Integrand size = 43, antiderivative size = 274 \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {8 a^4 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (28 A+17 B+12 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}}+\frac {2 a (9 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{63 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (63 A+117 B+97 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 (21 A+24 B+19 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{45 d \cos ^{\frac {3}{2}}(c+d x)} \]
-8/15*a^4*(21*A+24*B+19*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c) *EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/21*a^4*(28*A+17*B+12*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+2/63*a*(9*B+8*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2 /9*C*(a+a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(9/2)+2/315*(63*A+117*B+97 *C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(5/2)+4/45*(21*A+24*B+1 9*C)*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(3/2)+4/105*a^4*(287*A+2 53*B+193*C)*sin(d*x+c)/d/cos(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.43 (sec) , antiderivative size = 1748, normalized size of antiderivative = 6.38 \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Sqrt[Cos[c + d*x]]*(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)*(-1/120*((-183*A - 192*B - 152*C + 15*A*Cos [2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36*d) + (Sec [c]*Sec[c + d*x]^4*(7*C*Sin[c] + 9*B*Sin[d*x] + 36*C*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c] + 180*C*Sin[c] + 63*A*Sin[d*x] + 252*B *Sin[d*x] + 427*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(140*A*Sin[c] + 235*B*Sin[c] + 240*C*Sin[c] + 693*A*Sin[d*x] + 672*B*Sin[d*x] + 532*C*S in[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 252*B*Sin[c] + 4 27*C*Sin[c] + 420*A*Sin[d*x] + 705*B*Sin[d*x] + 720*C*Sin[d*x]))/(1260*d)) )/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^6* Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*S ec[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 - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqr t[-(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]) - (17*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*Se c[c + d*x])^4*(A + B*Sec[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]]]])/(21*d...
Time = 2.08 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.06, 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, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3500, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4600 |
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {11}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (9 B+8 C)+a (9 A-C) \cos (c+d x))}{2 \cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (9 B+8 C)+a (9 A-C) \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (9 B+8 C)+a (9 A-C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^3 \left ((63 A+117 B+97 C) a^2+3 (21 A-3 B-5 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(\cos (c+d x) a+a)^3 \left ((63 A+117 B+97 C) a^2+3 (21 A-3 B-5 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((63 A+117 B+97 C) a^2+3 (21 A-3 B-5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {(\cos (c+d x) a+a)^2 \left (21 (21 A+24 B+19 C) a^3+(126 A-81 B-86 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (21 (21 A+24 B+19 C) a^3+(126 A-81 B-86 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (\frac {2}{3} \int \frac {9 (\cos (c+d x) a+a) \left (a^4 (287 A+253 B+193 C)-a^4 (7 A+83 B+73 C) \cos (c+d x)\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \int \frac {(\cos (c+d x) a+a) \left (a^4 (287 A+253 B+193 C)-a^4 (7 A+83 B+73 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (287 A+253 B+193 C)-a^4 (7 A+83 B+73 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \int \frac {-\left ((7 A+83 B+73 C) \cos ^2(c+d x) a^5\right )+(287 A+253 B+193 C) a^5+\left (a^5 (287 A+253 B+193 C)-a^5 (7 A+83 B+73 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \int \frac {-\left ((7 A+83 B+73 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5\right )+(287 A+253 B+193 C) a^5+\left (a^5 (287 A+253 B+193 C)-a^5 (7 A+83 B+73 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \left (2 \int \frac {5 a^5 (28 A+17 B+12 C)-7 a^5 (21 A+24 B+19 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \left (2 \int \frac {5 a^5 (28 A+17 B+12 C)-7 a^5 (21 A+24 B+19 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \left (2 \left (5 a^5 (28 A+17 B+12 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-7 a^5 (21 A+24 B+19 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \left (2 \left (5 a^5 (28 A+17 B+12 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-7 a^5 (21 A+24 B+19 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \left (3 \left (2 \left (5 a^5 (28 A+17 B+12 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {14 a^5 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\right )+\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {2 a^2 (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{7} \left (\frac {2}{5} \left (\frac {14 (21 A+24 B+19 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \cos ^{\frac {3}{2}}(c+d x)}+3 \left (\frac {2 a^5 (287 A+253 B+193 C) \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+2 \left (\frac {10 a^5 (28 A+17 B+12 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {14 a^5 (21 A+24 B+19 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {2 a^3 (63 A+117 B+97 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )}{9 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((2*a ^2*(9*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2 )) + ((2*a^3*(63*A + 117*B + 97*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5 *d*Cos[c + d*x]^(5/2)) + (2*((14*(21*A + 24*B + 19*C)*(a^5 + a^5*Cos[c + d *x])*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)) + 3*(2*((-14*a^5*(21*A + 24*B + 19*C)*EllipticE[(c + d*x)/2, 2])/d + (10*a^5*(28*A + 17*B + 12*C)*Elliptic F[(c + d*x)/2, 2])/d) + (2*a^5*(287*A + 253*B + 193*C)*Sin[c + d*x])/(d*Sq rt[Cos[c + d*x]]))))/5)/7)/(9*a)
3.13.15.3.1 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_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
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. \(1399\) vs. \(2(302)=604\).
Time = 6.46 (sec) , antiderivative size = 1400, normalized size of antiderivative = 5.11
int(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
-32*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4*(3/16*A* (sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(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)*EllipticF(cos(1/2*d*x+1/2*c),2^( 1/2))+1/16*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(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)*EllipticF(cos(1/2*d *x+1/2*c),2^(1/2))+1/16*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(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)*(Ellip ticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))+1/ 16*C*(-1/144*cos(1/2*d*x+1/2*c)*(-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)^2-1/2)^5-7/180*cos(1/2*d*x+1/2*c)*(-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)^2-1/2)^3-1 4/15*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)/(-(-2*cos(1/2*d*x+1/2*c)^2+1) *sin(1/2*d*x+1/2*c)^2)^(1/2)+7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(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 )*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-7/15*(sin(1/2*d*x+1/2*c)^2)^(1/2)* (-2*cos(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)*(EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-EllipticE(cos(1/2*d*x+ 1/2*c),2^(1/2))))+(1/16*B+1/4*C)*(-1/56*cos(1/2*d*x+1/2*c)*(-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)^2-1/2)^4-5/42*co s(1/2*d*x+1/2*c)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)/(...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.11 \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (28 \, A + 17 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (28 \, A + 17 \, B + 12 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 42 i \, \sqrt {2} {\left (21 \, A + 24 \, B + 19 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 42 i \, \sqrt {2} {\left (21 \, A + 24 \, B + 19 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (21 \, {\left (99 \, A + 96 \, B + 76 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 15 \, {\left (28 \, A + 47 \, B + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 7 \, {\left (9 \, A + 36 \, B + 61 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 45 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 35 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )^{5}} \]
integrate(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="fricas")
-2/315*(30*I*sqrt(2)*(28*A + 17*B + 12*C)*a^4*cos(d*x + c)^5*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 30*I*sqrt(2)*(28*A + 17*B + 12*C)*a^4*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) + 42*I*sqrt(2)*(21*A + 24*B + 19*C)*a^4*cos(d*x + c)^5*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 42*I*sqrt(2)*(21*A + 24*B + 19*C)*a^4*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (21*(99*A + 96*B + 76*C)*a^4*cos(d*x + c)^4 + 15*(28*A + 47*B + 48*C)*a^4*cos(d*x + c)^3 + 7*(9*A + 36*B + 61*C)*a^4*cos(d*x + c)^2 + 45*(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)^5)
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(1/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \sqrt {\cos \left (d x + c\right )} \,d x } \]
integrate(cos(d*x+c)^(1/2)*(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*s qrt(cos(d*x + c)), x)
Time = 24.55 (sec) , antiderivative size = 724, normalized size of antiderivative = 2.64 \[ \int \sqrt {\cos (c+d x)} (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
(2*(A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 4*A*a^4*ellipticF(c/2 + (d*x)/2, 2 )))/d + (2*((34*A*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^( 1/2)) + (A*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)))* hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(5*d) + (8*((11*C*a^4*sin(c + d*x))/(cos(c + d*x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (3*C*a^4*sin(c + d*x) )/(cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2)))*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(21*d) - (8*((61*C*a^4*sin(c + d*x))/(cos(c + d*x)^(1/2) *(sin(c + d*x)^2)^(1/2)) + (5*C*a^4*sin(c + d*x))/(cos(c + d*x)^(5/2)*(sin (c + d*x)^2)^(1/2)))*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2))/(135*d) + (2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((289*C*a^4*sin(c + d*x)) /(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (66*C*a^4*sin(c + d*x))/(co s(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2)) + (5*C*a^4*sin(c + d*x))/(cos(c + d*x)^(9/2)*(sin(c + d*x)^2)^(1/2))))/(45*d) + (2*B*a^4*ellipticF(c/2 + (d *x)/2, 2))/d + (8*A*a^4*sin(c + d*x)*hypergeom([-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)) - (8*A*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 7/4, cos(c + d*x)^2))/(15*d*cos(c + d*x)^(1/2 )*(sin(c + d*x)^2)^(1/2)) + (8*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)) + (4*B* a^4*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2))/(d*cos(c + d *x)^(3/2)*(sin(c + d*x)^2)^(1/2)) + (8*B*a^4*sin(c + d*x)*hypergeom([-5...