Integrand size = 25, antiderivative size = 347 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \]
2*cot(d*x+c)*(1-sec(d*x+c))/a/d/(e*cot(d*x+c))^(1/2)+2*sin(d*x+c)/a/d/(e*c ot(d*x+c))^(1/2)+2*cos(d*x+c)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x )*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))/a/d/(e*cot(d*x+c))^(1/2)/sin(2*d*x+ 2*c)^(1/2)+1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/a/d*2^(1/2)/(e*cot(d*x+ c))^(1/2)/tan(d*x+c)^(1/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/a/d*2^(1 /2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)+1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2 )+tan(d*x+c))/a/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)-1/4*ln(1+2 ^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d *x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.94 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=-\frac {\csc (c+d x) \left (24 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\tan ^2(c+d x)\right )+8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-3 \cot ^{\frac {3}{2}}(c+d x) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right ) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{6 a d \sqrt {e \cot (c+d x)}} \]
-1/6*(Csc[c + d*x]*(24*Cot[c + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, - Tan[c + d*x]^2] + 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 3* Cot[c + d*x]^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2*S qrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqr t[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*(1 + Sqrt[Sec[c + d*x]^2])*Si n[(c + d*x)/2]^2)/(a*d*Sqrt[e*Cot[c + d*x]])
Time = 1.20 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.76, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.120, Rules used = {3042, 4388, 3042, 4376, 25, 3042, 4370, 27, 3042, 4372, 3042, 3093, 3042, 3095, 3042, 3052, 3042, 3119, 3957, 266, 826, 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.
\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) \sqrt {e \cot (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a \sec (c+d x)+a) \sqrt {e \cot (c+d x)}}dx\) |
\(\Big \downarrow \) 4388 |
\(\displaystyle \frac {\int \frac {\sqrt {\tan (c+d x)}}{\sec (c+d x) a+a}dx}{\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )}}{\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4376 |
\(\displaystyle \frac {\int -\frac {a-a \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}dx}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {a-a \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}dx}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-\cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4370 |
\(\displaystyle -\frac {2 \int -\frac {1}{2} (\sec (c+d x) a+a) \sqrt {\tan (c+d x)}dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {-\int (\sec (c+d x) a+a) \sqrt {\tan (c+d x)}dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-\int \sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 4372 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \int \sec (c+d x) \sqrt {\tan (c+d x)}dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \int \sec (c+d x) \sqrt {\tan (c+d x)}dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3093 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-2 \int \cos (c+d x) \sqrt {\tan (c+d x)}dx\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-2 \int \frac {\sqrt {\tan (c+d x)}}{\sec (c+d x)}dx\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3095 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3052 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {-a \int \sqrt {\tan (c+d x)}dx-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle -\frac {-\frac {a \int \frac {\sqrt {\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle -\frac {-\frac {2 a \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle -\frac {-\frac {2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {2 (a-a \sec (c+d x))}{d \sqrt {\tan (c+d x)}}-a \left (\frac {2 \cos (c+d x) \tan ^{\frac {3}{2}}(c+d x)}{d}-\frac {2 \cos (c+d x) \sqrt {\tan (c+d x)} E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{d \sqrt {\sin (2 c+2 d x)}}\right )}{a^2 \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}\) |
-(((-2*a*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x ]] + Tan[c + d*x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[ c + d*x]]/(2*Sqrt[2]))/2))/d - (2*(a - a*Sec[c + d*x]))/(d*Sqrt[Tan[c + d* x]]) - a*((-2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[Tan[c + d*x]] )/(d*Sqrt[Sin[2*c + 2*d*x]]) + (2*Cos[c + d*x]*Tan[c + d*x]^(3/2))/d))/(a^ 2*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]))
3.3.45.3.1 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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]] , x_Symbol] :> Simp[Sqrt[a*Sin[e + f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]) Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f}, x]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[a^2*(a*Sec[e + f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[a^2*((m - 2)/(m + n - 1)) Int[(a*Sec[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && ( GtQ[m, 1] || (EqQ[m, 1] && EqQ[n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[ 2*m, 2*n]
Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[Sqrt[Cos[e + f*x]]*(Sqrt[b*Tan[e + f*x]]/Sqrt[Sin[e + f*x]]) Int[ Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x _)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m Int[(a + b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && !IntegerQ[m]
Result contains complex when optimal does not.
Time = 8.32 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, \left (\operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 i \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{a d \sqrt {e \cot \left (d x +c \right )}}\) | \(252\) |
(-1/2-1/2*I)/a/d*2^(1/2)*(EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1 /2*I,1/2*2^(1/2))-I*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1 /2*2^(1/2))+2*I*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-I*E llipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-2*EllipticE((csc(d*x +c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))+EllipticF((csc(d*x+c)-cot(d*x+c)+1)^( 1/2),1/2*2^(1/2)))*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+ 1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)/(e*cot(d*x+c))^(1/2)*(cot(d*x+c)+cs c(d*x+c))
Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a} \]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]