Integrand size = 23, antiderivative size = 190 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d} \]
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Time = 0.20 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3963, 3957, 2917, 2644, 331, 335, 218, 212, 209, 2716, 2720} \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\frac {a e^2 \sqrt {\sin (c+d x)} \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}+\frac {a e^2 \sqrt {\sin (c+d x)} \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)}}{d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {e \csc (c+d x)}}{3 d} \]
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Rule 209
Rule 212
Rule 218
Rule 331
Rule 335
Rule 2644
Rule 2716
Rule 2720
Rule 2917
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)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\left (\left (e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx\right ) \\ & = \left (a 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 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 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {1}{3} \left (a 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 (a 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 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a 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 {\left (a 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 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a 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 {\left (2 a 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 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {2 a 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 {\left (a 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 (a 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 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 d}-\frac {2 a e^2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 d}+\frac {a e^2 \arctan \left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {a e^2 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}{d}+\frac {2 a 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} \\ \end{align*}
Time = 2.34 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.71 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=-\frac {a (e \csc (c+d x))^{5/2} \left (6 \arctan \left (\sqrt {\csc (c+d x)}\right )+4 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {\csc (c+d x)}+3 \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \log \left (1+\sqrt {\csc (c+d x)}\right )+4 \sqrt {\csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sqrt {\sin (c+d x)}\right )}{6 d \csc ^{\frac {5}{2}}(c+d x)} \]
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Result contains complex when optimal does not.
Time = 3.14 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.82
method | result | size |
default | \(\frac {a \sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \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 ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-1\right ) \csc \left (d x +c \right )^{2}}{6 d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2}}+\frac {a \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right ) \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-3 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-3 \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 \,\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )\right ) \csc \left (d x +c \right )}{3 d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(536\) |
parts | \(\frac {a \sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (2 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \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 ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )+\left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-1\right ) \csc \left (d x +c \right )^{2}}{6 d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2}}+\frac {a \,e^{2} \sqrt {e \csc \left (d x +c \right )}\, \left (3 \cos \left (d x +c \right ) \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-3 \cos \left (d x +c \right ) \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )-2 \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-3 \arctan \left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )+3 \,\operatorname {arctanh}\left (\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )\right ) \csc \left (d x +c \right )}{3 d \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(536\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.21 (sec) , antiderivative size = 682, normalized size of antiderivative = 3.59 \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\left [-\frac {6 \, a \sqrt {-e} e^{2} \arctan \left (-\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) + e\right )}}\right ) \sin \left (d x + c\right ) - 3 \, a \sqrt {-e} e^{2} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {-e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 8 i \, a \sqrt {-2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 16 \, {\left (a e^{2} \cos \left (d x + c\right ) + a e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{24 \, d \sin \left (d x + c\right )}, -\frac {6 \, a e^{\frac {5}{2}} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{4 \, {\left (e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) - 3 \, a e^{\frac {5}{2}} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (\cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} - {\left (7 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) + 8\right )} \sqrt {e} \sqrt {\frac {e}{\sin \left (d x + c\right )}} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 8 i \, a \sqrt {-2 i \, e} e^{2} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 16 \, {\left (a e^{2} \cos \left (d x + c\right ) + a e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}}{24 \, d \sin \left (d x + c\right )}\right ] \]
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Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
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\[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int { \left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (e \csc (c+d x))^{5/2} (a+a \sec (c+d x)) \, dx=\int \left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2} \,d x \]
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