Integrand size = 23, antiderivative size = 107 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \]
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Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2754, 2728, 212} \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{3 d}+\frac {a \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{2 d} \]
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Rule 212
Rule 2728
Rule 2754
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {1}{2} a \int \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {1}{4} a^2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx \\ & = \frac {a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d}-\frac {a^2 \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 )}{2 d} \\ & = -\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} d}+\frac {a \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.48 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {1}{2} (1-\sin (c+d x))\right ) \sec ^3(c+d x) (a (1+\sin (c+d x)))^{3/2}}{3 d} \]
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Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {\left (1+\sin \left (d x +c \right )\right ) \left (3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-10 a^{\frac {7}{2}}+6 \sin \left (d x +c \right ) a^{\frac {7}{2}}\right )}{12 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(107\) |
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (88) = 176\).
Time = 0.31 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.01 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {3 \, {\left (\sqrt {2} a \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \sqrt {2} a \cos \left (d x + c\right )\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \, {\left (3 \, a \sin \left (d x + c\right ) - 5 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{24 \, {\left (d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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none
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89 \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {2 \, {\left (3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - 3 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{24 \, d} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^4} \,d x \]
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