Integrand size = 19, antiderivative size = 67 \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\frac {6 b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {16, 2715, 2721, 2719} \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\frac {2 \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac {6 b E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}} \]
[In]
[Out]
Rule 16
Rule 2715
Rule 2719
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {\int (b \cos (c+d x))^{5/2} \, dx}{b} \\ & = \frac {2 (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{5} (3 b) \int \sqrt {b \cos (c+d x)} \, dx \\ & = \frac {2 (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\left (3 b \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}} \\ & = \frac {6 b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.97 \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\frac {(b \cos (c+d x))^{5/2} \left (6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} \sin (2 (c+d x))\right )}{5 b d \cos ^{\frac {5}{2}}(c+d x)} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(212\) vs. \(2(83)=166\).
Time = 2.67 (sec) , antiderivative size = 213, normalized size of antiderivative = 3.18
method | result | size |
default | \(-\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{2} \left (-8 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) b}\, d}\) | \(213\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.33 \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\frac {2 \, \sqrt {b \cos \left (d x + c\right )} b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{5 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
[In]
[Out]
\[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\int { \left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right ) \,d x } \]
[In]
[Out]
Timed out. \[ \int \cos (c+d x) (b \cos (c+d x))^{3/2} \, dx=\int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \]
[In]
[Out]