Optimal. Leaf size=87 \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]
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Rubi [A] time = 0.138701, antiderivative size = 87, 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, 88} \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{5 a^3}{4 d (a-a \sin (c+d x))}-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (\sin (c+d x)+1)}{8 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sin (c+d x))^2 \tan ^3(c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^3}{a^3 (a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \frac{x^3}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^2 \operatorname{Subst}\left (\int \left (\frac{a^2}{2 (a-x)^3}-\frac{5 a}{4 (a-x)^2}+\frac{7}{8 (a-x)}-\frac{1}{8 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac{7 a^2 \log (1-\sin (c+d x))}{8 d}-\frac{a^2 \log (1+\sin (c+d x))}{8 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{5 a^3}{4 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.381434, size = 91, normalized size = 1.05 \[ \frac{a^2 \left (3 \tanh ^{-1}(\sin (c+d x))-6 \tan (c+d x) \sec ^3(c+d x)+\tan (c+d x) \left (8 \tan ^2(c+d x)+3\right ) \sec (c+d x)-2 \left (-\tan ^4(c+d x)+\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.079, size = 173, normalized size = 2. \begin{align*}{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{4\,d}}-{\frac{{a}^{2} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d}}-{\frac{3\,{a}^{2}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}{a}^{2}}{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 = 1.08351, size = 97, normalized size = 1.11 \begin{align*} -\frac{a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 7 \, a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (5 \, a^{2} \sin \left (d x + c\right ) - 4 \, a^{2}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52263, size = 312, normalized size = 3.59 \begin{align*} -\frac{10 \, a^{2} \sin \left (d x + c\right ) - 8 \, 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 ) + 7 \,{\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 )}{8 \,{\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 [A] time = 1.28445, size = 105, normalized size = 1.21 \begin{align*} -\frac{2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) + 14 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{21 \, a^{2} \sin \left (d x + c\right )^{2} - 22 \, a^{2} \sin \left (d x + c\right ) + 5 \, a^{2}}{{\left (\sin \left (d x + c\right ) - 1\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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