Integrand size = 21, antiderivative size = 116 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {64 a^3 \cos (c+d x)}{21 d \sqrt {a+a \sin (c+d x)}}-\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2830, 2726, 2725} \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {64 a^3 \cos (c+d x)}{21 d \sqrt {a \sin (c+d x)+a}}-\frac {16 a^2 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{21 d}-\frac {2 a \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{7 d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 d} \]
[In]
[Out]
Rule 2725
Rule 2726
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {5}{7} \int (a+a \sin (c+d x))^{5/2} \, dx \\ & = -\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {1}{7} (8 a) \int (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d}+\frac {1}{21} \left (32 a^2\right ) \int \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {64 a^3 \cos (c+d x)}{21 d \sqrt {a+a \sin (c+d x)}}-\frac {16 a^2 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{21 d}-\frac {2 a \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{7 d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(116)=232\).
Time = 3.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.34 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a (1+\sin (c+d x)))^{5/2} \left (6 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}+10 \sqrt {2} \sqrt {1+\cos (c+d x)} \sin ^2\left (\frac {1}{2} (c+d x)\right )-45 \sqrt {2} \sqrt {1+\cos (c+d x)} \sin ^4\left (\frac {1}{2} (c+d x)\right )+3 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \left (5 \sqrt {2} \sqrt {1+\cos (c+d x)}+8 \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} \tan \left (\frac {1}{2} (c+d x)\right )\right )+10 \left (-4+2 \sqrt {2} \sqrt {1+\cos (c+d x)}-7 \cos ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \tan ^3\left (\frac {1}{2} (c+d x)\right )\right )\right )}{21 d \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} \left (1+\tan \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
[In]
[Out]
Time = 0.71 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.65
method | result | size |
default | \(\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{3} \left (\sin \left (d x +c \right )-1\right ) \left (3 \left (\sin ^{3}\left (d x +c \right )\right )+12 \left (\sin ^{2}\left (d x +c \right )\right )+23 \sin \left (d x +c \right )+46\right )}{21 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(75\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.21 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {2 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{4} + 12 \, a^{2} \cos \left (d x + c\right )^{3} - 17 \, a^{2} \cos \left (d x + c\right )^{2} - 58 \, a^{2} \cos \left (d x + c\right ) - 32 \, a^{2} + {\left (3 \, a^{2} \cos \left (d x + c\right )^{3} - 9 \, a^{2} \cos \left (d x + c\right )^{2} - 26 \, a^{2} \cos \left (d x + c\right ) + 32 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}} \sin {\left (c + d x \right )}\, dx \]
[In]
[Out]
\[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sin \left (d x + c\right ) \,d x } \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {\sqrt {2} {\left (315 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 77 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 21 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) + 3 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )\right )} \sqrt {a}}{84 \, d} \]
[In]
[Out]
Timed out. \[ \int \sin (c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \sin \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
[In]
[Out]