Optimal. Leaf size=140 \[ \frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {6 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^3(c+d x)}{a^2 d}+\frac {2 \sec (c+d x)}{a^2 d}+\frac {x}{a^2} \]
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Rubi [A] time = 0.30, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2875, 2873, 2607, 30, 2606, 194, 3473, 8} \[ \frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan (c+d x)}{a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {6 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^3(c+d x)}{a^2 d}+\frac {2 \sec (c+d x)}{a^2 d}+\frac {x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 194
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3473
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x) \tan ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sec ^2(c+d x) (a-a \sin (c+d x))^2 \tan ^6(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec ^2(c+d x) \tan ^6(c+d x)-2 a^2 \sec (c+d x) \tan ^7(c+d x)+a^2 \tan ^8(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec ^2(c+d x) \tan ^6(c+d x) \, dx}{a^2}+\frac {\int \tan ^8(c+d x) \, dx}{a^2}-\frac {2 \int \sec (c+d x) \tan ^7(c+d x) \, dx}{a^2}\\ &=\frac {\tan ^7(c+d x)}{7 a^2 d}-\frac {\int \tan ^6(c+d x) \, dx}{a^2}+\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,\tan (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \left (-1+x^2\right )^3 \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {\int \tan ^4(c+d x) \, dx}{a^2}-\frac {2 \operatorname {Subst}\left (\int \left (-1+3 x^2-3 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{a^2 d}+\frac {6 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {\int \tan ^2(c+d x) \, dx}{a^2}\\ &=\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{a^2 d}+\frac {6 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}+\frac {\int 1 \, dx}{a^2}\\ &=\frac {x}{a^2}+\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{a^2 d}+\frac {6 \sec ^5(c+d x)}{5 a^2 d}-\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {\tan (c+d x)}{a^2 d}+\frac {\tan ^3(c+d x)}{3 a^2 d}-\frac {\tan ^5(c+d x)}{5 a^2 d}+\frac {2 \tan ^7(c+d x)}{7 a^2 d}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 257, normalized size = 1.84 \[ \frac {2128 \sin (c+d x)+6720 c \sin (2 (c+d x))+6720 d x \sin (2 (c+d x))-9144 \sin (2 (c+d x))+456 \sin (3 (c+d x))+3360 c \sin (4 (c+d x))+3360 d x \sin (4 (c+d x))-4572 \sin (4 (c+d x))+1528 \sin (5 (c+d x))+42 (280 c+280 d x-381) \cos (c+d x)+5504 \cos (2 (c+d x))+2520 c \cos (3 (c+d x))+2520 d x \cos (3 (c+d x))-3429 \cos (3 (c+d x))+2752 \cos (4 (c+d x))-840 c \cos (5 (c+d x))-840 d x \cos (5 (c+d x))+1143 \cos (5 (c+d x))+4032}{13440 a^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^7} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 140, normalized size = 1.00 \[ \frac {105 \, d x \cos \left (d x + c\right )^{5} - 210 \, d x \cos \left (d x + c\right )^{3} - 172 \, \cos \left (d x + c\right )^{4} + 86 \, \cos \left (d x + c\right )^{2} - {\left (210 \, d x \cos \left (d x + c\right )^{3} + 191 \, \cos \left (d x + c\right )^{4} - 129 \, \cos \left (d x + c\right )^{2} + 25\right )} \sin \left (d x + c\right ) - 10}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 155, normalized size = 1.11 \[ \frac {\frac {840 \, {\left (d x + c\right )}}{a^{2}} + \frac {35 \, {\left (9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 10\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} + \frac {1365 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 9345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 26600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 39410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30261 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11837 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1886}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 230, normalized size = 1.64 \[ -\frac {1}{12 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {3}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2}}-\frac {4}{7 d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {2}{d \,a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {8}{5 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {1}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{12 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {11}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {13}{8 a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 421, normalized size = 3.01 \[ \frac {2 \, {\left (\frac {\frac {279 \, \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 {1048 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {364 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1554 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {980 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {280 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {420 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {105 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + 96}{a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} + \frac {105 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.89, size = 156, normalized size = 1.11 \[ \frac {x}{a^2}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {148\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {104\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {2096\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {88\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {186\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {64}{35}}{a^2\,d\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}^7} \]
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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