Optimal. Leaf size=71 \[ \frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a \sin (c+d x)}{d}+\frac {5 a \log (1-\sin (c+d x))}{4 d}-\frac {a \log (\sin (c+d x)+1)}{4 d} \]
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Rubi [A] time = 0.05, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2707, 88} \[ \frac {a^2}{2 d (a-a \sin (c+d x))}+\frac {a \sin (c+d x)}{d}+\frac {5 a \log (1-\sin (c+d x))}{4 d}-\frac {a \log (\sin (c+d x)+1)}{4 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int (a+a \sin (c+d x)) \tan ^3(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a-x)^2 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {a^2}{2 (a-x)^2}-\frac {5 a}{4 (a-x)}-\frac {a}{4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {5 a \log (1-\sin (c+d x))}{4 d}-\frac {a \log (1+\sin (c+d x))}{4 d}+\frac {a \sin (c+d x)}{d}+\frac {a^2}{2 d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 77, normalized size = 1.08 \[ -\frac {a \sin (c+d x) \tan ^2(c+d x)}{d}+\frac {a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d}-\frac {3 a \left (\tanh ^{-1}(\sin (c+d x))-\tan (c+d x) \sec (c+d x)\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 87, normalized size = 1.23 \[ -\frac {4 \, a \cos \left (d x + c\right )^{2} + {\left (a \sin \left (d x + c\right ) - a\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, {\left (a \sin \left (d x + c\right ) - a\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, a \sin \left (d x + c\right ) - 2 \, a}{4 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
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 [A] time = 0.15, size = 96, normalized size = 1.35 \[ \frac {a \left (\sin ^{5}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{2 d}+\frac {3 a \sin \left (d x +c \right )}{2 d}-\frac {3 a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 51, normalized size = 0.72 \[ -\frac {a \log \left (\sin \left (d x + c\right ) + 1\right ) - 5 \, a \log \left (\sin \left (d x + c\right ) - 1\right ) - 4 \, a \sin \left (d x + c\right ) + \frac {2 \, a}{\sin \left (d x + c\right ) - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.47, size = 154, normalized size = 2.17 \[ \frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{2\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{2\,d}+\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \sin {\left (c + d x \right )} \tan ^{3}{\left (c + d x \right )}\, dx + \int \tan ^{3}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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