Optimal. Leaf size=64 \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.11848, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2836, 12, 88, 206} \[ \frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 12
Rule 88
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^2 \tan ^2(c+d x) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{x^2}{a^2 (a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \frac{x^2}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{a}{2 (a-x)^3}-\frac{3}{4 (a-x)^2}+\frac{1}{4 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}+\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4}{4 d (a-a \sin (c+d x))^2}-\frac{3 a^3}{4 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.10705, size = 39, normalized size = 0.61 \[ \frac{a^2 \left (\frac{3 \sin (c+d x)-2}{(\sin (c+d x)-1)^2}+\tanh ^{-1}(\sin (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.075, size = 174, normalized size = 2.7 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5}{a}^{2}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d}}-{\frac{{a}^{2}\sin \left ( dx+c \right ) }{4\,d}}+{\frac{{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}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08203, size = 97, normalized size = 1.52 \begin{align*} \frac{a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) + \frac{2 \,{\left (3 \, a^{2} \sin \left (d x + c\right ) - 2 \, 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 [B] time = 1.38907, size = 308, normalized size = 4.81 \begin{align*} -\frac{6 \, a^{2} \sin \left (d x + c\right ) - 4 \, 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 ) +{\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 = 1.26176, size = 104, normalized size = 1.62 \begin{align*} \frac{2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 2 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac{3 \, a^{2} \sin \left (d x + c\right )^{2} + 6 \, 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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