Integrand size = 21, antiderivative size = 89 \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {1}{3 d (a+a \sin (c+d x))^{3/2}}-\frac {1}{2 a d \sqrt {a+a \sin (c+d x)}} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2746, 53, 65, 212} \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {1}{2 a d \sqrt {a \sin (c+d x)+a}}-\frac {1}{3 d (a \sin (c+d x)+a)^{3/2}} \]
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Rule 53
Rule 65
Rule 212
Rule 2746
Rubi steps \begin{align*} \text {integral}& = \frac {a \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{d} \\ & = -\frac {1}{3 d (a+a \sin (c+d x))^{3/2}}+\frac {\text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{2 d} \\ & = -\frac {1}{3 d (a+a \sin (c+d x))^{3/2}}-\frac {1}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{4 a d} \\ & = -\frac {1}{3 d (a+a \sin (c+d x))^{3/2}}-\frac {1}{2 a d \sqrt {a+a \sin (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{2 a d} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {1}{3 d (a+a \sin (c+d x))^{3/2}}-\frac {1}{2 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 = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.46 \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1+\sin (c+d x))\right )}{3 d (a+a \sin (c+d x))^{3/2}} \]
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Time = 0.56 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.80
method | result | size |
default | \(-\frac {a \left (\frac {1}{2 a^{2} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{3 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 a^{\frac {5}{2}}}\right )}{d}\) | \(71\) |
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Time = 0.30 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.48 \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {3 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (3 \, \sin \left (d x + c\right ) + 5\right )}}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - 2 \, a^{2} d \sin \left (d x + c\right ) - 2 \, a^{2} d\right )}} \]
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\[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=-\frac {\frac {3 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (3 \, a \sin \left (d x + c\right ) + 5 \, a\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}}{24 \, a d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (70) = 140\).
Time = 0.40 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.60 \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\frac {\sqrt {a} {\left (\frac {3 \, \sqrt {2} \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, \sqrt {2} \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}\right )}}{24 \, d} \]
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Timed out. \[ \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx=\int \frac {1}{\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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