Integrand size = 23, antiderivative size = 182 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
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Time = 0.24 (sec) , antiderivative size = 182, 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, 327, 335, 218, 212, 209, 2715, 2720} \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}-\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d e \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rule 209
Rule 212
Rule 218
Rule 327
Rule 335
Rule 2644
Rule 2715
Rule 2720
Rule 2917
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sec (c+d x)) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int (-a-a \cos (c+d x)) \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {a \int \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \int \sec (c+d x) \sin ^{\frac {3}{2}}(c+d x) \, dx}{e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {x^{3/2}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,\sin (c+d x)\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a}{d e \sqrt {e \csc (c+d x)}}-\frac {2 a \cos (c+d x)}{3 d e \sqrt {e \csc (c+d x)}}+\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{3 d e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ \end{align*}
Time = 9.74 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.74 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=-\frac {a \left (12+4 \cos (c+d x)+6 \arctan \left (\sqrt {\csc (c+d x)}\right ) \sqrt {\csc (c+d x)}+3 \sqrt {\csc (c+d x)} \log \left (1-\sqrt {\csc (c+d x)}\right )-3 \sqrt {\csc (c+d x)} \log \left (1+\sqrt {\csc (c+d x)}\right )+\frac {4 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )}{\sqrt {\sin (c+d x)}}\right )}{6 d e \sqrt {e \csc (c+d x)}} \]
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Result contains complex when optimal does not.
Time = 10.88 (sec) , antiderivative size = 547, normalized size of antiderivative = 3.01
method | result | size |
parts | \(-\frac {a \sqrt {2}\, \left (i \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 ) \cos \left (d x +c \right )+i \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 )-\cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\right ) \sin \left (d x +c \right )}{3 d \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) e \left (\cos \left (d x +c \right )+1\right )}+\frac {a \left (\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 ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \cos \left (d x +c \right )-\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \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 ) \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 ) \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}-\sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}\, \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 \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )}{d \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right ) e \left (\cos \left (d x +c \right )+1\right )}\) | \(547\) |
default | \(\text {Expression too large to display}\) | \(1058\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.40 (sec) , antiderivative size = 608, normalized size of antiderivative = 3.34 \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\left [-\frac {6 \, a \sqrt {-e} \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 ) + 3 \, a \sqrt {-e} \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 ) + 16 \, {\left (a \cos \left (d x + c\right ) + 3 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} {\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} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{24 \, d e^{2}}, -\frac {6 \, a \sqrt {e} \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 ) - 3 \, a \sqrt {e} \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 ) + 16 \, {\left (a \cos \left (d x + c\right ) + 3 \, a\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) + 8 i \, a \sqrt {2 i \, e} {\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} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )}{24 \, d e^{2}}\right ] \]
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\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
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\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \csc (c+d x))^{3/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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