Optimal. Leaf size=91 \[ -\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {3 \sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \]
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Rubi [A] time = 0.29, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2875, 2873, 2606, 14, 2607, 270} \[ -\frac {2 \tan ^7(c+d x)}{7 a^2 d}-\frac {2 \tan ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {3 \sec ^5(c+d x)}{5 a^2 d}+\frac {\sec ^3(c+d x)}{3 a^2 d} \]
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 (c+d x) \tan ^3(c+d x)}{(a+a \sin (c+d x))^2} \, dx &=\frac {\int \sec ^5(c+d x) (a-a \sin (c+d x))^2 \tan ^3(c+d x) \, dx}{a^4}\\ &=\frac {\int \left (a^2 \sec ^5(c+d x) \tan ^3(c+d x)-2 a^2 \sec ^4(c+d x) \tan ^4(c+d x)+a^2 \sec ^3(c+d x) \tan ^5(c+d x)\right ) \, dx}{a^4}\\ &=\frac {\int \sec ^5(c+d x) \tan ^3(c+d x) \, dx}{a^2}+\frac {\int \sec ^3(c+d x) \tan ^5(c+d x) \, dx}{a^2}-\frac {2 \int \sec ^4(c+d x) \tan ^4(c+d x) \, dx}{a^2}\\ &=\frac {\operatorname {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}+\frac {\operatorname {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\sec (c+d x)\right )}{a^2 d}-\frac {2 \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,\tan (c+d x)\right )}{a^2 d}\\ &=\frac {\sec ^3(c+d x)}{3 a^2 d}-\frac {3 \sec ^5(c+d x)}{5 a^2 d}+\frac {2 \sec ^7(c+d x)}{7 a^2 d}-\frac {2 \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 = 0.33, size = 126, normalized size = 1.38 \[ \frac {\sec ^3(c+d x) (448 \sin (c+d x)-104 \sin (2 (c+d x))-144 \sin (3 (c+d x))-52 \sin (4 (c+d x))+48 \sin (5 (c+d x))-182 \cos (c+d x)-736 \cos (2 (c+d x))-39 \cos (3 (c+d x))+192 \cos (4 (c+d x))+13 \cos (5 (c+d x))+672)}{6720 a^2 d (\sin (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 104, normalized size = 1.14 \[ -\frac {24 \, \cos \left (d x + c\right )^{4} - 47 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + 5\right )} \sin \left (d x + c\right ) + 25}{105 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 120, normalized size = 1.32 \[ -\frac {\frac {35 \, {\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}} - \frac {105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1015 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1330 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1302 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 469 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 67}{a^{2} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{7}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 130, normalized size = 1.43 \[ \frac {-\frac {1}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {4}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}+\frac {14}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {7}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 336, normalized size = 3.69 \[ \frac {4 \, {\left (\frac {4 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {91 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}}{105 \, {\left (a^{2} + \frac {4 \, a^{2} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a^{2} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {14 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {14 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {8 \, a^{2} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {3 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.07, size = 207, normalized size = 2.27 \[ \frac {\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{105}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}+\frac {4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}-\frac {32\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {52\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+4\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{a^2\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^3\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^7} \]
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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