Integrand size = 23, antiderivative size = 167 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {35 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}-\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.16 (sec) , antiderivative size = 167, 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 = {2760, 2766, 2729, 2728, 212} \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=-\frac {35 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{128 \sqrt {2} a^{5/2} d}+\frac {35 \sec (c+d x)}{96 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {35 \cos (c+d x)}{128 a d (a \sin (c+d x)+a)^{3/2}}-\frac {7 \sec (c+d x)}{48 a d (a \sin (c+d x)+a)^{3/2}}-\frac {\sec (c+d x)}{6 d (a \sin (c+d x)+a)^{5/2}} \]
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Rule 212
Rule 2728
Rule 2729
Rule 2760
Rule 2766
Rubi steps \begin{align*} \text {integral}& = -\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}+\frac {7 \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{12 a} \\ & = -\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{96 a^2} \\ & = -\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {35 \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx}{64 a} \\ & = -\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {35 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{256 a^2} \\ & = -\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{96 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {35 \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 a^2 d} \\ & = -\frac {35 \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} a^{5/2} d}-\frac {\sec (c+d x)}{6 d (a+a \sin (c+d x))^{5/2}}-\frac {35 \cos (c+d x)}{128 a d (a+a \sin (c+d x))^{3/2}}-\frac {7 \sec (c+d x)}{48 a d (a+a \sin (c+d x))^{3/2}}+\frac {35 \sec (c+d x)}{96 a^2 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 = 0.16 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.31 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},4,\frac {1}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec (c+d x) \sqrt {a (1+\sin (c+d x))}}{8 a^3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(140)=280\).
Time = 0.71 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.72
method | result | size |
default | \(-\frac {210 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{\frac {7}{2}}-105 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}\, a^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+490 \left (\cos ^{2}\left (d x +c \right )\right ) a^{\frac {7}{2}}-315 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}\, a^{3} \left (\cos ^{2}\left (d x +c \right )\right )-448 \sin \left (d x +c \right ) a^{\frac {7}{2}}+420 \sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \sin \left (d x +c \right ) a^{3}-320 a^{\frac {7}{2}}+420 \sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}}{768 a^{\frac {11}{2}} \left (1+\sin \left (d x +c \right )\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(288\) |
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Time = 0.30 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.68 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {105 \, \sqrt {2} {\left (3 \, \cos \left (d x + c\right )^{3} + {\left (\cos \left (d x + c\right )^{3} - 4 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) - 4 \, \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (245 \, \cos \left (d x + c\right )^{2} + 7 \, {\left (15 \, \cos \left (d x + c\right )^{2} - 32\right )} \sin \left (d x + c\right ) - 160\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1536 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right ) + {\left (a^{3} d \cos \left (d x + c\right )^{3} - 4 \, a^{3} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.31 \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\frac {\sqrt {a} {\left (\frac {105 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {105 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {96 \, \sqrt {2}}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (57 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 136 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 87 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{1536 \, d} \]
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Timed out. \[ \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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