Integrand size = 27, antiderivative size = 74 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a+a \sin (c+d x))^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \]
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Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750} \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 (a \sin (c+d x)+a)^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a \sin (c+d x)+a)^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \]
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Rule 2750
Rule 2751
Rubi steps \begin{align*} \text {integral}& = \frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{a} \\ & = \frac {2 (a+a \sin (c+d x))^{3/2}}{d e (e \cos (c+d x))^{5/2}}-\frac {4 (a+a \sin (c+d x))^{5/2}}{5 a d e (e \cos (c+d x))^{5/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=-\frac {2 a \sqrt {a (1+\sin (c+d x))} (-3+2 \sin (c+d x))}{5 d e^3 \sqrt {e \cos (c+d x)} \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \]
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Time = 2.56 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.70
method | result | size |
default | \(\frac {2 \left (2 \sin \left (d x +c \right )-3\right ) a \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{5 d \left (\sin \left (d x +c \right )-1\right ) \sqrt {e \cos \left (d x +c \right )}\, e^{3}}\) | \(52\) |
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (2 \, a \sin \left (d x + c\right ) - 3 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{5 \, {\left (d e^{4} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - d e^{4} \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 207 vs. \(2 (64) = 128\).
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 2.80 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {2 \, {\left (3 \, a^{\frac {3}{2}} \sqrt {e} - \frac {4 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, a^{\frac {3}{2}} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{5 \, {\left (e^{4} + \frac {2 \, e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d \sqrt {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]
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Timed out. \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\text {Timed out} \]
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Time = 5.94 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx=\frac {4\,a\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (5\,\sin \left (c+d\,x\right )+\cos \left (2\,c+2\,d\,x\right )-4\right )}{5\,d\,e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )+\cos \left (2\,c+2\,d\,x\right )-3\right )} \]
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