Integrand size = 23, antiderivative size = 211 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {99 \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2760, 2766, 2746, 53, 65, 212} \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {99 \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {99}{256 a d \sqrt {a \sin (c+d x)+a}}-\frac {33}{128 d (a \sin (c+d x)+a)^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a \sin (c+d x)+a}}-\frac {\sec ^4(c+d x)}{7 d (a \sin (c+d x)+a)^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a \sin (c+d x)+a}}-\frac {99 \sec ^2(c+d x)}{560 d (a \sin (c+d x)+a)^{3/2}} \]
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Rule 53
Rule 65
Rule 212
Rule 2746
Rule 2760
Rule 2766
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \int \frac {\sec ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{14 a} \\ & = -\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99}{112} \int \frac {\sec ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \int \frac {\sec ^3(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{160 a} \\ & = -\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99}{128} \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {(99 a) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{128 d} \\ & = -\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{256 d} \\ & = -\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{512 a d} \\ & = -\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{256 a d} \\ & = \frac {99 \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{256 \sqrt {2} a^{3/2} d}-\frac {33}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {99 \sec ^2(c+d x)}{560 d (a+a \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x)}{7 d (a+a \sin (c+d x))^{3/2}}-\frac {99}{256 a d \sqrt {a+a \sin (c+d x)}}+\frac {99 \sec ^2(c+d x)}{320 a d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^4(c+d x)}{56 a d \sqrt {a+a \sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {a^2 \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},3,-\frac {5}{2},\frac {1}{2} (1+\sin (c+d x))\right )}{28 d (a+a \sin (c+d x))^{7/2}} \]
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Time = 0.86 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {2 a^{5} \left (\frac {5}{32 a^{6} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{16 a^{5} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {3}{80 a^{4} \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{56 a^{3} \left (a +a \sin \left (d x +c \right )\right )^{\frac {7}{2}}}+\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}\, a \left (19 \sin \left (d x +c \right )-23\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {99 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 \sqrt {a}}}{32 a^{6}}\right )}{d}\) | \(152\) |
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Time = 0.31 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.98 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3465 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{4}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (5775 \, \cos \left (d x + c\right )^{4} - 1188 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 252 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 480\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35840 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 2 \, a^{2} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{4}\right )}} \]
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\[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3465 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{5} - 11550 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{4} a + 7392 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{3} a^{2} + 2112 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} a^{3} + 1408 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{4} + 1280 \, a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 4 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2}}}{35840 \, a d} \]
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Time = 0.32 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.18 \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a} {\left (\frac {3465 \, \sqrt {2} \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {70 \, {\left (19 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 21 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {16 \, \sqrt {2} {\left (350 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 70 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 21 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5\right )}}{a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{35840 \, d} \]
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Timed out. \[ \int \frac {\sec ^5(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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