Integrand size = 23, antiderivative size = 169 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} d}-\frac {7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \]
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Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 2766, 2729, 2728, 212} \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{16 \sqrt {2} d}-\frac {7 a^3 \cos (c+d x)}{16 d (a \sin (c+d x)+a)^{3/2}}+\frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a \sin (c+d x)+a}}+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d}+\frac {7 a \sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{30 d} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2766
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{10} (7 a) \int \sec ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{12} \left (7 a^2\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{8} \left (7 a^3\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {1}{32} \left (7 a^2\right ) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = -\frac {7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\left (7 a^2\right ) \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 )}{16 d} \\ & = -\frac {7 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} d}-\frac {7 a^3 \cos (c+d x)}{16 d (a+a \sin (c+d x))^{3/2}}+\frac {7 a^2 \sec (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}+\frac {7 a \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{30 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.14 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.32 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},2,-\frac {3}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^5(c+d x) (a (1+\sin (c+d x)))^{5/2}}{10 a d} \]
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Time = 0.50 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {210 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{\frac {7}{2}}+105 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a -350 \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {7}{2}}+105 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -168 \sin \left (d x +c \right ) a^{\frac {7}{2}}+72 a^{\frac {7}{2}}}{480 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(176\) |
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Time = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.47 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {105 \, {\left (\sqrt {2} a \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - \sqrt {2} a \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \, {\left (175 \, a \cos \left (d x + c\right )^{2} - 21 \, {\left (5 \, a \cos \left (d x + c\right )^{2} - 4 \, a\right )} \sin \left (d x + c\right ) - 36 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{960 \, {\left (d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{3}\right )}} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.85 \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {30 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + \frac {4 \, {\left (45 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 10 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - 105 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 105 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{960 \, d} \]
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Timed out. \[ \int \sec ^6(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^6} \,d x \]
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