Integrand size = 23, antiderivative size = 73 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {8 (a+a \sin (c+d x))^{13/2}}{13 a^3 d}-\frac {8 (a+a \sin (c+d x))^{15/2}}{15 a^4 d}+\frac {2 (a+a \sin (c+d x))^{17/2}}{17 a^5 d} \]
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Time = 0.06 (sec) , antiderivative size = 73, 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 = {2746, 45} \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{13/2}}{13 a^3 d} \]
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Rule 45
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{11/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {\text {Subst}\left (\int \left (4 a^2 (a+x)^{11/2}-4 a (a+x)^{13/2}+(a+x)^{15/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = \frac {8 (a+a \sin (c+d x))^{13/2}}{13 a^3 d}-\frac {8 (a+a \sin (c+d x))^{15/2}}{15 a^4 d}+\frac {2 (a+a \sin (c+d x))^{17/2}}{17 a^5 d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.74 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 a^3 (1+\sin (c+d x))^6 \sqrt {a (1+\sin (c+d x))} \left (331-494 \sin (c+d x)+195 \sin ^2(c+d x)\right )}{3315 d} \]
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Time = 58.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {17}{2}}}{17}-\frac {8 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {15}{2}}}{15}+\frac {8 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {13}{2}}}{13}}{d \,a^{5}}\) | \(56\) |
default | \(\frac {\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {17}{2}}}{17}-\frac {8 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {15}{2}}}{15}+\frac {8 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {13}{2}}}{13}}{d \,a^{5}}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (61) = 122\).
Time = 0.28 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.75 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (195 \, a^{3} \cos \left (d x + c\right )^{8} - 1072 \, a^{3} \cos \left (d x + c\right )^{6} + 56 \, a^{3} \cos \left (d x + c\right )^{4} + 128 \, a^{3} \cos \left (d x + c\right )^{2} + 1024 \, a^{3} - 4 \, {\left (169 \, a^{3} \cos \left (d x + c\right )^{6} - 126 \, a^{3} \cos \left (d x + c\right )^{4} - 160 \, a^{3} \cos \left (d x + c\right )^{2} - 256 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{3315 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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none
Time = 0.17 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.75 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {2 \, {\left (195 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {17}{2}} - 884 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a + 1020 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{2}\right )}}{3315 \, a^{5} d} \]
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none
Time = 0.31 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.48 \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {512 \, \sqrt {2} {\left (195 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{17} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 442 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 255 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{3315 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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