Integrand size = 43, antiderivative size = 269 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (8 A+7 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {8 a^4 (17 A+28 B+35 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^4 (83 A+7 B-175 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (3 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (A-7 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{7 d}+\frac {4 (27 A-42 B-175 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{105 d} \]
8/5*a^4*(8*A+7*B)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Elliptic E(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/21*a^4*(17*A+28*B+35*C)*(cos(1/2*d*x+1/2 *c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))/d+2/ 3*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(3/2)+2/3*a*(3*B+8*C)*(a+a* cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(1/2)+4/105*a^4*(83*A+7*B-175*C)*sin (d*x+c)*cos(d*x+c)^(1/2)/d+2/7*(A-7*B-21*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x +c)*cos(d*x+c)^(1/2)/d+4/105*(27*A-42*B-175*C)*(a^4+a^4*cos(d*x+c))*sin(d* x+c)*cos(d*x+c)^(1/2)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.35 (sec) , antiderivative size = 1451, normalized size of antiderivative = 5.39 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Se c[c + d*x] + C*Sec[c + d*x]^2)*(-1/40*((32*A + 23*B - 20*C + 32*A*Cos[2*c] + 33*B*Cos[2*c] + 20*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((191*A + 112*B + 28* C)*Cos[d*x]*Sin[c])/(336*d) + ((4*A + B)*Cos[2*d*x]*Sin[2*c])/(40*d) + (A* Cos[3*d*x]*Sin[3*c])/(112*d) + ((191*A + 112*B + 28*C)*Cos[c]*Sin[d*x])/(3 36*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*( C*Sin[c] + 3*B*Sin[d*x] + 12*C*Sin[d*x]))/(12*d) + ((4*A + B)*Cos[2*c]*Sin [2*d*x])/(40*d) + (A*Cos[3*c]*Sin[3*d*x])/(112*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (17*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPF Q[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcT an[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]* Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(2 1*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Se c[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c ]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x ] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^6*Csc[c...
Time = 1.98 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.07, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4600, 3042, 3522, 27, 3042, 3454, 27, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3227, 3042, 3119, 3120}
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.
| \(\displaystyle \int \cos ^{\frac {7}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{7/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (3 B+8 C)+a (3 A-7 C) \cos (c+d x))}{2 \cos ^{\frac {3}{2}}(c+d x)}dx}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (3 B+8 C)+a (3 A-7 C) \cos (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)}dx}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (3 B+8 C)+a (3 A-7 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 \left ((3 A+21 B+49 C) a^2+3 (A-7 B-21 C) \cos (c+d x) a^2\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 \left ((3 A+21 B+49 C) a^2+3 (A-7 B-21 C) \cos (c+d x) a^2\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((3 A+21 B+49 C) a^2+3 (A-7 B-21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((12 A+63 B+140 C) a^3+(27 A-42 B-175 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((12 A+63 B+140 C) a^3+(27 A-42 B-175 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left ((29 A+91 B+175 C) a^4+(83 A+7 B-175 C) \cos (c+d x) a^4\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \int \frac {(\cos (c+d x) a+a) \left ((29 A+91 B+175 C) a^4+(83 A+7 B-175 C) \cos (c+d x) a^4\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((29 A+91 B+175 C) a^4+(83 A+7 B-175 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \int \frac {(83 A+7 B-175 C) \cos ^2(c+d x) a^5+(29 A+91 B+175 C) a^5+\left ((83 A+7 B-175 C) a^5+(29 A+91 B+175 C) a^5\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \int \frac {(83 A+7 B-175 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(29 A+91 B+175 C) a^5+\left ((83 A+7 B-175 C) a^5+(29 A+91 B+175 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {5 (17 A+28 B+35 C) a^5+21 (8 A+7 B) \cos (c+d x) a^5}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {5 (17 A+28 B+35 C) a^5+21 (8 A+7 B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \left (5 a^5 (17 A+28 B+35 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+21 a^5 (8 A+7 B) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \left (5 a^5 (17 A+28 B+35 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+21 a^5 (8 A+7 B) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {3}{5} \left (\frac {2}{3} \left (5 a^5 (17 A+28 B+35 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {42 a^5 (8 A+7 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2}{7} \left (\frac {2 (27 A-42 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^5 \cos (c+d x)+a^5\right )}{5 d}+\frac {3}{5} \left (\frac {2 a^5 (83 A+7 B-175 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2}{3} \left (\frac {10 a^5 (17 A+28 B+35 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {42 a^5 (8 A+7 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 a^3 (A-7 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}+\frac {2 a^2 (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt {\cos (c+d x)}}}{3 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac {3}{2}}(c+d x)}\) |
(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + ((6*a ^3*(A - 7*B - 21*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x] )/(7*d) + (2*a^2*(3*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[ Cos[c + d*x]]) + (2*((2*(27*A - 42*B - 175*C)*Sqrt[Cos[c + d*x]]*(a^5 + a^ 5*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (3*((2*((42*a^5*(8*A + 7*B)*Elliptic E[(c + d*x)/2, 2])/d + (10*a^5*(17*A + 28*B + 35*C)*EllipticF[(c + d*x)/2, 2])/d))/3 + (2*a^5*(83*A + 7*B - 175*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/ (3*d)))/5))/7)/(3*a)
3.13.12.3.1 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[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[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_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + a*d)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp [a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B *(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0 ])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(b*d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^m* (c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(a*d*m + b*c*(n + 1)) + (c*C - B*d)*( a*c*m + b*d*(n + 1)) + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2* (n + 1)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ [m, -2^(-1)] && (LtQ[n, -1] || EqQ[m + n + 2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) *(x_)]^2), x_Symbol] :> Simp[d^(m + 2) Int[(b + a*Cos[e + f*x])^m*(d*Cos[ e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr eeQ[{a, b, d, e, f, A, B, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Leaf count of result is larger than twice the leaf count of optimal. \(863\) vs. \(2(297)=594\).
Time = 432.82 (sec) , antiderivative size = 864, normalized size of antiderivative = 3.21
int(cos(d*x+c)^(7/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-4/105*(-240*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*cos(1/ 2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+24*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x +1/2*c)^2)^(1/2)*(48*A+7*B)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)-4*(-2* sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(577*A+203*B+35*C)*sin(1/ 2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+4*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1 /2*c)^2)^(1/2)*(391*A+224*B+245*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c) -2*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(167*A+133*B+245*C )*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-4*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/ 2)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c )^2)^(1/2)*(85*A*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-168*A*EllipticE(cos (1/2*d*x+1/2*c),2^(1/2))+140*B*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-147*B *EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+175*C*EllipticF(cos(1/2*d*x+1/2*c), 2^(1/2)))*sin(1/2*d*x+1/2*c)^2+170*A*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+ 1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1 /2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-336*A*(-2*sin(1/2*d*x+1/2*c)^4+s in(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2 *c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+280*B*(-2*sin(1/2*d*x +1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin( 1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-294*B*(-2* sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*(sin(1/2*d*x+1/2*c)^2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.06 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (10 i \, \sqrt {2} {\left (17 \, A + 28 \, B + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 10 i \, \sqrt {2} {\left (17 \, A + 28 \, B + 35 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 42 i \, \sqrt {2} {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} {\left (8 \, A + 7 \, B\right )} a^{4} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (15 \, A a^{4} \cos \left (d x + c\right )^{4} + 21 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 5 \, {\left (47 \, A + 28 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 35 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{105 \, d \cos \left (d x + c\right )^{2}} \]
integrate(cos(d*x+c)^(7/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-2/105*(10*I*sqrt(2)*(17*A + 28*B + 35*C)*a^4*cos(d*x + c)^2*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 10*I*sqrt(2)*(17*A + 28*B + 35*C)*a^4*cos(d*x + c)^2*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) - 42*I*sqrt(2)*(8*A + 7*B)*a^4*cos(d*x + c)^2*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 42*I*sqr t(2)*(8*A + 7*B)*a^4*cos(d*x + c)^2*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (15*A*a^4*cos(d*x + c)^4 + 2 1*(4*A + B)*a^4*cos(d*x + c)^3 + 5*(47*A + 28*B + 7*C)*a^4*cos(d*x + c)^2 + 105*(B + 4*C)*a^4*cos(d*x + c) + 35*C*a^4)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^2)
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(7/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {7}{2}} \,d x } \]
integrate(cos(d*x+c)^(7/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="giac")
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^4*c os(d*x + c)^(7/2), x)
Time = 20.13 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.74 \[ \int \cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (4\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+3\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,A\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\frac {2\,\left (12\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+19\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+C\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {4\,B\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,B\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {8\,A\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,B\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {8\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(4*A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 3*A*a^4*ellipticF(c/2 + (d*x)/2, 2) + 2*A*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (2*(12*C*a^4*ellipticE (c/2 + (d*x)/2, 2) + 19*C*a^4*ellipticF(c/2 + (d*x)/2, 2) + C*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) + (4*B*a^4*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d + (12*B*a^4*ellipticE(c/2 + (d*x)/2, 2))/d + (8*B*a^4*ellipticF(c/2 + (d*x)/2, 2))/d - (8*A*a^4*cos( c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/( 7*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^4*cos(c + d*x)^(9/2)*sin(c + d*x)*hyp ergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) + ( 2*B*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) - (2*B*a^4*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^ (1/2)) + (8*C*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2) )/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (2*C*a^4*sin(c + d*x)*hy pergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(3*d*cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2))