Integrand size = 23, antiderivative size = 106 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d} \]
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Time = 0.13 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2755, 2754, 2746, 65, 212} \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=-\frac {a^{7/2} \text {arctanh}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a \sin (c+d x)+a)^{5/2}}{2 d} \]
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Rule 65
Rule 212
Rule 2746
Rule 2754
Rule 2755
Rubi steps \begin{align*} \text {integral}& = \frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {1}{4} a^2 \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx \\ & = -\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {1}{16} a^3 \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {a^4 \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{16 d} \\ & = -\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d}-\frac {a^4 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{8 d} \\ & = -\frac {a^{7/2} \text {arctanh}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{8 \sqrt {2} d}-\frac {a^2 \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{8 d}+\frac {a \sec ^4(c+d x) (a+a \sin (c+d x))^{5/2}}{2 d} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {-\sqrt {2} a^{7/2} \text {arctanh}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+2 a^3 \sqrt {a (1+\sin (c+d x))} (3+\sin (c+d x))}{16 d (-1+\sin (c+d x))^2} \]
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Time = 0.36 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71
\[-\frac {2 a^{5} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \left (3+\sin \left (d x +c \right )\right )}{16 \left (a \sin \left (d x +c \right )-a \right )^{2}}+\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{32 a^{\frac {3}{2}}}\right )}{d}\]
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.37 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {{\left (\sqrt {2} a^{3} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} a^{3} \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{3}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (a^{3} \sin \left (d x + c\right ) + 3 \, a^{3}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{32 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
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Timed out. \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\text {Timed out} \]
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Time = 0.26 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.25 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left ({\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{5} + 2 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{6}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{2} - 4 \, {\left (a \sin \left (d x + c\right ) + a\right )} a + 4 \, a^{2}}}{32 \, a d} \]
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Time = 0.31 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.41 \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\frac {\sqrt {2} a^{\frac {7}{2}} {\left (\frac {4 \, {\left (\frac {1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\frac {1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{2} - 4} - \log \left ({\left | \frac {1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + \log \left ({\left | \frac {1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{64 \, d} \]
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Timed out. \[ \int \sec ^5(c+d x) (a+a \sin (c+d x))^{7/2} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2}}{{\cos \left (c+d\,x\right )}^5} \,d x \]
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