Integrand size = 23, antiderivative size = 83 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
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Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2747, 711} \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^{3/2} \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = \frac {\text {Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^{3/2}+2 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d} \\ & = -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.70 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\frac {(a+b \sin (c+d x))^{5/2} \left (-16 a^2+91 b^2+35 b^2 \cos (2 (c+d x))+40 a b \sin (c+d x)\right )}{315 b^3 d} \]
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Time = 0.58 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {2 \left (\frac {\left (a +b \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {2 a \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a^{2}-b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,b^{3}}\) | \(62\) |
default | \(-\frac {2 \left (\frac {\left (a +b \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {2 a \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (a^{2}-b^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}\right )}{d \,b^{3}}\) | \(62\) |
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Time = 0.28 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (35 \, b^{4} \cos \left (d x + c\right )^{4} + 8 \, a^{4} - 60 \, a^{2} b^{2} - 28 \, b^{4} - {\left (3 \, a^{2} b^{2} + 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (25 \, a b^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} b + 38 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b^{3} d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (76) = 152\).
Time = 4.46 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.78 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\begin {cases} a^{\frac {3}{2}} x \cos ^{3}{\left (c \right )} & \text {for}\: b = 0 \wedge d = 0 \\a^{\frac {3}{2}} \cdot \left (\frac {2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {\sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}\right ) & \text {for}\: b = 0 \\x \left (a + b \sin {\left (c \right )}\right )^{\frac {3}{2}} \cos ^{3}{\left (c \right )} & \text {for}\: d = 0 \\- \frac {16 a^{4} \sqrt {a + b \sin {\left (c + d x \right )}}}{315 b^{3} d} + \frac {8 a^{3} \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )}}{315 b^{2} d} + \frac {8 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )}}{21 b d} + \frac {2 a^{2} \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}}{5 b d} + \frac {152 a \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{3}{\left (c + d x \right )}}{315 d} + \frac {4 a \sqrt {a + b \sin {\left (c + d x \right )}} \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {8 b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{4}{\left (c + d x \right )}}{45 d} + \frac {2 b \sqrt {a + b \sin {\left (c + d x \right )}} \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.73 \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 90 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 63 \, {\left (a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}\right )}}{315 \, b^{3} d} \]
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\[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{3} \,d x } \]
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sin (c+d x))^{3/2} \, dx=\int {\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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