Optimal. Leaf size=77 \[ \frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}+\frac{a^2 (A-B) \tanh ^{-1}(\sin (c+d x))}{4 d} \]
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Rubi [A] time = 0.118901, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ \frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}+\frac{a^2 (A-B) \tanh ^{-1}(\sin (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 77
Rule 206
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a-x)^3 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 \operatorname{Subst}\left (\int \left (\frac{A+B}{2 a (a-x)^3}+\frac{A-B}{4 a^2 (a-x)^2}+\frac{A-B}{4 a^2 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}+\frac{\left (a^3 (A-B)\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac{a^2 (A-B) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4 (A+B)}{4 d (a-a \sin (c+d x))^2}+\frac{a^3 (A-B)}{4 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.136888, size = 75, normalized size = 0.97 \[ \frac{a^5 \left (\frac{A-B}{4 a^2 (a-a \sin (c+d x))}+\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{4 a^3}+\frac{A+B}{4 a (a-a \sin (c+d x))^2}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 281, normalized size = 3.7 \begin{align*}{\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}A\sin \left ( dx+c \right ) }{8\,d}}+{\frac{{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{{a}^{2}A}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{B{a}^{2}\sin \left ( dx+c \right ) }{4\,d}}-{\frac{B{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{4\,d}}+{\frac{3\,{a}^{2}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{B{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 = 1.03936, size = 117, normalized size = 1.52 \begin{align*} \frac{{\left (A - B\right )} a^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (A - B\right )} a^{2} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left ({\left (A - B\right )} a^{2} \sin \left (d x + c\right ) - 2 \, A 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 = 1.81465, size = 385, normalized size = 5. \begin{align*} \frac{2 \,{\left (A - B\right )} a^{2} \sin \left (d x + c\right ) - 4 \, A a^{2} +{\left ({\left (A - B\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (A - B\right )} a^{2} \sin \left (d x + c\right ) - 2 \,{\left (A - B\right )} a^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left ({\left (A - B\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \,{\left (A - B\right )} a^{2} \sin \left (d x + c\right ) - 2 \,{\left (A - B\right )} 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 = 1.36705, size = 176, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (A a^{2} - B a^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 2 \,{\left (A a^{2} - B a^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) + \frac{3 \, A a^{2} \sin \left (d x + c\right )^{2} - 3 \, B a^{2} \sin \left (d x + c\right )^{2} - 10 \, A a^{2} \sin \left (d x + c\right ) + 10 \, B a^{2} \sin \left (d x + c\right ) + 11 \, A a^{2} - 3 \, B 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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