Optimal. Leaf size=150 \[ -\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^6(c+d x)}{6 a d}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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Rubi [A] time = 0.23, antiderivative size = 150, 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, 2611, 3768, 3770, 2607, 14} \[ -\frac {\tan ^8(c+d x)}{8 a d}-\frac {\tan ^6(c+d x)}{6 a d}+\frac {3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {\tan ^3(c+d x) \sec ^5(c+d x)}{8 a d}-\frac {\tan (c+d x) \sec ^5(c+d x)}{16 a d}+\frac {\tan (c+d x) \sec ^3(c+d x)}{64 a d}+\frac {3 \tan (c+d x) \sec (c+d x)}{128 a d} \]
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 ^3(c+d x) \tan ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int \sec ^5(c+d x) \tan ^4(c+d x) \, dx}{a}-\frac {\int \sec ^4(c+d x) \tan ^5(c+d x) \, dx}{a}\\ &=\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}-\frac {3 \int \sec ^5(c+d x) \tan ^2(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 {\sec ^5(c+d x) \tan (c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^3(c+d x)}{8 a d}+\frac {\int \sec ^5(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 {\sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^3(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 {3 \int \sec ^3(c+d x) \, dx}{64 a}\\ &=\frac {3 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^3(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 {3 \int \sec (c+d x) \, dx}{128 a}\\ &=\frac {3 \tanh ^{-1}(\sin (c+d x))}{128 a d}+\frac {3 \sec (c+d x) \tan (c+d x)}{128 a d}+\frac {\sec ^3(c+d x) \tan (c+d x)}{64 a d}-\frac {\sec ^5(c+d x) \tan (c+d x)}{16 a d}+\frac {\sec ^5(c+d x) \tan ^3(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 = 0.67, size = 101, normalized size = 0.67 \[ \frac {\frac {-9 \sin ^6(c+d x)-9 \sin ^5(c+d x)+24 \sin ^4(c+d x)-72 \sin ^3(c+d x)-39 \sin ^2(c+d x)+25 \sin (c+d x)+16}{(\sin (c+d x)-1)^3 (\sin (c+d x)+1)^4}+9 \tanh ^{-1}(\sin (c+d x))}{384 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 167, normalized size = 1.11 \[ -\frac {18 \, \cos \left (d x + c\right )^{6} - 6 \, \cos \left (d x + c\right )^{4} + 36 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 9 \, {\left (\cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{6}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (9 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 56\right )} \sin \left (d x + c\right ) - 16}{768 \, {\left (a d \cos \left (d x + c\right )^{6} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 136, normalized size = 0.91 \[ \frac {\frac {36 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac {36 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac {2 \, {\left (33 \, \sin \left (d x + c\right )^{3} - 87 \, \sin \left (d x + c\right )^{2} + 39 \, \sin \left (d x + c\right ) - 1\right )}}{a {\left (\sin \left (d x + c\right ) - 1\right )}^{3}} - \frac {75 \, \sin \left (d x + c\right )^{4} + 396 \, \sin \left (d x + c\right )^{3} + 786 \, \sin \left (d x + c\right )^{2} + 556 \, \sin \left (d x + c\right ) + 139}{a {\left (\sin \left (d x + c\right ) + 1\right )}^{4}}}{3072 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 162, normalized size = 1.08 \[ -\frac {1}{96 a d \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {3}{128 a d \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {1}{128 a d \left (\sin \left (d x +c \right )-1\right )}-\frac {3 \ln \left (\sin \left (d x +c \right )-1\right )}{256 a d}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{4}}+\frac {1}{24 a d \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {1}{64 a d \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{32 a d \left (1+\sin \left (d x +c \right )\right )}+\frac {3 \ln \left (1+\sin \left (d x +c \right )\right )}{256 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 175, normalized size = 1.17 \[ -\frac {\frac {2 \, {\left (9 \, \sin \left (d x + c\right )^{6} + 9 \, \sin \left (d x + c\right )^{5} - 24 \, \sin \left (d x + c\right )^{4} + 72 \, \sin \left (d x + c\right )^{3} + 39 \, \sin \left (d x + c\right )^{2} - 25 \, \sin \left (d x + c\right ) - 16\right )}}{a \sin \left (d x + c\right )^{7} + a \sin \left (d x + c\right )^{6} - 3 \, a \sin \left (d x + c\right )^{5} - 3 \, a \sin \left (d x + c\right )^{4} + 3 \, a \sin \left (d x + c\right )^{3} + 3 \, a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a} - \frac {9 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac {9 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{768 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 17.15, size = 388, normalized size = 2.59 \[ \frac {3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a\,d}+\frac {-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{32}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{32}+\frac {387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{48}+\frac {299\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{48}+\frac {43\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{48}+\frac {387\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{64}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32}+\frac {7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{32}-\frac {3\,\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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