Optimal. Leaf size=18 \[ \frac {\cos (e+f x) \sin ^6(e+f x)}{f} \]
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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {3090}
\begin {gather*} \frac {\sin ^6(e+f x) \cos (e+f x)}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 3090
Rubi steps
\begin {align*} \int \sin ^5(e+f x) \left (6-7 \sin ^2(e+f x)\right ) \, dx &=\frac {\cos (e+f x) \sin ^6(e+f x)}{f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(18)=36\).
time = 0.03, size = 59, normalized size = 3.28 \begin {gather*} \frac {5 \cos (e+f x)}{64 f}-\frac {9 \cos (3 (e+f x))}{64 f}+\frac {5 \cos (5 (e+f x))}{64 f}-\frac {\cos (7 (e+f x))}{64 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(70\) vs.
\(2(18)=36\).
time = 0.29, size = 71, normalized size = 3.94
method | result | size |
norman | \(\frac {-\frac {64 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {64 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{7}}\) | \(50\) |
risch | \(\frac {5 \cos \left (f x +e \right )}{64 f}-\frac {\cos \left (7 f x +7 e \right )}{64 f}+\frac {5 \cos \left (5 f x +5 e \right )}{64 f}-\frac {9 \cos \left (3 f x +3 e \right )}{64 f}\) | \(55\) |
derivativedivides | \(\frac {\left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )-\frac {6 \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(71\) |
default | \(\frac {\left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )-\frac {6 \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}}{f}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (20) = 40\).
time = 0.30, size = 46, normalized size = 2.56 \begin {gather*} -\frac {\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs.
\(2 (20) = 40\).
time = 0.38, size = 46, normalized size = 2.56 \begin {gather*} -\frac {\cos \left (f x + e\right )^{7} - 3 \, \cos \left (f x + e\right )^{5} + 3 \, \cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (15) = 30\).
time = 0.64, size = 141, normalized size = 7.83 \begin {gather*} \begin {cases} \frac {7 \sin ^{6}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {14 \sin ^{4}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {6 \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {56 \sin ^{2}{\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {8 \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} + \frac {16 \cos ^{7}{\left (e + f x \right )}}{5 f} - \frac {16 \cos ^{5}{\left (e + f x \right )}}{5 f} & \text {for}\: f \neq 0 \\x \left (6 - 7 \sin ^{2}{\left (e \right )}\right ) \sin ^{5}{\left (e \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (20) = 40\).
time = 0.47, size = 58, normalized size = 3.22 \begin {gather*} -\frac {\cos \left (7 \, f x + 7 \, e\right )}{64 \, f} + \frac {5 \, \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} - \frac {9 \, \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac {5 \, \cos \left (f x + e\right )}{64 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.47, size = 23, normalized size = 1.28 \begin {gather*} -\frac {\cos \left (e+f\,x\right )\,{\left ({\cos \left (e+f\,x\right )}^2-1\right )}^3}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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