Optimal. Leaf size=134 \[ -\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}-\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}+\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.24, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2835, 2611, 3768, 3770, 2607, 30} \[ -\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}-\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}+\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}-\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec (c+d x) \tan ^6(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^3(c+d x) \tan ^6(c+d x) \, dx}{a}-\frac {\int \sec ^2(c+d x) \tan ^7(c+d x) \, dx}{a}\\ &=\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {5 \int \sec ^3(c+d x) \tan ^4(c+d x) \, dx}{8 a}-\frac {\operatorname {Subst}\left (\int x^7 \, dx,x,\tan (c+d x)\right )}{a d}\\ &=-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}+\frac {5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a}\\ &=\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=-\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}-\frac {5 \int \sec (c+d x) \, dx}{128 a}\\ &=-\frac {5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac {5 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {5 \sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {5 \sec ^3(c+d x) \tan ^3(c+d x)}{48 a d}+\frac {\sec ^3(c+d x) \tan ^5(c+d x)}{8 a d}-\frac {\tan ^8(c+d x)}{8 a d}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 101, normalized size = 0.75 \[ -\frac {\frac {-15 \sin ^6(c+d x)+177 \sin ^5(c+d x)+104 \sin ^4(c+d x)-184 \sin ^3(c+d x)-129 \sin ^2(c+d x)+63 \sin (c+d x)+48}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 167, normalized size = 1.25 \[ \frac {30 \, \cos \left (d x + c\right )^{6} + 118 \, \cos \left (d x + c\right )^{4} - 68 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (177 \, \cos \left (d x + c\right )^{4} - 170 \, \cos \left (d x + c\right )^{2} + 56\right )} \sin \left (d x + c\right ) + 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 136, normalized size = 1.01 \[ -\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )^{2} - 111 \, \sin \left (d x + c\right ) + 57\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 980 \, \sin \left (d x + c\right )^{3} + 1662 \, \sin \left (d x + c\right )^{2} + 1140 \, \sin \left (d x + c\right ) + 285}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 162, normalized size = 1.21 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {7}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {15}{128 a d \left (\sin \left (d x +c \right )-1\right )}+\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{12 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {11}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5}{32 a d \left (1+\sin \left (d x +c \right )\right )}-\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 175, normalized size = 1.31 \[ \frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} - 177 \, \sin \left (d x + c\right )^{5} - 104 \, \sin \left (d x + c\right )^{4} + 184 \, \sin \left (d x + c\right )^{3} + 129 \, \sin \left (d x + c\right )^{2} - 63 \, \sin \left (d x + c\right ) - 48\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 17.12, size = 388, normalized size = 2.90 \[ \frac {\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{96}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{96}+\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{192}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{16}+\frac {289\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {33\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{16}+\frac {113\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{192}-\frac {85\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{96}-\frac {35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}+\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+30\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+9\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d} \]
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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