Integrand size = 23, antiderivative size = 197 \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {63 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d} \]
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Time = 0.21 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2754, 2766, 2760, 2729, 2728, 212} \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=-\frac {63 a^2 \cos (c+d x)}{128 d (a \sin (c+d x)+a)^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a \sin (c+d x)+a)^{3/2}}-\frac {63 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{128 \sqrt {2} d}+\frac {\sec ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a \sin (c+d x)+a}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a \sin (c+d x)+a}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2760
Rule 2766
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{10} (9 a) \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{20} \left (21 a^2\right ) \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{32} (21 a) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{64} \left (63 a^2\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{256} (63 a) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {(63 a) \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 )}{128 d} \\ & = -\frac {63 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.31 \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},3,-\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^5(c+d x) (1+\sin (c+d x))^2 \sqrt {a (1+\sin (c+d x))}}{20 d} \]
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Time = 0.43 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.29
method | result | size |
default | \(-\frac {-630 a^{\frac {9}{2}} \left (\cos ^{4}\left (d x +c \right )\right )-420 a^{\frac {9}{2}} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-315 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, a^{2} \left (\cos ^{2}\left (d x +c \right )\right )+84 a^{\frac {9}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+630 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a^{2}-288 a^{\frac {9}{2}} \sin \left (d x +c \right )+630 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+32 a^{\frac {9}{2}}}{1280 a^{\frac {7}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(254\) |
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Time = 0.30 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.07 \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {315 \, \sqrt {2} \sqrt {a} \cos \left (d x + c\right )^{5} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (315 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (35 \, \cos \left (d x + c\right )^{2} + 24\right )} \sin \left (d x + c\right ) - 16\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, d \cos \left (d x + c\right )^{5}} \]
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Timed out. \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int { \sqrt {a \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{6} \,d x } \]
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Time = 0.36 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.21 \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\frac {\sqrt {2} {\left (315 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 315 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {10 \, {\left (15 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 17 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {16 \, {\left (30 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}\right )} \sqrt {a}}{2560 \, d} \]
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Timed out. \[ \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^6} \,d x \]
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