Optimal. Leaf size=127 \[ \frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {9 \tan ^7(c+d x)}{7 a^3 d}+\frac {6 \tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {5 \sec ^7(c+d x)}{7 a^3 d}-\frac {\sec ^5(c+d x)}{5 a^3 d} \]
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Rubi [A] time = 0.36, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2875, 2873, 2607, 270, 2606, 14} \[ \frac {4 \tan ^9(c+d x)}{9 a^3 d}+\frac {9 \tan ^7(c+d x)}{7 a^3 d}+\frac {6 \tan ^5(c+d x)}{5 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {5 \sec ^7(c+d x)}{7 a^3 d}-\frac {\sec ^5(c+d x)}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x) \tan ^2(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \sec ^8(c+d x) (a-a \sin (c+d x))^3 \tan ^2(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (a^3 \sec ^8(c+d x) \tan ^2(c+d x)-3 a^3 \sec ^7(c+d x) \tan ^3(c+d x)+3 a^3 \sec ^6(c+d x) \tan ^4(c+d x)-a^3 \sec ^5(c+d x) \tan ^5(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \sec ^8(c+d x) \tan ^2(c+d x) \, dx}{a^3}-\frac {\int \sec ^5(c+d x) \tan ^5(c+d x) \, dx}{a^3}-\frac {3 \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx}{a^3}+\frac {3 \int \sec ^6(c+d x) \tan ^4(c+d x) \, dx}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {\operatorname {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {\operatorname {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^3 d}+\frac {3 \operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=-\frac {\sec ^5(c+d x)}{5 a^3 d}+\frac {5 \sec ^7(c+d x)}{7 a^3 d}-\frac {4 \sec ^9(c+d x)}{9 a^3 d}+\frac {\tan ^3(c+d x)}{3 a^3 d}+\frac {6 \tan ^5(c+d x)}{5 a^3 d}+\frac {9 \tan ^7(c+d x)}{7 a^3 d}+\frac {4 \tan ^9(c+d x)}{9 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 185, normalized size = 1.46 \[ \frac {73728 \sin (c+d x)-7263 \sin (2 (c+d x))+512 \sin (3 (c+d x))-3228 \sin (4 (c+d x))-1536 \sin (5 (c+d x))+269 \sin (6 (c+d x))-9684 \cos (c+d x)-6912 \cos (2 (c+d x))-538 \cos (3 (c+d x))-3072 \cos (4 (c+d x))+1614 \cos (5 (c+d x))+256 \cos (6 (c+d x))+32256}{322560 d (a \sin (c+d x)+a)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 130, normalized size = 1.02 \[ -\frac {8 \, \cos \left (d x + c\right )^{6} - 36 \, \cos \left (d x + c\right )^{4} + 15 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (12 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 35\right )} \sin \left (d x + c\right ) + 35}{315 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} + {\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 172, normalized size = 1.35 \[ -\frac {\frac {105 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {945 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 10080 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 23940 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 42840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 41958 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 32592 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 14148 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5112 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 673}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{10080 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 190, normalized size = 1.50 \[ \frac {-\frac {1}{24 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {4}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {60}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {34}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {99}{10 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {23}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{4 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {3}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 442, normalized size = 3.48 \[ \frac {4 \, {\left (\frac {66 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {132 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {232 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {18 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {108 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {84 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {504 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {315 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {210 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 11\right )}}{315 \, {\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 = 14.87, size = 279, normalized size = 2.20 \[ \frac {\frac {44\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{315}+\frac {88\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}+\frac {176\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{105}+\frac {928\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{315}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{35}+\frac {48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{35}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{15}+\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}}{a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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