Integrand size = 23, antiderivative size = 103 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}+\frac {3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d} \]
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Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2754, 2746, 65, 212} \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}+\frac {\sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{4 d}+\frac {3 a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{16 d} \]
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Rule 65
Rule 212
Rule 2746
Rule 2754
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac {1}{8} (3 a) \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \\ & = \frac {3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac {1}{32} \left (3 a^2\right ) \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = \frac {3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{32 d} \\ & = \frac {3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{16 d} \\ & = \frac {3 a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}+\frac {3 a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{16 d}+\frac {\sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{4 d} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.07 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {3 \sqrt {2} a^{5/2} \text {arctanh}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+2 a^2 (7-3 \sin (c+d x)) \sqrt {a (1+\sin (c+d x))}}{32 d (-1+\sin (c+d x))^2} \]
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Time = 0.38 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.04
\[-\frac {2 a^{5} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{8 a \left (a \sin \left (d x +c \right )-a \right )^{2}}-\frac {3 \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \left (a \sin \left (d x +c \right )-a \right )}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{8 a}\right )}{d}\]
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Time = 0.30 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.43 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\frac {3 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} a^{2} \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, {\left (3 \, a^{2} \sin \left (d x + c\right ) - 7 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{64 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
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Timed out. \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.30 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {3 \, \sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4} - 10 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{5}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )} a + 4 \, a^{2}}}{64 \, a d} \]
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Time = 0.31 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.09 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=-\frac {\sqrt {2} a^{\frac {5}{2}} {\left (\frac {2 \, {\left (3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - 3 \, \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{64 \, d} \]
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Timed out. \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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