Integrand size = 36, antiderivative size = 44 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 B \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d} \]
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Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 2715, 2720} \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 B \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d} \]
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Rule 21
Rule 2715
Rule 2720
Rubi steps \begin{align*} \text {integral}& = B \int \cos ^{\frac {3}{2}}(c+d x) \, dx \\ & = \frac {2 B \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} B \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {2 B \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 B \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\frac {2 B \left (\operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\sqrt {\cos (c+d x)} \sin (c+d x)\right )}{3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(64)=128\).
Time = 3.36 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.09
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, B \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, F\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d}\) | \(180\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.61 \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\frac {2 \, B \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - i \, \sqrt {2} B {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + i \, \sqrt {2} B {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{3 \, d} \]
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Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (B b \cos \left (d x + c\right ) + B a\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\int { \frac {{\left (B b \cos \left (d x + c\right ) + B a\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\cos ^{\frac {3}{2}}(c+d x) (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (B\,a+B\,b\,\cos \left (c+d\,x\right )\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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