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 = 0.07, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 373
Rule 3013
Rule 4044
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 0.44, size = 45, normalized size = 0.90 \[ \frac {{\left (3 \, A \cos \left (d x + c\right )^{4} + {\left (4 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{2} + 8 \, A + 10 \, C\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 57, normalized size = 1.14 \[ \frac {3 \, A \sin \left (d x + c\right )^{5} - 10 \, A \sin \left (d x + c\right )^{3} - 5 \, C \sin \left (d x + c\right )^{3} + 15 \, A \sin \left (d x + c\right ) + 15 \, C \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.50, size = 54, normalized size = 1.08 \[ \frac {\frac {A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {C \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 43, normalized size = 0.86 \[ \frac {3 \, A \sin \left (d x + c\right )^{5} - 5 \, {\left (2 \, A + C\right )} \sin \left (d x + c\right )^{3} + 15 \, {\left (A + C\right )} \sin \left (d x + c\right )}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 43, normalized size = 0.86 \[ \frac {\frac {A\,{\sin \left (c+d\,x\right )}^5}{5}+\left (-\frac {2\,A}{3}-\frac {C}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (A+C\right )\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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