Integrand size = 10, antiderivative size = 42 \[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2715, 2720} \[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}+\frac {2 \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
[In]
[Out]
Rule 2715
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}+\frac {1}{3} \int \frac {1}{\sqrt {\cos (a+b x)}} \, dx \\ & = \frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )}{3 b}+\frac {2 \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86 \[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \left (\operatorname {EllipticF}\left (\frac {1}{2} (a+b x),2\right )+\sqrt {\cos (a+b x)} \sin (a+b x)\right )}{3 b} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(178\) vs. \(2(62)=124\).
Time = 2.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 4.26
method | result | size |
default | \(-\frac {2 \sqrt {\left (-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (4 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+\sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, F\left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )\right )}{3 \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\, \sin \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, b}\) | \(179\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.62 \[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {2 \, \sqrt {\cos \left (b x + a\right )} \sin \left (b x + a\right ) - i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + i \, \sqrt {2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right )}{3 \, b} \]
[In]
[Out]
\[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\int \cos ^{\frac {3}{2}}{\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\int { \cos \left (b x + a\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \cos ^{\frac {3}{2}}(a+b x) \, dx=\frac {2\,\mathrm {F}\left (\frac {a}{2}+\frac {b\,x}{2}\middle |2\right )}{3\,b}+\frac {2\,\sqrt {\cos \left (a+b\,x\right )}\,\sin \left (a+b\,x\right )}{3\,b} \]
[In]
[Out]