Optimal. Leaf size=69 \[ \frac {4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {2 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b d} \]
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Rubi [A]
time = 0.05, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2648, 2721,
2720} \begin {gather*} \frac {4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {2 \sin (a+b x) \sqrt {d \cos (a+b x)}}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2648
Rule 2720
Rule 2721
Rubi steps
\begin {align*} \int \frac {\sin ^2(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx &=-\frac {2 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b d}+\frac {2}{3} \int \frac {1}{\sqrt {d \cos (a+b x)}} \, dx\\ &=-\frac {2 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b d}+\frac {\left (2 \sqrt {\cos (a+b x)}\right ) \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx}{3 \sqrt {d \cos (a+b x)}}\\ &=\frac {4 \sqrt {\cos (a+b x)} F\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{3 b \sqrt {d \cos (a+b x)}}-\frac {2 \sqrt {d \cos (a+b x)} \sin (a+b x)}{3 b d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.07, size = 58, normalized size = 0.84 \begin {gather*} \frac {d \cos ^2(a+b x)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {5}{2};\sin ^2(a+b x)\right ) \sin ^3(a+b x)}{3 b (d \cos (a+b x))^{3/2}} \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. \(187\) vs.
\(2(85)=170\).
time = 0.15, size = 188, normalized size = 2.72
method | result | size |
default | \(\frac {4 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (2 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(188\) |
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 = 0.13, size = 78, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (i \, \sqrt {2} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) - i \, \sqrt {2} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) + \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )\right )}}{3 \, b d} \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 (a + b x \right )}}{\sqrt {d \cos {\left (a + b x \right )}}}\, 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 (a+b\,x\right )}^2}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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