Optimal. Leaf size=152 \[ \frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.24, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2835, 2607, 14, 2611, 3768, 3770} \[ \frac {\tan ^8(c+d x)}{8 a d}+\frac {\tan ^6(c+d x)}{6 a d}+\frac {5 \tanh ^{-1}(\sin (c+d x))}{128 a d}-\frac {\tan ^5(c+d x) \sec ^3(c+d x)}{8 a d}+\frac {5 \tan ^3(c+d x) \sec ^3(c+d x)}{48 a d}-\frac {5 \tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac {5 \tan (c+d x) \sec (c+d x)}{128 a d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2607
Rule 2611
Rule 2835
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x) \tan ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^4(c+d x) \tan ^5(c+d x) \, dx}{a}-\frac {\int \sec ^3(c+d x) \tan ^6(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^5 \left (1+x^2\right ) \, 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 {5 \int \sec ^3(c+d x) \tan ^2(c+d x) \, dx}{16 a}+\frac {\operatorname {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,\tan (c+d x)\right )}{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 ^6(c+d x)}{6 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 ^6(c+d x)}{6 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 ^6(c+d x)}{6 a d}+\frac {\tan ^8(c+d x)}{8 a d}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 92, normalized size = 0.61 \[ \frac {-\frac {15}{\sin (c+d x)-1}-\frac {15}{(\sin (c+d x)-1)^2}+\frac {30}{(\sin (c+d x)+1)^2}-\frac {4}{(\sin (c+d x)-1)^3}-\frac {24}{(\sin (c+d x)+1)^3}+\frac {6}{(\sin (c+d x)+1)^4}+15 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 167, normalized size = 1.10 \[ -\frac {30 \, \cos \left (d x + c\right )^{6} - 266 \, \cos \left (d x + c\right )^{4} + 316 \, \cos \left (d x + c\right )^{2} - 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 22 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) - 112}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 136, normalized size = 0.89 \[ \frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (55 \, \sin \left (d x + c\right )^{3} - 225 \, \sin \left (d x + c\right )^{2} + 225 \, \sin \left (d x + c\right ) - 71\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {125 \, \sin \left (d x + c\right )^{4} + 500 \, \sin \left (d x + c\right )^{3} + 510 \, \sin \left (d x + c\right )^{2} + 212 \, \sin \left (d x + c\right ) + 29}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.40, size = 144, normalized size = 0.95 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {5}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {5 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}+\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {1}{16 a d \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {5}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 173, normalized size = 1.14 \[ -\frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{6} + 15 \, \sin \left (d x + c\right )^{5} + 88 \, \sin \left (d x + c\right )^{4} - 8 \, \sin \left (d x + c\right )^{3} - 63 \, \sin \left (d x + c\right )^{2} + \sin \left (d x + c\right ) + 16\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 17.21, size = 388, normalized size = 2.55 \[ \frac {5\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d}+\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 {413\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{48}+\frac {157\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {413\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48}-\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 )} \]
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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