Optimal. Leaf size=193 \[ \frac {2 a^3 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d}+\frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {\left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac {x \left (8 a^4-4 a^2 b^2-b^4\right )}{8 b^5}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 b d} \]
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Rubi [A] time = 0.61, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3050, 3049, 3023, 2735, 2659, 205} \[ \frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}+\frac {2 a^3 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d}-\frac {\left (4 a^2-b^2\right ) \sin (c+d x) \cos (c+d x)}{8 b^3 d}-\frac {x \left (-4 a^2 b^2+8 a^4-b^4\right )}{8 b^5}+\frac {a \sin (c+d x) \cos ^2(c+d x)}{3 b^2 d}-\frac {\sin (c+d x) \cos ^3(c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2659
Rule 2735
Rule 3023
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)} \, dx &=-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {\cos ^2(c+d x) \left (-3 a+b \cos (c+d x)+4 a \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{4 b}\\ &=\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {\cos (c+d x) \left (8 a^2-a b \cos (c+d x)-3 \left (4 a^2-b^2\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{12 b^2}\\ &=-\frac {\left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {-3 a \left (4 a^2-b^2\right )+b \left (4 a^2+3 b^2\right ) \cos (c+d x)+8 a \left (3 a^2-b^2\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^3}\\ &=\frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {\left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\int \frac {-3 a b \left (4 a^2-b^2\right )-3 \left (8 a^4-4 a^2 b^2-b^4\right ) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{24 b^4}\\ &=-\frac {\left (8 a^4-4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {\left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\left (a^3 \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{b^5}\\ &=-\frac {\left (8 a^4-4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {\left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}+\frac {\left (2 a^3 \left (a^2-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 {\left (8 a^4-4 a^2 b^2-b^4\right ) x}{8 b^5}+\frac {2 a^3 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^5 d}+\frac {a \left (3 a^2-b^2\right ) \sin (c+d x)}{3 b^4 d}-\frac {\left (4 a^2-b^2\right ) \cos (c+d x) \sin (c+d x)}{8 b^3 d}+\frac {a \cos ^2(c+d x) \sin (c+d x)}{3 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{4 b d}\\ \end {align*}
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Mathematica [A] time = 0.99, size = 168, normalized size = 0.87 \[ \frac {-96 a^4 c-96 a^4 d x-24 a^2 b^2 \sin (2 (c+d x))+24 a b \left (4 a^2-b^2\right ) \sin (c+d x)+48 a^2 b^2 c+48 a^2 b^2 d x+192 a^3 \sqrt {b^2-a^2} \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )+8 a b^3 \sin (3 (c+d x))-3 b^4 \sin (4 (c+d x))+12 b^4 c+12 b^4 d x}{96 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 369, normalized size = 1.91 \[ \left [\frac {12 \, \sqrt {-a^{2} + b^{2}} a^{3} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 3 \, {\left (8 \, a^{4} - 4 \, a^{2} b^{2} - b^{4}\right )} d x - {\left (6 \, b^{4} \cos \left (d x + c\right )^{3} - 8 \, a b^{3} \cos \left (d x + c\right )^{2} - 24 \, a^{3} b + 8 \, a b^{3} + 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}, \frac {24 \, \sqrt {a^{2} - b^{2}} a^{3} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - 3 \, {\left (8 \, a^{4} - 4 \, a^{2} b^{2} - b^{4}\right )} d x - {\left (6 \, b^{4} \cos \left (d x + c\right )^{3} - 8 \, a b^{3} \cos \left (d x + c\right )^{2} - 24 \, a^{3} b + 8 \, a b^{3} + 3 \, {\left (4 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, b^{5} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 370, normalized size = 1.92 \[ -\frac {\frac {3 \, {\left (8 \, a^{4} - 4 \, a^{2} b^{2} - b^{4}\right )} {\left (d x + c\right )}}{b^{5}} + \frac {48 \, {\left (a^{5} - a^{3} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} b^{5}} - \frac {2 \, {\left (24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 32 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 72 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 32 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} b^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 653, normalized size = 3.38 \[ \frac {2 a^{5} \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 {2 a^{3} \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 {2 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {6 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {\left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {8 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {8 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{3 d \,b^{2} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3}}{d \,b^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}{d \,b^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d b \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{d \,b^{5}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}}{d \,b^{3}}+\frac {\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d b} \]
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 = 2.59, size = 240, normalized size = 1.24 \[ \frac {a^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {a^2\,\sin \left (2\,c+2\,d\,x\right )}{4}}{b^3\,d}-\frac {\frac {a\,\sin \left (c+d\,x\right )}{4}-\frac {a\,\sin \left (3\,c+3\,d\,x\right )}{12}}{b^2\,d}+\frac {\frac {\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {\sin \left (4\,c+4\,d\,x\right )}{32}}{b\,d}+\frac {a^3\,\sin \left (c+d\,x\right )}{b^4\,d}-\frac {2\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b^5\,d}-\frac {2\,a^3\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}{a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sqrt {b^2-a^2}}{b^5\,d} \]
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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