Integrand size = 23, antiderivative size = 310 \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {6 a e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}-\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d} \]
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Time = 0.42 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3966, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719} \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {a e^{5/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {a e^{5/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {6 a e^2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}-\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (3 a \sec (c+d x)+5 a) (e \tan (c+d x))^{3/2}}{15 d} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2652
Rule 2693
Rule 2695
Rule 2719
Rule 3557
Rule 3966
Rule 3969
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}-\frac {1}{5} \left (2 e^2\right ) \int \left (\frac {5 a}{2}+\frac {3}{2} a \sec (c+d x)\right ) \sqrt {e \tan (c+d x)} \, dx \\ & = \frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}-\frac {1}{5} \left (3 a e^2\right ) \int \sec (c+d x) \sqrt {e \tan (c+d x)} \, dx-\left (a e^2\right ) \int \sqrt {e \tan (c+d x)} \, dx \\ & = -\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}+\frac {1}{5} \left (6 a e^2\right ) \int \cos (c+d x) \sqrt {e \tan (c+d x)} \, dx-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{d} \\ & = -\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}-\frac {\left (2 a e^3\right ) \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}+\frac {\left (6 a e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{5 \sqrt {\sin (c+d x)}} \\ & = -\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}+\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}+\frac {\left (6 a e^2 \cos (c+d x) \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{5 \sqrt {\sin (2 c+2 d x)}} \\ & = \frac {6 a e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}-\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}-\frac {\left (a e^{5/2}\right ) \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} d}-\frac {\left (a e^{5/2}\right ) \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} d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 d}-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 d} \\ & = -\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {6 a e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}-\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d}-\frac {\left (a e^{5/2}\right ) \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} d}+\frac {\left (a e^{5/2}\right ) \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} d} \\ & = \frac {a e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {6 a e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}-\frac {6 a e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 e (5 a+3 a \sec (c+d x)) (e \tan (c+d x))^{3/2}}{15 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.58 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.60 \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\frac {a (1+\cos (c+d x)) \csc (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \left (12+20 \cos (c+d x)-36 \cos ^2(c+d x)+\frac {24 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\tan ^2(c+d x)\right )}{\sqrt {\sec ^2(c+d x)}}+15 \arcsin (\cos (c+d x)-\sin (c+d x)) \cot ^2(c+d x) \sqrt {\sin (2 (c+d x))}+15 \cot ^2(c+d x) \log \left (\cos (c+d x)+\sin (c+d x)+\sqrt {\sin (2 (c+d x))}\right ) \sqrt {\sin (2 (c+d x))}\right ) (e \tan (c+d x))^{5/2}}{60 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(579\) vs. \(2(274)=548\).
Time = 6.00 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.87
method | result | size |
parts | \(\frac {2 a e \left (\frac {\left (e \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \tan \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \tan \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {a \sqrt {2}\, \left (-6 \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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{3}+3 \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 ) \cos \left (d x +c \right )^{3}-6 \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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{2}+3 \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 ) \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )^{3} \sqrt {2}-4 \sqrt {2}\, \cos \left (d x +c \right )^{2}+\sqrt {2}\right ) e^{2} \sqrt {e \tan \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )}{5 d \left (\cos \left (d x +c \right )^{2}-1\right )}\) | \(580\) |
default | \(\text {Expression too large to display}\) | \(1152\) |
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Timed out. \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=a \left (\int \left (e \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx + \int \left (e \tan {\left (c + d x \right )}\right )^{\frac {5}{2}} \sec {\left (c + d x \right )}\, dx\right ) \]
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Exception generated. \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x)) (e \tan (c+d x))^{5/2} \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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