Optimal. Leaf size=63 \[ -\frac {2 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {4 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \]
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Rubi [A]
time = 0.04, antiderivative size = 63, 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 = {2702, 276}
\begin {gather*} -\frac {2 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {4 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rule 2702
Rubi steps
\begin {align*} \int (b \sec (e+f x))^{5/2} \sin ^5(e+f x) \, dx &=\frac {b^5 \text {Subst}\left (\int \frac {\left (-1+\frac {x^2}{b^2}\right )^2}{x^{7/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac {b^5 \text {Subst}\left (\int \left (\frac {1}{x^{7/2}}-\frac {2}{b^2 x^{3/2}}+\frac {\sqrt {x}}{b^4}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=-\frac {2 b^5}{5 f (b \sec (e+f x))^{5/2}}+\frac {4 b^3}{f \sqrt {b \sec (e+f x)}}+\frac {2 b (b \sec (e+f x))^{3/2}}{3 f}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 42, normalized size = 0.67 \begin {gather*} \frac {b (151+108 \cos (2 (e+f x))-3 \cos (4 (e+f x))) (b \sec (e+f x))^{3/2}}{60 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. \(521\) vs.
\(2(53)=106\).
time = 0.26, size = 522, normalized size = 8.29
method | result | size |
default | \(-\frac {\left (-1+\cos \left (f x +e \right )\right )^{2} \left (6 \left (\cos ^{4}\left (f x +e \right )\right )+15 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1\right )}{\sin \left (f x +e \right )^{2}}\right ) \left (\cos ^{2}\left (f x +e \right )\right )-15 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1}{\sin \left (f x +e \right )^{2}}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+15 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1\right )}{\sin \left (f x +e \right )^{2}}\right ) \cos \left (f x +e \right )-15 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \ln \left (-\frac {2 \left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-\left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )-2 \sqrt {-\frac {\cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-1}{\sin \left (f x +e \right )^{2}}\right ) \cos \left (f x +e \right )-60 \left (\cos ^{2}\left (f x +e \right )\right )-10\right ) \cos \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right )^{2} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {5}{2}}}{15 f \sin \left (f x +e \right )^{4}}\) | \(522\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 55, normalized size = 0.87 \begin {gather*} \frac {2 \, {\left (5 \, \left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {3}{2}} - \frac {3 \, {\left (b^{4} - \frac {10 \, b^{4}}{\cos \left (f x + e\right )^{2}}\right )}}{\left (\frac {b}{\cos \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )} b}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 61, normalized size = 0.97 \begin {gather*} -\frac {2 \, {\left (3 \, b^{2} \cos \left (f x + e\right )^{4} - 30 \, b^{2} \cos \left (f x + e\right )^{2} - 5 \, b^{2}\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}}}{15 \, f \cos \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.55, size = 80, normalized size = 1.27 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {b \cos \left (f x + e\right )} b^{2} \cos \left (f x + e\right )^{2} - 30 \, \sqrt {b \cos \left (f x + e\right )} b^{2} - \frac {5 \, b^{3}}{\sqrt {b \cos \left (f x + e\right )} \cos \left (f x + e\right )}\right )} \mathrm {sgn}\left (\cos \left (f x + e\right )\right )}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\sin \left (e+f\,x\right )}^5\,{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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