Optimal. Leaf size=133 \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}-\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {39 a \log (1-\sin (c+d x))}{16 d}-\frac {9 a \log (\sin (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.11, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2836, 12, 88} \[ \frac {a^3}{8 d (a-a \sin (c+d x))^2}-\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a \sin (c+d x)+a)}-\frac {a \sin ^2(c+d x)}{2 d}-\frac {a \sin (c+d x)}{d}-\frac {39 a \log (1-\sin (c+d x))}{16 d}-\frac {9 a \log (\sin (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \sin (c+d x) (a+a \sin (c+d x)) \tan ^5(c+d x) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x^6}{a^6 (a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{(a-x)^3 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a+\frac {a^4}{4 (a-x)^3}-\frac {5 a^3}{4 (a-x)^2}+\frac {39 a^2}{16 (a-x)}-x+\frac {a^3}{8 (a+x)^2}-\frac {9 a^2}{16 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=-\frac {39 a \log (1-\sin (c+d x))}{16 d}-\frac {9 a \log (1+\sin (c+d x))}{16 d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin ^2(c+d x)}{2 d}+\frac {a^3}{8 d (a-a \sin (c+d x))^2}-\frac {5 a^2}{4 d (a-a \sin (c+d x))}-\frac {a^2}{8 d (a+a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 133, normalized size = 1.00 \[ -\frac {a \sin (c+d x) \tan ^4(c+d x)}{d}-\frac {a \left (2 \sin ^2(c+d x)-\sec ^4(c+d x)+6 \sec ^2(c+d x)+12 \log (\cos (c+d x))\right )}{4 d}-\frac {5 a \left (6 \tan (c+d x) \sec ^3(c+d x)-8 \tan ^3(c+d x) \sec (c+d x)-3 \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 172, normalized size = 1.29 \[ \frac {8 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - 9 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 39 \, {\left (a \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - a \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, a \cos \left (d x + c\right )^{4} + 6 \, a \cos \left (d x + c\right )^{2} - 3 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 113, normalized size = 0.85 \[ -\frac {16 \, a \sin \left (d x + c\right )^{2} + 18 \, a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 78 \, a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + 32 \, a \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, a \sin \left (d x + c\right ) + 7 \, a\right )}}{\sin \left (d x + c\right ) + 1} - \frac {117 \, a \sin \left (d x + c\right )^{2} - 194 \, a \sin \left (d x + c\right ) + 81 \, a}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 205, normalized size = 1.54 \[ \frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a \left (\sin ^{8}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}-\frac {a \left (\sin ^{6}\left (d x +c \right )\right )}{2 d}-\frac {3 a \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {3 a \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {3 a \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a \left (\sin ^{7}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {3 a \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {5 a \left (\sin ^{3}\left (d x +c \right )\right )}{8 d}-\frac {15 a \sin \left (d x +c \right )}{8 d}+\frac {15 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 106, normalized size = 0.80 \[ -\frac {8 \, a \sin \left (d x + c\right )^{2} + 9 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) + 39 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, a \sin \left (d x + c\right )^{2} + 3 \, a \sin \left (d x + c\right ) - 10 \, a\right )}}{\sin \left (d x + c\right )^{3} - \sin \left (d x + c\right )^{2} - \sin \left (d x + c\right ) + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.82, size = 286, normalized size = 2.15 \[ \frac {-\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {15\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {9\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{8\,d}-\frac {39\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{8\,d}+\frac {3\,a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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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