Optimal. Leaf size=73 \[ -\frac {(A+3 C) \cos ^5(e+f x)}{5 f}+\frac {(2 A+3 C) \cos ^3(e+f x)}{3 f}-\frac {(A+C) \cos (e+f x)}{f}+\frac {C \cos ^7(e+f x)}{7 f} \]
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Rubi [A] time = 0.06, antiderivative size = 73, 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} \[ -\frac {(A+3 C) \cos ^5(e+f x)}{5 f}+\frac {(2 A+3 C) \cos ^3(e+f x)}{3 f}-\frac {(A+C) \cos (e+f x)}{f}+\frac {C \cos ^7(e+f x)}{7 f} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3013
Rubi steps
\begin {align*} \int \sin ^5(e+f x) \left (A+C \sin ^2(e+f x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right )^2 \left (A+C-C x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (A \left (1+\frac {C}{A}\right )-(2 A+3 C) x^2+(A+3 C) x^4-C x^6\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {(A+C) \cos (e+f x)}{f}+\frac {(2 A+3 C) \cos ^3(e+f x)}{3 f}-\frac {(A+3 C) \cos ^5(e+f x)}{5 f}+\frac {C \cos ^7(e+f x)}{7 f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 109, normalized size = 1.49 \[ -\frac {5 A \cos (e+f x)}{8 f}+\frac {5 A \cos (3 (e+f x))}{48 f}-\frac {A \cos (5 (e+f x))}{80 f}-\frac {35 C \cos (e+f x)}{64 f}+\frac {7 C \cos (3 (e+f x))}{64 f}-\frac {7 C \cos (5 (e+f x))}{320 f}+\frac {C \cos (7 (e+f x))}{448 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 60, normalized size = 0.82 \[ \frac {15 \, C \cos \left (f x + e\right )^{7} - 21 \, {\left (A + 3 \, C\right )} \cos \left (f x + e\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \cos \left (f x + e\right )^{3} - 105 \, {\left (A + C\right )} \cos \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 97, normalized size = 1.33 \[ \frac {C \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} - \frac {{\left (4 \, A + 7 \, C\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac {{\left (20 \, A + 21 \, C\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} - \frac {{\left (16 \, A + 23 \, C\right )} \cos \left (f x + e\right )}{64 \, f} - \frac {3 \, {\left (2 \, A + C\right )} \cos \left (f x + e\right )}{16 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 74, normalized size = 1.01 \[ \frac {-\frac {C \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-\frac {A \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 60, normalized size = 0.82 \[ \frac {15 \, C \cos \left (f x + e\right )^{7} - 21 \, {\left (A + 3 \, C\right )} \cos \left (f x + e\right )^{5} + 35 \, {\left (2 \, A + 3 \, C\right )} \cos \left (f x + e\right )^{3} - 105 \, {\left (A + C\right )} \cos \left (f x + e\right )}{105 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.15, size = 58, normalized size = 0.79 \[ \frac {\frac {C\,{\cos \left (e+f\,x\right )}^7}{7}+\left (-\frac {A}{5}-\frac {3\,C}{5}\right )\,{\cos \left (e+f\,x\right )}^5+\left (\frac {2\,A}{3}+C\right )\,{\cos \left (e+f\,x\right )}^3+\left (-A-C\right )\,\cos \left (e+f\,x\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 7.13, size = 153, normalized size = 2.10 \[ \begin {cases} - \frac {A \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {4 A \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {8 A \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {C \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 C \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {8 C \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {16 C \cos ^{7}{\left (e + f x \right )}}{35 f} & \text {for}\: f \neq 0 \\x \left (A + C \sin ^{2}{\relax (e )}\right ) \sin ^{5}{\relax (e )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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