Integrand size = 25, antiderivative size = 270 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {2 a^2 e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {7 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d} \]
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Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3963, 3957, 2952, 2716, 2720, 2644, 331, 335, 218, 212, 209, 2650, 2651} \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\frac {2 a^2 e^2 \sqrt {\sin (c+d x)} \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}+\frac {2 a^2 e^2 \sqrt {\sin (c+d x)} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {5 a^2 e^2 \tan (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sec (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {7 a^2 e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {e \csc (c+d x)}}{3 d} \]
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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 2952
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^2}{\sin ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x))^2 \sec ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)}\right ) \, dx \\ & = \left (a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)} \, dx+\left (a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx+\left (2 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {1}{3} \left (a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx+\frac {1}{3} \left (5 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\sec ^2(c+d x)}{\sqrt {\sin (c+d x)}} \, dx+\frac {\left (2 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{x^{5/2} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {2 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac {1}{6} \left (5 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx+\frac {\left (2 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {7 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac {\left (4 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d} \\ & = -\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {7 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d}+\frac {\left (2 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d}+\frac {\left (2 a^2 e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d} \\ & = -\frac {2 a^2 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {4 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a^2 e^2 \csc (c+d x) \sqrt {e \csc (c+d x)} \sec (c+d x)}{3 d}+\frac {2 a^2 e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a^2 e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {7 a^2 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d}+\frac {5 a^2 e^2 \sqrt {e \csc (c+d x)} \tan (c+d x)}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.27 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.72 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 e^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {e \csc (c+d x)} \left (-7+6 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}-6 \text {arctanh}\left (\sqrt {\csc (c+d x)}\right ) \sqrt {\cos ^2(c+d x)} \sqrt {\csc (c+d x)}+4 \csc ^2(c+d x)+4 \sqrt {\cos ^2(c+d x)} \csc ^2(c+d x)+7 \sqrt {-\cot ^2(c+d x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\csc ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \csc ^{-1}(\csc (c+d x))\right ) \tan (c+d x)}{3 d} \]
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Result contains complex when optimal does not.
Time = 11.70 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.25
method | result | size |
default | \(\frac {a^{2} e^{2} \sqrt {2}\, \left (6 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+6 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-5 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-6 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-7 \sqrt {2}\, \cos \left (d x +c \right )+3 \sqrt {2}\right ) \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \csc \left (d x +c \right )}\, \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\) | \(607\) |
parts | \(\text {Expression too large to display}\) | \(1052\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 806, normalized size of antiderivative = 2.99 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Too large to display} \]
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Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
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Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]
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\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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