Integrand size = 23, antiderivative size = 274 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}-\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}-\frac {a \sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d}+\frac {a \sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d} \]
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Time = 0.24 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {3985, 3969, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2694, 2653, 2720} \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=-\frac {a \sqrt {\tan (c+d x)} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)}}{\sqrt {2} d}+\frac {a \sqrt {\tan (c+d x)} \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \sqrt {e \cot (c+d x)}}{\sqrt {2} d}-\frac {a \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right ) \sqrt {e \cot (c+d x)}}{d} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2653
Rule 2694
Rule 2720
Rule 3557
Rule 3969
Rule 3985
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {a+a \sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx+\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{\sqrt {\cos (c+d x)}}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \left (a \sqrt {e \cot (c+d x)} \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx+\frac {\left (2 a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}-\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d} \\ & = \frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}-\frac {a \sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d}+\frac {a \sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d}+\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = \frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}-\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}-\frac {a \sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d}+\frac {a \sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.83 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.62 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\frac {a (1+\cos (c+d x)) \sqrt {e \cot (c+d x)} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (4 \sqrt [4]{-1} \sqrt {\cot (c+d x)} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\cot (c+d x)}\right ),-1\right )+\sqrt {\csc ^2(c+d x)} \left (-\arcsin (\cos (c+d x)-\sin (c+d x))+\log \left (\cos (c+d x)+\sin (c+d x)+\sqrt {\sin (2 (c+d x))}\right )\right ) \sqrt {\sin (2 (c+d x))}\right )}{4 d \sqrt {\csc ^2(c+d x)}} \]
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Result contains complex when optimal does not.
Time = 9.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a \sqrt {2}\, \left (i \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {e \cot \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (1+\sec \left (d x +c \right )\right )}{d}\) | \(141\) |
parts | \(-\frac {a e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (1+\sec \left (d x +c \right )\right )}{d}\) | \(241\) |
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Timed out. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\text {Timed out} \]
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\[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=a \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\int { \sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\int \sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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