Integrand size = 25, antiderivative size = 315 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {\sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}+\frac {\sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a d}+\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d}-\frac {\sqrt {e} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a d}+\frac {2 e (1-\sec (c+d x))}{a d \sqrt {e \tan (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{a d \sqrt {\sin (2 c+2 d x)}}+\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{a d e} \]
-1/2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a/d*2^(1/2)+1/ 2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))*e^(1/2)/a/d*2^(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))*e^(1/2)/a/d*2^( 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))*e^(1/ 2)/a/d*2^(1/2)+2*e*(1-sec(d*x+c))/a/d/(e*tan(d*x+c))^(1/2)+2*cos(d*x+c)*(s in(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))*(e*tan(d*x+c))^(1/2)/a/d/sin(2*d*x+2*c)^(1/2)+2*cos(d*x+c)*(e*tan(d *x+c))^(3/2)/a/d/e
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.58 (sec) , antiderivative size = 2715, normalized size of antiderivative = 8.62 \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\text {Result too large to show} \]
(Cos[c/2 + (d*x)/2]^2*Sec[c + d*x]*((-2*Cos[c/2]*Cos[d*x]*Sec[2*c]*(4*Sin[ c/2] + Sin[(3*c)/2] + Sin[(5*c)/2]))/(d*(1 + 2*Cos[c])) - (4*Sec[c/2]*Sec[ c/2 + (d*x)/2]*Sin[(d*x)/2])/d - ((-2 - 5*Cos[c] - 6*Cos[2*c] + Cos[3*c])* Sec[2*c]*Sin[d*x])/(d*(1 + 2*Cos[c])) - (4*Tan[c/2])/d)*Sqrt[e*Tan[c + d*x ]])/(a + a*Sec[c + d*x]) + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*Ar cTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sq rt[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[e*T an[c + d*x]])/(2*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^( (2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[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))]*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]*Sqr t[e*Tan[c + d*x]])/(2*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x)))) /(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(1 + 2*Cos[c])*(a + a*Sec[c + d*x])*Sqrt[Tan[c + d*x]]) - ((-(E^((6*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))]*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[...
Time = 1.29 (sec) , antiderivative size = 298, normalized size of antiderivative = 0.95, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.040, Rules used = {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.
| \(\displaystyle \int \frac {\sqrt {e \tan (c+d x)}}{a \sec (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}{a \csc \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {e^2 \int -\frac {a-a \sec (c+d x)}{(e \tan (c+d x))^{3/2}}dx}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e^2 \int \frac {a-a \sec (c+d x)}{(e \tan (c+d x))^{3/2}}dx}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \int \frac {a-a \csc \left (c+d x+\frac {\pi }{2}\right )}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{a^2}\) |
| \(\Big \downarrow \) 4370 |
| \(\displaystyle -\frac {e^2 \left (\frac {2 \int -\frac {1}{2} (\sec (c+d x) a+a) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\int (\sec (c+d x) a+a) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\int \sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )} \left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )dx}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 4372 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \int \sec (c+d x) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \int \sec (c+d x) \sqrt {e \tan (c+d x)}dx}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3093 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-2 \int \cos (c+d x) \sqrt {e \tan (c+d x)}dx\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-2 \int \frac {\sqrt {e \tan (c+d x)}}{\sec (c+d x)}dx\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3095 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)} \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}dx}{\sqrt {\sin (c+d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3052 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) \sqrt {e \tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) \sqrt {e \tan (c+d x)} \int \sqrt {\sin (2 c+2 d x)}dx}{\sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle -\frac {e^2 \left (-\frac {a \int \sqrt {e \tan (c+d x)}dx+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {a e \int \frac {\sqrt {e \tan (c+d x)}}{\tan ^2(c+d x) e^2+e^2}d(e \tan (c+d x))}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \int \frac {e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \int \frac {e^2 \tan ^2(c+d x)+e}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \tan ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )-\frac {1}{2} \int \frac {e-e^2 \tan ^2(c+d x)}{e^4 \tan ^4(c+d x)+e^2}d\sqrt {e \tan (c+d x)}\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)-\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \tan (c+d x)}}{e^2 \tan ^2(c+d x)+\sqrt {2} e^{3/2} \tan (c+d x)+e}d\sqrt {e \tan (c+d x)}}{2 \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {e^2 \left (-\frac {\frac {2 a e \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {e} \tan (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \tan (c+d x)\right )}{\sqrt {2} \sqrt {e}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (\sqrt {2} e^{3/2} \tan (c+d x)+e^2 \tan ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}\right )\right )}{d}+a \left (\frac {2 \cos (c+d x) (e \tan (c+d x))^{3/2}}{d e}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{d \sqrt {\sin (2 c+2 d x)}}\right )}{e^2}-\frac {2 (a-a \sec (c+d x))}{d e \sqrt {e \tan (c+d x)}}\right )}{a^2}\) |
-((e^2*((-2*(a - a*Sec[c + d*x]))/(d*e*Sqrt[e*Tan[c + d*x]]) - ((2*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 + (Log[e - Sqrt[2]*e^(3 /2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e]) - Log[e + Sqrt[ 2]*e^(3/2)*Tan[c + d*x] + e^2*Tan[c + d*x]^2]/(2*Sqrt[2]*Sqrt[e]))/2))/d + a*((-2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(d *Sqrt[Sin[2*c + 2*d*x]]) + (2*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(d*e))) /e^2))/a^2)
3.2.23.3.1 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]
Result contains complex when optimal does not.
Time = 2.36 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, \left (2 i \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {e \tan \left (d x +c \right )}\, \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{a d}\) | \(256\) |
(1/2-1/2*I)/a/d*2^(1/2)*(2*I*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2 *2^(1/2))-I*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-I*Ellip ticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-EllipticPi((c sc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,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)))*(e*tan(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(co t(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(cot(d*x+c)+csc (d*x+c))
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sqrt {e \tan {\left (c + d x \right )}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a} \,d x } \]
\[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\int { \frac {\sqrt {e \tan \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \tan (c+d x)}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]