Integrand size = 23, antiderivative size = 122 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=-\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
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Time = 0.18 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3963, 3957, 2917, 2644, 335, 304, 209, 212, 2719} \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=-\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}+\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 2644
Rule 2719
Rule 2917
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sec (c+d x)) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int (-a-a \cos (c+d x)) \sec (c+d x) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {a \int \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \int \sec (c+d x) \sqrt {\sin (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \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 \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 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {a \arctan \left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {a \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.76 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\frac {a \left (-4 \cot (c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},\csc ^2(c+d x)\right )+\sqrt {-\cot ^2(c+d x)} \sqrt {\csc (c+d x)} \left (2 \arctan \left (\sqrt {\csc (c+d x)}\right )-\log \left (1-\sqrt {\csc (c+d x)}\right )+\log \left (1+\sqrt {\csc (c+d x)}\right )\right )\right )}{2 d \sqrt {-\cot ^2(c+d x)} \sqrt {e \csc (c+d x)}} \]
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Result contains complex when optimal does not.
Time = 9.39 (sec) , antiderivative size = 544, normalized size of antiderivative = 4.46
method | result | size |
parts | \(-\frac {a \sqrt {2}\, \left (2 \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 {EllipticE}\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 )-\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 )+2 \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 {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\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 )+\sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \csc \left (d x +c \right )}{d \sqrt {e \csc \left (d x +c \right )}}+\frac {a \left (\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 )+\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 )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \csc \left (d x +c \right )}\, \sqrt {\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right )^{2}}}}\) | \(544\) |
default | \(\text {Expression too large to display}\) | \(1235\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.38 (sec) , antiderivative size = 554, normalized size of antiderivative = 4.54 \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\left [-\frac {2 \, 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 ) + 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 ) - 8 \, a \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 8 \, a \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{8 \, d e}, \frac {2 \, 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 ) + 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 ) + 8 \, a \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 8 \, a \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{8 \, d e}\right ] \]
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\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=a \left (\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\sqrt {e \csc {\left (c + d x \right )}}}\, dx\right ) \]
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\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\sqrt {e \csc \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {a+a \sec (c+d x)}{\sqrt {e \csc (c+d x)}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}} \,d x \]
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