Integrand size = 25, antiderivative size = 194 \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=-\frac {2 a^2 e^{5/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{5/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {9 a^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d} \]
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Time = 0.47 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3957, 2952, 2715, 2721, 2719, 2644, 327, 335, 304, 209, 212, 2646} \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=-\frac {2 a^2 e^{5/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{5/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {9 a^2 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d} \]
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Rule 209
Rule 212
Rule 304
Rule 327
Rule 335
Rule 2644
Rule 2646
Rule 2715
Rule 2719
Rule 2721
Rule 2952
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int (-a-a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \, dx \\ & = \int \left (a^2 (e \sin (c+d x))^{5/2}+2 a^2 \sec (c+d x) (e \sin (c+d x))^{5/2}+a^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2}\right ) \, dx \\ & = a^2 \int (e \sin (c+d x))^{5/2} \, dx+a^2 \int \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \, dx+\left (2 a^2\right ) \int \sec (c+d x) (e \sin (c+d x))^{5/2} \, dx \\ & = -\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {x^{5/2}}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {1}{5} \left (3 a^2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx-\frac {1}{2} \left (3 a^2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx \\ & = -\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac {\left (2 a^2 e\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d}+\frac {\left (3 a^2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 \sqrt {\sin (c+d x)}}-\frac {\left (3 a^2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{2 \sqrt {\sin (c+d x)}} \\ & = -\frac {9 a^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac {\left (4 a^2 e\right ) \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d} \\ & = -\frac {9 a^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d}+\frac {\left (2 a^2 e^3\right ) \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}-\frac {\left (2 a^2 e^3\right ) \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d} \\ & = -\frac {2 a^2 e^{5/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a^2 e^{5/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}-\frac {9 a^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 d \sqrt {\sin (c+d x)}}-\frac {4 a^2 e (e \sin (c+d x))^{3/2}}{3 d}-\frac {2 a^2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 d}+\frac {a^2 e \sec (c+d x) (e \sin (c+d x))^{3/2}}{d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 17.64 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.06 \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\frac {2 a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \sec ^4\left (\frac {1}{2} \arcsin (\sin (c+d x))\right ) (e \sin (c+d x))^{5/2} \left (-15 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}+15 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {\cos ^2(c+d x)}-9 \sin ^{\frac {3}{2}}(c+d x)-10 \sqrt {\cos ^2(c+d x)} \sin ^{\frac {3}{2}}(c+d x)+9 \sqrt {\cos ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\sin ^2(c+d x)\right ) \sin ^{\frac {3}{2}}(c+d x)+3 \sin ^{\frac {7}{2}}(c+d x)\right )}{15 d \sin ^{\frac {5}{2}}(c+d x)} \]
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Time = 19.13 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.37
method | result | size |
default | \(\frac {a^{2} \left (-60 \sqrt {e \sin \left (d x +c \right )}\, e^{\frac {5}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right )+60 \sqrt {e \sin \left (d x +c \right )}\, e^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right ) \cos \left (d x +c \right )+54 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) e^{3}-27 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) e^{3}+12 e^{3} \cos \left (d x +c \right )^{4}+40 e^{3} \cos \left (d x +c \right )^{3}-42 e^{3} \cos \left (d x +c \right )^{2}-40 \cos \left (d x +c \right ) e^{3}+30 e^{3}\right )}{30 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(265\) |
parts | \(-\frac {a^{2} e^{3} \left (6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{4}+2 \sin \left (d x +c \right )^{2}\right )}{5 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a^{2} e^{3} \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}+2\right )}{2 \sqrt {-e \sin \left (d x +c \right ) \left (\sin \left (d x +c \right )-1\right ) \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {2 a^{2} \left (-\frac {2 e \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-e^{\frac {5}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+e^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )\right )}{d}\) | \(407\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 769, normalized size of antiderivative = 3.96 \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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\[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \]
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