Integrand size = 21, antiderivative size = 51 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {(a+b) \cos (c+d x)}{d}+\frac {(a+2 b) \cos ^3(c+d x)}{3 d}-\frac {b \cos ^5(c+d x)}{5 d} \]
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Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3092, 380} \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=\frac {(a+2 b) \cos ^3(c+d x)}{3 d}-\frac {(a+b) \cos (c+d x)}{d}-\frac {b \cos ^5(c+d x)}{5 d} \]
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Rule 380
Rule 3092
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (a \left (1+\frac {b}{a}\right )-(a+2 b) x^2+b x^4\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {(a+b) \cos (c+d x)}{d}+\frac {(a+2 b) \cos ^3(c+d x)}{3 d}-\frac {b \cos ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.51 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {3 a \cos (c+d x)}{4 d}-\frac {5 b \cos (c+d x)}{8 d}+\frac {a \cos (3 (c+d x))}{12 d}+\frac {5 b \cos (3 (c+d x))}{48 d}-\frac {b \cos (5 (c+d x))}{80 d} \]
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Time = 1.37 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.06
method | result | size |
derivativedivides | \(\frac {-\frac {b \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}-\frac {a \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(54\) |
default | \(\frac {-\frac {b \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5}-\frac {a \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(54\) |
parallelrisch | \(\frac {\left (20 a +25 b \right ) \cos \left (3 d x +3 c \right )-3 b \cos \left (5 d x +5 c \right )+\left (-180 a -150 b \right ) \cos \left (d x +c \right )-160 a -128 b}{240 d}\) | \(56\) |
parts | \(-\frac {a \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3 d}-\frac {b \left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{5 d}\) | \(56\) |
risch | \(-\frac {3 a \cos \left (d x +c \right )}{4 d}-\frac {5 b \cos \left (d x +c \right )}{8 d}-\frac {b \cos \left (5 d x +5 c \right )}{80 d}+\frac {a \cos \left (3 d x +3 c \right )}{12 d}+\frac {5 \cos \left (3 d x +3 c \right ) b}{48 d}\) | \(71\) |
norman | \(\frac {-\frac {20 a +16 b}{15 d}-\frac {4 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 \left (14 a +16 b \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (20 a +16 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(93\) |
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Time = 0.25 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {3 \, b \cos \left (d x + c\right )^{5} - 5 \, {\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a + b\right )} \cos \left (d x + c\right )}{15 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (44) = 88\).
Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 2.10 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=\begin {cases} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {2 a \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {b \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {4 b \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {8 b \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin ^{2}{\left (c \right )}\right ) \sin ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.84 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {3 \, b \cos \left (d x + c\right )^{5} - 5 \, {\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{3} + 15 \, {\left (a + b\right )} \cos \left (d x + c\right )}{15 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.31 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {b \cos \left (d x + c\right )^{5}}{5 \, d} + \frac {a \cos \left (d x + c\right )^{3}}{3 \, d} + \frac {2 \, b \cos \left (d x + c\right )^{3}}{3 \, d} - \frac {a \cos \left (d x + c\right )}{d} - \frac {b \cos \left (d x + c\right )}{d} \]
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Time = 13.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int \sin ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx=-\frac {\frac {b\,{\cos \left (c+d\,x\right )}^5}{5}+\left (-\frac {a}{3}-\frac {2\,b}{3}\right )\,{\cos \left (c+d\,x\right )}^3+\left (a+b\right )\,\cos \left (c+d\,x\right )}{d} \]
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