Integrand size = 23, antiderivative size = 73 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \]
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Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2754, 2746, 65, 212} \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rule 65
Rule 212
Rule 2746
Rule 2754
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {1}{4} a \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{4 d} \\ & = \frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{2 d} \\ & = \frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.99 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {a \left (\sqrt {2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right )-\frac {2 \sqrt {a (1+\sin (c+d x))}}{-1+\sin (c+d x)}\right )}{4 d} \]
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Time = 0.39 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {2 a^{3} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \left (a \sin \left (d x +c \right )-a \right )}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{d}\) | \(70\) |
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Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.36 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {{\left (\sqrt {2} a \sin \left (d x + c\right ) - \sqrt {2} a\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{8 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
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Timed out. \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.29 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} a^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}}{a \sin \left (d x + c\right ) - a}}{8 \, a d} \]
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Time = 0.30 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} a^{\frac {3}{2}} {\left (\frac {2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{8 \, d} \]
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Timed out. \[ \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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