Optimal. Leaf size=50 \[ -\frac{(2 A+C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{A \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0676302, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4044, 3013, 373} \[ -\frac{(2 A+C) \sin ^3(c+d x)}{3 d}+\frac{(A+C) \sin (c+d x)}{d}+\frac{A \sin ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4044
Rule 3013
Rule 373
Rubi steps
\begin{align*} \int \cos ^5(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\int \cos ^3(c+d x) \left (C+A \cos ^2(c+d x)\right ) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (A+C-A x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (A \left (1+\frac{C}{A}\right )-(2 A+C) x^2+A x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{(A+C) \sin (c+d x)}{d}-\frac{(2 A+C) \sin ^3(c+d x)}{3 d}+\frac{A \sin ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.021858, size = 71, normalized size = 1.42 \[ \frac{A \sin ^5(c+d x)}{5 d}-\frac{2 A \sin ^3(c+d x)}{3 d}+\frac{A \sin (c+d x)}{d}-\frac{C \sin ^3(c+d x)}{3 d}+\frac{C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 54, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{A\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) }+{\frac{C \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.921677, size = 58, normalized size = 1.16 \begin{align*} \frac{3 \, A \sin \left (d x + c\right )^{5} - 5 \,{\left (2 \, A + C\right )} \sin \left (d x + c\right )^{3} + 15 \,{\left (A + C\right )} \sin \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.476398, size = 113, normalized size = 2.26 \begin{align*} \frac{{\left (3 \, A \cos \left (d x + c\right )^{4} +{\left (4 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, A + 10 \, C\right )} \sin \left (d x + c\right )}{15 \, d} \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 = 1.21075, size = 77, normalized size = 1.54 \begin{align*} \frac{3 \, A \sin \left (d x + c\right )^{5} - 10 \, A \sin \left (d x + c\right )^{3} - 5 \, C \sin \left (d x + c\right )^{3} + 15 \, A \sin \left (d x + c\right ) + 15 \, C \sin \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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