Optimal. Leaf size=123 \[ \frac {4 \tan ^3(c+d x)}{63 a^3 d}+\frac {4 \tan (c+d x)}{21 a^3 d}-\frac {\sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.16, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2859, 2672, 3767} \[ \frac {4 \tan ^3(c+d x)}{63 a^3 d}+\frac {4 \tan (c+d x)}{21 a^3 d}-\frac {\sec ^3(c+d x)}{21 d \left (a^3 \sin (c+d x)+a^3\right )}-\frac {\sec ^3(c+d x)}{21 a d (a \sin (c+d x)+a)^2}+\frac {\sec ^3(c+d x)}{9 d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2859
Rule 3767
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x) \tan (c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}+\frac {\int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx}{3 a}\\ &=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}+\frac {5 \int \frac {\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx}{21 a^2}\\ &=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {4 \int \sec ^4(c+d x) \, dx}{21 a^3}\\ &=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}-\frac {4 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{21 a^3 d}\\ &=\frac {\sec ^3(c+d x)}{9 d (a+a \sin (c+d x))^3}-\frac {\sec ^3(c+d x)}{21 a d (a+a \sin (c+d x))^2}-\frac {\sec ^3(c+d x)}{21 d \left (a^3+a^3 \sin (c+d x)\right )}+\frac {4 \tan (c+d x)}{21 a^3 d}+\frac {4 \tan ^3(c+d x)}{63 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 185, normalized size = 1.50 \[ \frac {9216 \sin (c+d x)+675 \sin (2 (c+d x))+512 \sin (3 (c+d x))+300 \sin (4 (c+d x))-1536 \sin (5 (c+d x))-25 \sin (6 (c+d x))+900 \cos (c+d x)-6912 \cos (2 (c+d x))+50 \cos (3 (c+d x))-3072 \cos (4 (c+d x))-150 \cos (5 (c+d x))+256 \cos (6 (c+d x))+10752}{64512 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 130, normalized size = 1.06 \[ -\frac {8 \, \cos \left (d x + c\right )^{6} - 36 \, \cos \left (d x + c\right )^{4} + 15 \, \cos \left (d x + c\right )^{2} - {\left (24 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{2} - 7\right )} \sin \left (d x + c\right ) + 14}{63 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 172, normalized size = 1.40 \[ -\frac {\frac {21 \, {\left (15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 756 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4200 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 11340 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 14994 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 13356 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6768 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2196 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 209}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{2016 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.60, size = 190, normalized size = 1.54 \[ \frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}-\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}+\frac {64}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {40}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {27}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {39}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {59}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {13}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {5}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 442, normalized size = 3.59 \[ \frac {2 \, {\left (\frac {6 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {75 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {128 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {162 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {42 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {189 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {126 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {63 \, \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}}{63 \, {\left (a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {12 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {27 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {36 \, a^{3} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {27 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {12 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.00, size = 279, normalized size = 2.27 \[ \frac {\frac {2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{63}+\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{21}+\frac {50\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{21}+\frac {256\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{63}+\frac {36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}-\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{7}-\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+6\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \]
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 \[ \frac {\int \frac {\sin {\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\sin ^{3}{\left (c + d x \right )} + 3 \sin ^{2}{\left (c + d x \right )} + 3 \sin {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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