Integrand size = 25, antiderivative size = 234 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\frac {2 a^2 \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3} \]
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Time = 0.53 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3957, 2952, 2716, 2721, 2720, 2644, 331, 335, 218, 212, 209, 2650, 2651} \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\frac {2 a^2 \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {7 a^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d e^2 \sqrt {e \sin (c+d x)}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}} \]
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Rule 209
Rule 212
Rule 218
Rule 331
Rule 335
Rule 2644
Rule 2650
Rule 2651
Rule 2716
Rule 2720
Rule 2721
Rule 2952
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx \\ & = \int \left (\frac {a^2}{(e \sin (c+d x))^{5/2}}+\frac {2 a^2 \sec (c+d x)}{(e \sin (c+d x))^{5/2}}+\frac {a^2 \sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}}\right ) \, dx \\ & = a^2 \int \frac {1}{(e \sin (c+d x))^{5/2}} \, dx+a^2 \int \frac {\sec ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx+\left (2 a^2\right ) \int \frac {\sec (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx \\ & = -\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {a^2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2}+\frac {\left (5 a^2\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{3 e^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e} \\ & = -\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e^3}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{6 e^2}+\frac {\left (a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e^2 \sqrt {e \sin (c+d x)}} \\ & = -\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {2 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^3}+\frac {\left (5 a^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{6 e^2 \sqrt {e \sin (c+d x)}} \\ & = -\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^2}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^2} \\ & = \frac {2 a^2 \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}+\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{5/2}}-\frac {4 a^2}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}-\frac {2 a^2 \sec (c+d x)}{3 d e (e \sin (c+d x))^{3/2}}+\frac {7 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d e^2 \sqrt {e \sin (c+d x)}}+\frac {5 a^2 \sec (c+d x) \sqrt {e \sin (c+d x)}}{3 d e^3} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 18.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.72 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=-\frac {a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \left (3+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},1,\frac {1}{4},\sin ^2(c+d x)\right )+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {3}{2},\frac {1}{4},\sin ^2(c+d x)\right )+3 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\sin ^2(c+d x)\right )\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \arcsin (\sin (c+d x))\right ) \sqrt {e \sin (c+d x)}}{3 d e^3} \]
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Time = 17.50 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.29
method | result | size |
default | \(\frac {a^{2} \left (7 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) e^{\frac {7}{2}}-14 e^{\frac {7}{2}} \cos \left (d x +c \right )^{4}-8 e^{\frac {7}{2}} \cos \left (d x +c \right )^{3}+12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right )^{3} e^{2}+12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right )^{3} e^{2}+20 e^{\frac {7}{2}} \cos \left (d x +c \right )^{2}+8 e^{\frac {7}{2}} \cos \left (d x +c \right )-12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right ) e^{2}-12 \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right ) e^{2}-6 e^{\frac {7}{2}}\right )}{6 e^{\frac {9}{2}} \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \left (\cos \left (d x +c \right )^{2}-1\right ) d}\) | \(301\) |
parts | \(-\frac {a^{2} \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {5}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{3 e^{2} \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}-\frac {a^{2} \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (5 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {5}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+10 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-6 \sin \left (d x +c \right )\right )}{6 e^{2} \sin \left (d x +c \right )^{2} \sqrt {-e \sin \left (d x +c \right ) \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 a^{2} \left (-\frac {2}{3 e \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {\arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {5}{2}}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {5}{2}}}\right )}{d}\) | \(330\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 849, normalized size of antiderivative = 3.63 \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (c+d x))^2}{(e \sin (c+d x))^{5/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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