Integrand size = 21, antiderivative size = 69 \[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\frac {4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}-\frac {2 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d} \]
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Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2648, 2721, 2719} \[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\frac {4 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {d \cos (a+b x)}}{5 b \sqrt {\cos (a+b x)}}-\frac {2 \sin (a+b x) (d \cos (a+b x))^{3/2}}{5 b d} \]
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Rule 2648
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d}+\frac {2}{5} \int \sqrt {d \cos (a+b x)} \, dx \\ & = -\frac {2 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d}+\frac {\left (2 \sqrt {d \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 \sqrt {\cos (a+b x)}} \\ & = \frac {4 \sqrt {d \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b \sqrt {\cos (a+b x)}}-\frac {2 (d \cos (a+b x))^{3/2} \sin (a+b x)}{5 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\frac {d \sqrt [4]{\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {3}{2},\frac {5}{2},\sin ^2(a+b x)\right ) \sin ^3(a+b x)}{3 b \sqrt {d \cos (a+b x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(193\) vs. \(2(85)=170\).
Time = 0.50 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.81
method | result | size |
default | \(\frac {4 \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, d \left (4 \left (\cos ^{7}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8 \left (\cos ^{5}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+5 \left (\cos ^{3}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {1-2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, E\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-\cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{5 \sqrt {-d \left (2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {d \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b}\) | \(194\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.26 \[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=-\frac {2 \, {\left (\sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right ) - i \, \sqrt {2} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right )\right ) + i \, \sqrt {2} \sqrt {d} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )\right )\right )}}{5 \, b} \]
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\[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\int \sqrt {d \cos {\left (a + b x \right )}} \sin ^{2}{\left (a + b x \right )}\, dx \]
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\[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )^{2} \,d x } \]
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\[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\int { \sqrt {d \cos \left (b x + a\right )} \sin \left (b x + a\right )^{2} \,d x } \]
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Timed out. \[ \int \sqrt {d \cos (a+b x)} \sin ^2(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^2\,\sqrt {d\,\cos \left (a+b\,x\right )} \,d x \]
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