(c+d x)+2 b^4 \left (a^2-b^2\right ) \sin ^3(c+d x)\right )-a^2 \left (2 \left (12 a^4-13 a^2 b^2+b^4\right ) (c+d x)+\left (18 a^2 b^2-17 b^4\right ) \sin (2 (c+d x))\right )}{(a+b \sin (c+d x))^2}}{4 (a-b) b^5 (a+b) d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.72, size = 349, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}}{d}\) | \(349\) |
default | \(\frac {-\frac {2 \left (\frac {\frac {b^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}+3 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}+3 a b}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\left (12 a^{2}-b^{2}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{5}}+\frac {2 a \left (\frac {-\frac {a \,b^{2} \left (5 a^{2}-4 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {b \left (6 a^{4}+7 a^{2} b^{2}-10 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a \,b^{2} \left (19 a^{2}-16 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a^{2}-b^{2}\right )}-\frac {a^{2} b \left (6 a^{2}-5 b^{2}\right )}{2 \left (a^{2}-b^{2}\right )}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {\left (12 a^{4}-19 a^{2} b^{2}+6 b^{4}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{b^{5}}}{d}\) | \(349\) |
risch | \(-\frac {6 x \,a^{2}}{b^{5}}+\frac {x}{2 b^{3}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}-\frac {3 a \,{\mathrm e}^{i \left (d x +c \right )}}{2 b^{4} d}-\frac {3 a \,{\mathrm e}^{-i \left (d x +c \right )}}{2 b^{4} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {i a^{2} \left (-8 i a^{3} b \,{\mathrm e}^{3 i \left (d x +c \right )}+7 i a \,b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+20 i a^{3} b \,{\mathrm e}^{i \left (d x +c \right )}-17 i a \,b^{3} {\mathrm e}^{i \left (d x +c \right )}+14 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-5 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-7 a^{2} b^{2}+6 b^{4}\right )}{\left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right ) d \,b^{5}}+\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}-\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}+\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a +a^{2}-b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}-\frac {6 i a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{5}}+\frac {19 i a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d \,b^{3}}-\frac {3 i a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i \left (\sqrt {a^{2}-b^{2}}\, a -a^{2}+b^{2}\right )}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) d b}\) | \(773\) |
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 [A]
time = 0.45, size = 1058, normalized size = 3.98 \begin {gather*} \left [-\frac {2 \, {\left (12 \, a^{6} b^{2} - 25 \, a^{4} b^{4} + 14 \, a^{2} b^{6} - b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (12 \, a^{8} - 13 \, a^{6} b^{2} - 11 \, a^{4} b^{4} + 13 \, a^{2} b^{6} - b^{8}\right )} d x + {\left (12 \, a^{7} - 7 \, a^{5} b^{2} - 13 \, a^{3} b^{4} + 6 \, a b^{6} - {\left (12 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{6} b - 19 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (-\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} - 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (12 \, a^{7} b - 19 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + 4 \, a b^{7}\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (12 \, a^{7} b - 25 \, a^{5} b^{3} + 14 \, a^{3} b^{5} - a b^{7}\right )} d x + {\left (18 \, a^{6} b^{2} - 36 \, a^{4} b^{4} + 19 \, a^{2} b^{6} - b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{4} b^{7} - 2 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b^{6} - 2 \, a^{3} b^{8} + a b^{10}\right )} d \sin \left (d x + c\right ) - {\left (a^{6} b^{5} - a^{4} b^{7} - a^{2} b^{9} + b^{11}\right )} d\right )}}, -\frac {{\left (12 \, a^{6} b^{2} - 25 \, a^{4} b^{4} + 14 \, a^{2} b^{6} - b^{8}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (a^{5} b^{3} - 2 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right )^{3} - {\left (12 \, a^{8} - 13 \, a^{6} b^{2} - 11 \, a^{4} b^{4} + 13 \, a^{2} b^{6} - b^{8}\right )} d x - {\left (12 \, a^{7} - 7 \, a^{5} b^{2} - 13 \, a^{3} b^{4} + 6 \, a b^{6} - {\left (12 \, a^{5} b^{2} - 19 \, a^{3} b^{4} + 6 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (12 \, a^{6} b - 19 \, a^{4} b^{3} + 6 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) - {\left (12 \, a^{7} b - 19 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + 4 \, a b^{7}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (12 \, a^{7} b - 25 \, a^{5} b^{3} + 14 \, a^{3} b^{5} - a b^{7}\right )} d x + {\left (18 \, a^{6} b^{2} - 36 \, a^{4} b^{4} + 19 \, a^{2} b^{6} - b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{7} - 2 \, a^{2} b^{9} + b^{11}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{5} b^{6} - 2 \, a^{3} b^{8} + a b^{10}\right )} d \sin \left (d x + c\right ) - {\left (a^{6} b^{5} - a^{4} b^{7} - a^{2} b^{9} + b^{11}\right )} d\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. 535 vs.
\(2 (251) = 502\).
time = 0.53, size = 535, normalized size = 2.01 \begin {gather*} \frac {\frac {2 \, {\left (12 \, a^{5} - 19 \, a^{3} b^{2} + 6 \, a b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (6 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 14 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 54 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 45 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 15 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 44 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 90 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 87 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 42 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 39 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, a^{5} - 11 \, a^{3} b^{2}\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2}} - \frac {{\left (12 \, a^{2} - b^{2}\right )} {\left (d x + c\right )}}{b^{5}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 17.72, size = 2500, normalized size = 9.40 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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