Integrand size = 25, antiderivative size = 290 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d \sqrt {e}}+\frac {2 e (1-\sec (c+d x))}{3 a d (e \tan (c+d x))^{3/2}}-\frac {\operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d \sqrt {e \tan (c+d x)}} \]
-1/2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a/d*2^(1/2)/e^(1/2)+1/ 2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a/d*2^(1/2)/e^(1/2)-1/4*l n(e^(1/2)-2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a/d*2^(1/2)/e^( 1/2)+1/4*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a/d*2 ^(1/2)/e^(1/2)+1/3*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticF (cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*sin(2*d*x+2*c)^(1/2)/a/d/(e*tan(d*x +c))^(1/2)+2/3*e*(1-sec(d*x+c))/a/d/(e*tan(d*x+c))^(3/2)
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}} \]
(2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^(( 4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x) )]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTa nh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(2*d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + (Sqrt[ ((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^ ((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)* Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[ (-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2] ^2*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(2*d*E^((2*I)*c)*(-1 + E^((2* I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) - (Sqrt[((-I)*(- 1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^2* (3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqr t[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(2*c + d*x))*( -1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])*Sqrt[e*Tan[c + d*x]]) + ...
Time = 1.10 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.97, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.960, Rules used = {3042, 4376, 25, 3042, 4370, 27, 3042, 4372, 3042, 3094, 3042, 3053, 3042, 3120, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) \sqrt {e \tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right ) \sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4376 |
\(\displaystyle \frac {e^2 \int -\frac {a-a \sec (c+d x)}{(e \tan (c+d x))^{5/2}}dx}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {e^2 \int \frac {a-a \sec (c+d x)}{(e \tan (c+d x))^{5/2}}dx}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {e^2 \int \frac {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{a^2}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle -\frac {e^2 \left (\frac {2 \int -\frac {3 a-a \sec (c+d x)}{2 \sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e^2 \left (-\frac {\int \frac {3 a-a \sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {e^2 \left (-\frac {\int \frac {3 a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 4372 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-a \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-a \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}}dx}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3094 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3053 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle -\frac {e^2 \left (-\frac {3 a \int \frac {1}{\sqrt {e \tan (c+d x)}}dx-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {3 a e \int \frac {1}{\sqrt {e \tan (c+d x)} \left (\tan ^2(c+d x) e^2+e^2\right )}d(e \tan (c+d x))}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \int \frac {1}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 755 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\int \frac {e^2 \tan ^2(c+d x)+e}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {e^2 \left (-\frac {\frac {6 a e \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d}-\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {e \tan (c+d x)}}}{3 e^2}-\frac {2 (a-a \sec (c+d x))}{3 d e (e \tan (c+d x))^{3/2}}\right )}{a^2}\) |
-((e^2*((-2*(a - a*Sec[c + d*x]))/(3*d*e*(e*Tan[c + d*x])^(3/2)) - ((6*a*e *((-(ArcTan[1 - Sqrt[2]*Sqrt[e]*Tan[c + d*x]]/(Sqrt[2]*Sqrt[e])) + ArcTan[ 1 + Sqrt[2]*Sqrt[e]*Tan[c + d*x]]/(Sqrt[2]*Sqrt[e]))/(2*e) + (-1/2*Log[e - Sqrt[2]*e^(3/2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(Sqrt[2]*Sqrt[e]) + Lo g[e + Sqrt[2]*e^(3/2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e ]))/(2*e)))/d - (a*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[e*Tan[c + d*x]]))/(3*e^2)))/a^2)
3.2.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ )]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b *Cos[e + f*x]]) Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f }, x]
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]]) Int[ 1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_)), x_Symbol] :> Simp[(-(e*Cot[c + d*x])^(m + 1))*((a + b*Csc[c + d*x])/( d*e*(m + 1))), x] - Simp[1/(e^2*(m + 1)) Int[(e*Cot[c + d*x])^(m + 2)*(a* (m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && L tQ[m, -1]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(e*Cot[c + d*x])^m, x], x] + Simp[b Int[ (e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
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}}}\) | \(611\) |
-1/6/a/d*2^(1/2)*(3*I*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*co t(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d *x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*I*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)* (2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*Elliptic Pi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-8*(csc(d*x+c)-co t(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+ c))^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))+3*(csc(d* x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-c sc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2* 2^(1/2))+3*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^( 1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1 /2),1/2+1/2*I,1/2*2^(1/2))+2*(1-cos(d*x+c))^3*csc(d*x+c)^3-2*csc(d*x+c)+2* cot(d*x+c))*(1-cos(d*x+c))/((1-cos(d*x+c))^3*csc(d*x+c)^3+cot(d*x+c)-csc(d *x+c))^(1/2)/((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c)) ^(1/2)/(-e/((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(-cot(d*x+c)+csc(d*x+c)))^(1/ 2)*csc(d*x+c)
Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\frac {\int \frac {1}{\sqrt {e \tan {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \tan {\left (c + d x \right )}}}\, dx}{a} \]
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]