Optimal. Leaf size=98 \[ \frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2707, 3854,
3856, 2720} \begin {gather*} \frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2707
Rule 2720
Rule 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {\sin ^2(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {2}{7} \int \frac {1}{(b \sec (e+f x))^{3/2}} \, dx\\ &=-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}+\frac {2 \int \sqrt {b \sec (e+f x)} \, dx}{21 b^2}\\ &=-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}+\frac {\left (2 \sqrt {\cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{21 b^2}\\ &=\frac {4 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {b \sec (e+f x)}}{21 b^2 f}-\frac {2 b \sin (e+f x)}{7 f (b \sec (e+f x))^{5/2}}+\frac {4 \sin (e+f x)}{21 b f \sqrt {b \sec (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 71, normalized size = 0.72 \begin {gather*} \frac {\sec ^2(e+f x) \left (16 \sqrt {\cos (e+f x)} F\left (\left .\frac {1}{2} (e+f x)\right |2\right )+2 \sin (2 (e+f x))-3 \sin (4 (e+f x))\right )}{84 f (b \sec (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.23, size = 153, normalized size = 1.56
method | result | size |
default | \(-\frac {2 \left (\cos \left (f x +e \right )+1\right )^{2} \left (-1+\cos \left (f x +e \right )\right ) \left (2 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (f x +e \right )\right )}{\sin \left (f x +e \right )}, i\right ) \sin \left (f x +e \right )+3 \left (\cos ^{4}\left (f x +e \right )\right )-3 \left (\cos ^{3}\left (f x +e \right )\right )-2 \left (\cos ^{2}\left (f x +e \right )\right )+2 \cos \left (f x +e \right )\right )}{21 f \cos \left (f x +e \right )^{2} \sin \left (f x +e \right )^{3} \left (\frac {b}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.15, size = 107, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left ({\left (3 \, \cos \left (f x + e\right )^{3} - 2 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {b}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) - i \, \sqrt {2} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}}{21 \, b^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{2}{\left (e + f x \right )}}{\left (b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {{\sin \left (e+f\,x\right )}^2}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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