Optimal. Leaf size=73 \[ \frac{a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.11585, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2855, 2669, 3767, 8} \[ \frac{a^2 (A-2 B) \tan (c+d x)}{3 d}+\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a \sin (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2855
Rule 2669
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac{1}{3} (a (A-2 B)) \int \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx\\ &=\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac{1}{3} \left (a^2 (A-2 B)\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}-\frac{\left (a^2 (A-2 B)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a^2 (A-2 B) \sec (c+d x)}{3 d}+\frac{(A+B) \sec ^3(c+d x) (a+a \sin (c+d x))^2}{3 d}+\frac{a^2 (A-2 B) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0183405, size = 121, normalized size = 1.66 \[ -\frac{a^2 A \tan ^3(c+d x)}{3 d}+\frac{2 a^2 A \sec ^3(c+d x)}{3 d}+\frac{a^2 A \tan (c+d x) \sec ^2(c+d x)}{d}+\frac{2 a^2 B \tan ^3(c+d x)}{3 d}-\frac{a^2 B \sec ^3(c+d x)}{3 d}+\frac{a^2 B \tan ^2(c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.095, size = 162, normalized size = 2.2 \begin{align*}{\frac{1}{d} \left ({\frac{{a}^{2}A \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+B{a}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{3\,\cos \left ( dx+c \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{3}} \right ) +{\frac{2\,{a}^{2}A}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}+{\frac{2\,B{a}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{a}^{2}A \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) +{\frac{B{a}^{2}}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01528, size = 146, normalized size = 2. \begin{align*} \frac{A a^{2} \tan \left (d x + c\right )^{3} + 2 \, B a^{2} \tan \left (d x + c\right )^{3} +{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} - \frac{{\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} B a^{2}}{\cos \left (d x + c\right )^{3}} + \frac{2 \, A a^{2}}{\cos \left (d x + c\right )^{3}} + \frac{B a^{2}}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92604, size = 294, normalized size = 4.03 \begin{align*} -\frac{{\left (A - 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, A - B\right )} a^{2} \cos \left (d x + c\right ) +{\left (A + B\right )} a^{2} -{\left ({\left (A - 2 \, B\right )} a^{2} \cos \left (d x + c\right ) -{\left (A + B\right )} a^{2}\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) +{\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \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 = 1.33111, size = 105, normalized size = 1.44 \begin{align*} -\frac{2 \,{\left (3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, A a^{2} - B a^{2}\right )}}{3 \, d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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