Integrand size = 23, antiderivative size = 221 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {231 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.27 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2766, 2760, 2729, 2728, 212} \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=-\frac {231 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}-\frac {11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a \sin (c+d x)+a}}-\frac {77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2760
Rule 2766
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{10} (11 a) \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {33}{40} \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{80} (77 a) \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {77}{128} \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{256} (231 a) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {231 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{1024} \\ & = -\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {231 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{512 d} \\ & = -\frac {231 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 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 = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.28 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},4,-\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^5(c+d x) (1+\sin (c+d x))^3}{40 d \sqrt {a (1+\sin (c+d x))}} \]
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Time = 0.83 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.49
method | result | size |
default | \(-\frac {-6930 \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right ) a^{\frac {11}{2}}-2310 \left (\cos ^{4}\left (d x +c \right )\right ) a^{\frac {11}{2}}-3696 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {11}{2}}-3465 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-528 \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {11}{2}}-10395 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (\cos ^{2}\left (d x +c \right )\right )-2816 \sin \left (d x +c \right ) a^{\frac {11}{2}}+13860 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a^{3}-256 a^{\frac {11}{2}}+13860 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{15360 a^{\frac {11}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(330\) |
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Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.13 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {3465 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{5}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (1155 \, \cos \left (d x + c\right )^{4} + 264 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} + 168 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30720 \, {\left (a d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{5}\right )}} \]
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\[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int { \frac {\sec \left (d x + c\right )^{6}}{\sqrt {a \sin \left (d x + c\right ) + a}} \,d x } \]
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Time = 0.38 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.19 \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\frac {\frac {3465 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {10 \, \sqrt {2} {\left (213 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 472 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 267 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {32 \, \sqrt {2} {\left (150 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, \sqrt {a}\right )}}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{30720 \, d} \]
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Timed out. \[ \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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