Integrand size = 31, antiderivative size = 233 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a+a \sin (c+d x)}}-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d} \]
[Out]
Time = 0.53 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2958, 3055, 3060, 2849, 2838, 2830, 2725} \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {38 a^2 \sin ^4(c+d x) \cos (c+d x)}{1287 d \sqrt {a \sin (c+d x)+a}}-\frac {862 a^2 \sin ^3(c+d x) \cos (c+d x)}{9009 d \sqrt {a \sin (c+d x)+a}}-\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a \sin (c+d x)+a}}+\frac {2 \sin ^4(c+d x) \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d}+\frac {6 a \sin ^4(c+d x) \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{143 d}-\frac {1724 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{15015 d}+\frac {3448 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{45045 d} \]
[In]
[Out]
Rule 2725
Rule 2830
Rule 2838
Rule 2849
Rule 2958
Rule 3055
Rule 3060
Rubi steps \begin{align*} \text {integral}& = \frac {\int \sin ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2} \\ & = \frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {2 \int \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac {5 a^2}{2}-\frac {3}{2} a^2 \sin (c+d x)\right ) \, dx}{13 a^2} \\ & = \frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {4 \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {31 a^3}{4}+\frac {19}{4} a^3 \sin (c+d x)\right ) \, dx}{143 a^2} \\ & = -\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(431 a) \int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{1287} \\ & = -\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(862 a) \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{3003} \\ & = -\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {1724 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{15015} \\ & = -\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac {(862 a) \int \sqrt {a+a \sin (c+d x)} \, dx}{6435} \\ & = -\frac {1724 a^2 \cos (c+d x)}{6435 d \sqrt {a+a \sin (c+d x)}}-\frac {862 a^2 \cos (c+d x) \sin ^3(c+d x)}{9009 d \sqrt {a+a \sin (c+d x)}}-\frac {38 a^2 \cos (c+d x) \sin ^4(c+d x)}{1287 d \sqrt {a+a \sin (c+d x)}}+\frac {3448 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{45045 d}+\frac {6 a \cos (c+d x) \sin ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{143 d}-\frac {1724 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{15015 d}+\frac {2 \cos (c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d} \\ \end{align*}
Time = 2.79 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.52 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 \sqrt {a (1+\sin (c+d x))} (281816-194160 \cos (2 (c+d x))+22680 \cos (4 (c+d x))+381174 \sin (c+d x)-77665 \sin (3 (c+d x))+3465 \sin (5 (c+d x)))}{360360 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) a^{2} \left (\sin \left (d x +c \right )-1\right )^{2} \left (3465 \left (\sin ^{5}\left (d x +c \right )\right )+11340 \left (\sin ^{4}\left (d x +c \right )\right )+15085 \left (\sin ^{3}\left (d x +c \right )\right )+12930 \left (\sin ^{2}\left (d x +c \right )\right )+10344 \sin \left (d x +c \right )+6896\right )}{45045 \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(97\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.81 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (3465 \, a \cos \left (d x + c\right )^{7} - 4410 \, a \cos \left (d x + c\right )^{6} - 14140 \, a \cos \left (d x + c\right )^{5} + 7330 \, a \cos \left (d x + c\right )^{4} + 15299 \, a \cos \left (d x + c\right )^{3} - 568 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) - {\left (3465 \, a \cos \left (d x + c\right )^{6} + 7875 \, a \cos \left (d x + c\right )^{5} - 6265 \, a \cos \left (d x + c\right )^{4} - 13595 \, a \cos \left (d x + c\right )^{3} + 1704 \, a \cos \left (d x + c\right )^{2} + 2272 \, a \cos \left (d x + c\right ) + 4544 \, a\right )} \sin \left (d x + c\right ) + 4544 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45045 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
[In]
[Out]
Timed out. \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right )^{3} \,d x } \]
[In]
[Out]
none
Time = 0.33 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.82 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {16 \, \sqrt {2} {\left (27720 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 114660 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 190190 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 160875 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72072 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15015 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )} \sqrt {a}}{45045 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^3\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
[In]
[Out]