Optimal. Leaf size=78 \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}-\frac {3 a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \sin (c+d x)}{d}-\frac {3 a^3 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 43} \[ \frac {a^5}{2 d (a-a \sin (c+d x))^2}-\frac {3 a^4}{d (a-a \sin (c+d x))}-\frac {a^3 \sin (c+d x)}{d}-\frac {3 a^3 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2836
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^3 \tan ^3(c+d x) \, dx &=\frac {a^5 \operatorname {Subst}\left (\int \frac {x^3}{a^3 (a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {x^3}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (-1+\frac {a^3}{(a-x)^3}-\frac {3 a^2}{(a-x)^2}+\frac {3 a}{a-x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \log (1-\sin (c+d x))}{d}-\frac {a^3 \sin (c+d x)}{d}+\frac {a^5}{2 d (a-a \sin (c+d x))^2}-\frac {3 a^4}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 53, normalized size = 0.68 \[ -\frac {a^3 \left (2 \sin (c+d x)+\frac {5-6 \sin (c+d x)}{(\sin (c+d x)-1)^2}+6 \log (1-\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 110, normalized size = 1.41 \[ \frac {4 \, a^{3} \cos \left (d x + c\right )^{2} + a^{3} - 6 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) - 2 \, a^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{3} \cos \left (d x + c\right )^{2} + a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 178, normalized size = 2.28 \[ \frac {6 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 12 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {25 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 108 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 170 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 108 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.32, size = 237, normalized size = 3.04 \[ \frac {a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{3} \left (\sin ^{7}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}-\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d}-\frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a^{3} \sin \left (d x +c \right )}{d}+\frac {3 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {3 a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {3 a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{3} \left (\sin ^{5}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {a^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 70, normalized size = 0.90 \[ -\frac {6 \, a^{3} \log \left (\sin \left (d x + c\right ) - 1\right ) + 2 \, a^{3} \sin \left (d x + c\right ) - \frac {6 \, a^{3} \sin \left (d x + c\right ) - 5 \, a^{3}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.13, size = 205, normalized size = 2.63 \[ \frac {3\,a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {6\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d}-\frac {6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+20\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-18\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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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