Optimal. Leaf size=189 \[ -\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {a x \left (4 a^2-b^2\right )}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d} \]
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Rubi [A] time = 0.62, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3048, 3050, 3049, 3023, 2735, 2659, 205} \[ -\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d \sqrt {a-b} \sqrt {a+b}}+\frac {a x \left (4 a^2-b^2\right )}{b^5}+\frac {2 a \sin (c+d x) \cos (c+d x)}{b^3 d}+\frac {\sin (c+d x) \cos ^3(c+d x)}{b d (a+b \cos (c+d x))}-\frac {4 \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 2735
Rule 3023
Rule 3048
Rule 3049
Rule 3050
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (1-\cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx &=\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos ^2(c+d x) \left (-3 \left (a^2-b^2\right )+4 \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (8 a \left (a^2-b^2\right )-b \left (a^2-b^2\right ) \cos (c+d x)-12 a \left (a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {-12 a^2 \left (a^2-b^2\right )+4 a b \left (a^2-b^2\right ) \cos (c+d x)+2 \left (a^2-b^2\right ) \left (12 a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^3 \left (a^2-b^2\right )}\\ &=-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\int \frac {-12 a^2 b \left (a^2-b^2\right )-6 a \left (a^2-b^2\right ) \left (4 a^2-b^2\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{6 b^4 \left (a^2-b^2\right )}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\left (a^2 \left (4 a^2-3 b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}-\frac {\left (2 a^2 \left (4 a^2-3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^5 d}\\ &=\frac {a \left (4 a^2-b^2\right ) x}{b^5}-\frac {2 a^2 \left (4 a^2-3 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^5 \sqrt {a+b} d}-\frac {\left (12 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a \cos (c+d x) \sin (c+d x)}{b^3 d}-\frac {4 \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}+\frac {\cos ^3(c+d x) \sin (c+d x)}{b d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A] time = 2.38, size = 217, normalized size = 1.15 \[ \frac {\frac {48 a^2 \left (4 a^2-3 b^2\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+\frac {96 a^4 c+96 a^4 d x-24 a^2 b^2 \sin (2 (c+d x))+12 a b \left (b^2-8 a^2\right ) \sin (c+d x)+24 a b \left (4 a^2-b^2\right ) (c+d x) \cos (c+d x)-24 a^2 b^2 c-24 a^2 b^2 d x+4 a b^3 \sin (3 (c+d x))+2 b^4 \sin (2 (c+d x))-b^4 \sin (4 (c+d x))}{a+b \cos (c+d x)}}{24 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 629, normalized size = 3.33 \[ \left [\frac {6 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x \cos \left (d x + c\right ) + 6 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x + 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}, \frac {3 \, {\left (4 \, a^{5} b - 5 \, a^{3} b^{3} + a b^{5}\right )} d x \cos \left (d x + c\right ) + 3 \, {\left (4 \, a^{6} - 5 \, a^{4} b^{2} + a^{2} b^{4}\right )} d x - 3 \, {\left (4 \, a^{5} - 3 \, a^{3} b^{2} + {\left (4 \, a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (12 \, a^{5} b - 13 \, a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} - b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{3} b^{3} - a b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (6 \, a^{4} b^{2} - 7 \, a^{2} b^{4} + b^{6}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3 \, {\left ({\left (a^{2} b^{6} - b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{3} b^{5} - a b^{7}\right )} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 280, normalized size = 1.48 \[ -\frac {\frac {6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )} b^{4}} - \frac {3 \, {\left (4 \, a^{3} - a b^{2}\right )} {\left (d x + c\right )}}{b^{5}} - \frac {6 \, {\left (4 \, a^{4} - 3 \, a^{2} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} + \frac {2 \, {\left (9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3} b^{4}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 403, normalized size = 2.13 \[ -\frac {2 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,b^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b \right )}-\frac {8 a^{4} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{5} \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {6 a^{2} \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d \,b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {12 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{5}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{d \,b^{3}} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.73, size = 1652, normalized size = 8.74 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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