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 = 0.118999, antiderivative size = 78, normalized size of antiderivative = 1., 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 2836
Rule 12
Rule 43
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*}
Mathematica [A] time = 0.219748, 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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Maple [B] time = 0.091, size = 237, normalized size = 3. \begin{align*}{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{7}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d}}-{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{d}}-3\,{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d}}+3\,{\frac{{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{3\,{a}^{3} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-3\,{\frac{{a}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{3\,{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{5}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991603, size = 95, normalized size = 1.22 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44588, size = 269, normalized size = 3.45 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.21339, size = 240, normalized size = 3.08 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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