Optimal. Leaf size=119 \[ \frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {9 a^3}{4 d (a-a \sin (c+d x))}-\frac {a^2 \sin ^2(c+d x)}{2 d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {9 a^3}{4 d (a-a \sin (c+d x))}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac {a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^2 \tan ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a+\frac {a^4}{2 (a-x)^3}-\frac {9 a^3}{4 (a-x)^2}+\frac {31 a^2}{8 (a-x)}-x-\frac {a^2}{8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {31 a^2 \log (1-\sin (c+d x))}{8 d}-\frac {a^2 \log (1+\sin (c+d x))}{8 d}-\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {a^4}{4 d (a-a \sin (c+d x))^2}-\frac {9 a^3}{4 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 75, normalized size = 0.63 \[ -\frac {a^2 \left (4 \sin ^2(c+d x)+16 \sin (c+d x)-\frac {18}{\sin (c+d x)-1}-\frac {2}{(\sin (c+d x)-1)^2}+31 \log (1-\sin (c+d x))+\log (\sin (c+d x)+1)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 168, normalized size = 1.41 \[ \frac {4 \, a^{2} \cos \left (d x + c\right )^{4} + 22 \, a^{2} \cos \left (d x + c\right )^{2} - 12 \, a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 31 \, {\left (a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \sin \left (d x + c\right ) - 2 \, a^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, a^{2} \cos \left (d x + c\right )^{2} - 5 \, a^{2}\right )} \sin \left (d x + c\right )}{8 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.20, size = 261, normalized size = 2.19 \[ \frac {a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {a^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}-\frac {a^{2} \left (\sin ^{6}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{2} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {4 a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{2} \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}-\frac {3 a^{2} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d}-\frac {5 a^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {15 a^{2} \sin \left (d x +c \right )}{4 d}+\frac {15 a^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 96, normalized size = 0.81 \[ -\frac {4 \, a^{2} \sin \left (d x + c\right )^{2} + a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 31 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + 16 \, a^{2} \sin \left (d x + c\right ) - \frac {2 \, {\left (9 \, a^{2} \sin \left (d x + c\right ) - 8 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.57, size = 283, normalized size = 2.38 \[ \frac {4\,a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{4\,d}-\frac {\frac {15\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}-22\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}-36\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {61\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-22\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {15\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {31\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int 2 \sin {\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \sin ^{2}{\left (c + d x \right )} \tan ^{5}{\left (c + d x \right )}\, dx + \int \tan ^{5}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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