Integrand size = 23, antiderivative size = 152 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d} \]
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Time = 0.08 (sec) , antiderivative size = 152, 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} \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d} \]
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Rule 711
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{\sqrt {a+x}} \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (a^2-b^2\right )^2}{\sqrt {a+x}}-4 \left (a^3-a b^2\right ) \sqrt {a+x}+2 \left (3 a^2-b^2\right ) (a+x)^{3/2}-4 a (a+x)^{5/2}+(a+x)^{7/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {2 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}{b^5 d}-\frac {8 a \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac {4 \left (3 a^2-b^2\right ) (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac {8 a (a+b \sin (c+d x))^{7/2}}{7 b^5 d}+\frac {2 (a+b \sin (c+d x))^{9/2}}{9 b^5 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.78 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {\sqrt {a+b \sin (c+d x)} \left (1024 a^4-2496 a^2 b^2+2121 b^4-4 \left (48 a^2 b^2-91 b^4\right ) \cos (2 (c+d x))+35 b^4 \cos (4 (c+d x))-512 a^3 b \sin (c+d x)+1104 a b^3 \sin (c+d x)+80 a b^3 \sin (3 (c+d x))\right )}{1260 b^5 d} \]
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Time = 0.40 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {8 a \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a +b \right )^{2}+\left (-2 a -2 b \right ) \left (-2 a +2 b \right )+\left (a -b \right )^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a +b \right )^{2} \left (-2 a +2 b \right )+\left (-2 a -2 b \right ) \left (a -b \right )^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {a +b \sin \left (d x +c \right )}}{d \,b^{5}}\) | \(149\) |
default | \(\frac {\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {9}{2}}}{9}-\frac {8 a \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a +b \right )^{2}+\left (-2 a -2 b \right ) \left (-2 a +2 b \right )+\left (a -b \right )^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (\left (a +b \right )^{2} \left (-2 a +2 b \right )+\left (-2 a -2 b \right ) \left (a -b \right )^{2}\right ) \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 \left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {a +b \sin \left (d x +c \right )}}{d \,b^{5}}\) | \(149\) |
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Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {2 \, {\left (35 \, b^{4} \cos \left (d x + c\right )^{4} + 128 \, a^{4} - 288 \, a^{2} b^{2} + 224 \, b^{4} - 8 \, {\left (6 \, a^{2} b^{2} - 7 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (5 \, a b^{3} \cos \left (d x + c\right )^{2} - 8 \, a^{3} b + 16 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{315 \, b^{5} d} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.05 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {2 \, {\left (315 \, \sqrt {b \sin \left (d x + c\right ) + a} - \frac {42 \, {\left (3 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} - 10 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2}\right )}}{b^{2}} + \frac {35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4}}{b^{4}}\right )}}{315 \, b d} \]
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Time = 0.38 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\frac {2 \, {\left (35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 180 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{4} - 126 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} b^{2} + 420 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a b^{2} - 630 \, \sqrt {b \sin \left (d x + c\right ) + a} a^{2} b^{2} + 315 \, \sqrt {b \sin \left (d x + c\right ) + a} b^{4}\right )}}{315 \, b^{5} d} \]
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Timed out. \[ \int \frac {\cos ^5(c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^5}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]
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