Integrand size = 25, antiderivative size = 290 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}} \]
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Time = 0.45 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3973, 3967, 3969, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2694, 2653, 2720} \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a d \sqrt {e}}-\frac {\log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {\log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{3 a d \sqrt {e \tan (c+d x)}} \]
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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 3967
Rule 3969
Rule 3973
Rubi steps \begin{align*} \text {integral}& = \frac {e^2 \int \frac {-a+a \sec (c+d x)}{(e \tan (c+d x))^{5/2}} \, dx}{a^2} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}+\frac {2 \int \frac {\frac {3 a}{2}-\frac {1}{2} a \sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{3 a}+\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{a} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{a d}-\frac {\sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 a \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}+\frac {(2 e) \text {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a d}-\frac {\left (\sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{3 a \sqrt {e \tan (c+d x)}} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a d}+\frac {\text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a d} \\ & = \frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a d}+\frac {\text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a d}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}} \\ & = -\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.24 (sec) , antiderivative size = 1253, normalized size of antiderivative = 4.32 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\frac {2 e^{-i (c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (1+e^{2 i (c+d x)}\right ) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec (c+d x) \sqrt {\tan (c+d x)}}{3 d (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (e^{4 i c} \sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec (c+d x) \sqrt {\tan (c+d x)}}{2 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (\sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 e^{4 i c} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec (c+d x) \sqrt {\tan (c+d x)}}{2 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}-\frac {e^{-i (2 c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 \left (-1+e^{4 i (c+d x)}\right )+e^{4 i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec (c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}+\frac {e^{-i d x} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3-3 e^{4 i (c+d x)}+e^{2 i (c+2 d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec (c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}+\frac {\cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c+d x) \left (-\frac {4}{3 d}+\frac {2 (3-2 \cos (c)+3 \cos (2 c)) \cos (d x) \sec (2 c)}{3 d}+\frac {2 \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}-\frac {2 \sec (2 c) (-2 \sin (c)+3 \sin (2 c)) \sin (d x)}{3 d}\right ) \tan (c+d x)}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}}+\frac {4 \sqrt [4]{-1} \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ),-1\right ) \sec ^4(c+d x) \sqrt {\tan (c+d x)}}{3 d (a+a \sec (c+d x)) \sqrt {e \tan (c+d x)} \left (1+\tan ^2(c+d x)\right )^{3/2}} \]
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Result contains complex when optimal does not.
Time = 3.75 (sec) , antiderivative size = 611, normalized size of antiderivative = 2.11
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (3 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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 )-3 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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 )-8 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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 )+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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 )+2 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right ) \csc \left (d x +c \right )}{6 a d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \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 )}\, \sqrt {-\frac {e \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(611\) |
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\frac {\int \frac {1}{\sqrt {e \tan {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \tan {\left (c + d x \right )}}}\, dx}{a} \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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