Optimal. Leaf size=50 \[ -\frac {(A+2 C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {C \sin ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3013, 373} \[ -\frac {(A+2 C) \sin ^3(c+d x)}{3 d}+\frac {(A+C) \sin (c+d x)}{d}+\frac {C \sin ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3013
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (A+C-C x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(A+2 C) x^2+C x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac {(A+C) \sin (c+d x)}{d}-\frac {(A+2 C) \sin ^3(c+d x)}{3 d}+\frac {C \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 71, normalized size = 1.42 \[ -\frac {A \sin ^3(c+d x)}{3 d}+\frac {A \sin (c+d x)}{d}+\frac {C \sin ^5(c+d x)}{5 d}-\frac {2 C \sin ^3(c+d x)}{3 d}+\frac {C \sin (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 45, normalized size = 0.90 \[ \frac {{\left (3 \, C \cos \left (d x + c\right )^{4} + {\left (5 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{2} + 10 \, A + 8 \, C\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 57, normalized size = 1.14 \[ \frac {3 \, C \sin \left (d x + c\right )^{5} - 5 \, A \sin \left (d x + c\right )^{3} - 10 \, C \sin \left (d x + c\right )^{3} + 15 \, A \sin \left (d x + c\right ) + 15 \, C \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 54, normalized size = 1.08 \[ \frac {\frac {C \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 43, normalized size = 0.86 \[ \frac {3 \, C \sin \left (d x + c\right )^{5} - 5 \, {\left (A + 2 \, C\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (A + C\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 43, normalized size = 0.86 \[ \frac {\frac {C\,{\sin \left (c+d\,x\right )}^5}{5}+\left (-\frac {A}{3}-\frac {2\,C}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (A+C\right )\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.70, size = 105, normalized size = 2.10 \[ \begin {cases} \frac {2 A \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {A \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac {8 C \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 C \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {C \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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