Optimal. Leaf size=129 \[ \frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac {23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]
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Rubi [A] time = 0.29, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3872, 2869, 2757, 2633, 2635, 8} \[ \frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {\sin (c+d x) \cos ^5(c+d x)}{6 a^3 d}-\frac {23 \sin (c+d x) \cos ^3(c+d x)}{24 a^3 d}-\frac {23 \sin (c+d x) \cos (c+d x)}{16 a^3 d}-\frac {23 x}{16 a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2757
Rule 2869
Rule 3872
Rubi steps
\begin {align*} \int \frac {\sin ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos ^3(c+d x) \sin ^6(c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac {\int \cos ^3(c+d x) (-a+a \cos (c+d x))^3 \, dx}{a^6}\\ &=-\frac {\int \left (-a^3 \cos ^3(c+d x)+3 a^3 \cos ^4(c+d x)-3 a^3 \cos ^5(c+d x)+a^3 \cos ^6(c+d x)\right ) \, dx}{a^6}\\ &=\frac {\int \cos ^3(c+d x) \, dx}{a^3}-\frac {\int \cos ^6(c+d x) \, dx}{a^3}-\frac {3 \int \cos ^4(c+d x) \, dx}{a^3}+\frac {3 \int \cos ^5(c+d x) \, dx}{a^3}\\ &=-\frac {3 \cos ^3(c+d x) \sin (c+d x)}{4 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {5 \int \cos ^4(c+d x) \, dx}{6 a^3}-\frac {9 \int \cos ^2(c+d x) \, dx}{4 a^3}-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{a^3 d}\\ &=\frac {4 \sin (c+d x)}{a^3 d}-\frac {9 \cos (c+d x) \sin (c+d x)}{8 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {5 \int \cos ^2(c+d x) \, dx}{8 a^3}-\frac {9 \int 1 \, dx}{8 a^3}\\ &=-\frac {9 x}{8 a^3}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}-\frac {5 \int 1 \, dx}{16 a^3}\\ &=-\frac {23 x}{16 a^3}+\frac {4 \sin (c+d x)}{a^3 d}-\frac {23 \cos (c+d x) \sin (c+d x)}{16 a^3 d}-\frac {23 \cos ^3(c+d x) \sin (c+d x)}{24 a^3 d}-\frac {\cos ^5(c+d x) \sin (c+d x)}{6 a^3 d}-\frac {7 \sin ^3(c+d x)}{3 a^3 d}+\frac {3 \sin ^5(c+d x)}{5 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.96, size = 111, normalized size = 0.86 \[ \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (5040 \sin (c+d x)-1890 \sin (2 (c+d x))+760 \sin (3 (c+d x))-270 \sin (4 (c+d x))+72 \sin (5 (c+d x))-10 \sin (6 (c+d x))+9 \tan \left (\frac {c}{2}\right )-2760 d x\right )}{240 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 70, normalized size = 0.54 \[ -\frac {345 \, d x + {\left (40 \, \cos \left (d x + c\right )^{5} - 144 \, \cos \left (d x + c\right )^{4} + 230 \, \cos \left (d x + c\right )^{3} - 272 \, \cos \left (d x + c\right )^{2} + 345 \, \cos \left (d x + c\right ) - 544\right )} \sin \left (d x + c\right )}{240 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 113, normalized size = 0.88 \[ -\frac {\frac {345 \, {\left (d x + c\right )}}{a^{3}} - \frac {2 \, {\left (1575 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 3165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5814 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 4554 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1955 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 345 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{3}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 222, normalized size = 1.72 \[ \frac {105 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {211 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {969 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {759 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {391 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {23 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {23 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.50, size = 292, normalized size = 2.26 \[ \frac {\frac {\frac {345 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {1955 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {4554 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5814 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {3165 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {1575 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}}}{a^{3} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{3} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {345 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 106, normalized size = 0.82 \[ \frac {\frac {105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {211\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}+\frac {969\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}+\frac {759\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {391\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {23\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{a^3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}-\frac {23\,x}{16\,a^3} \]
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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