Integrand size = 43, antiderivative size = 271 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {8 a^4 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {8 a^4 (35 A+28 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}-\frac {4 a^4 (175 A+287 B+253 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (7 B+8 C) (a+a \cos (c+d x))^3 \sin (c+d x)}{35 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (35 A+77 B+73 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 (175 A+238 B+197 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{105 d \sqrt {\cos (c+d x)}} \]
-8/5*a^4*(7*B+8*C)*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*Ellipti cE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/21*a^4*(35*A+28*B+17*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 /35*a*(7*B+8*C)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(5/2)+2/7*C*(a+ a*cos(d*x+c))^4*sin(d*x+c)/d/cos(d*x+c)^(7/2)+2/105*(35*A+77*B+73*C)*(a^2+ a^2*cos(d*x+c))^2*sin(d*x+c)/d/cos(d*x+c)^(3/2)+4/105*(175*A+238*B+197*C)* (a^4+a^4*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(1/2)-4/105*a^4*(175*A+287*B+ 253*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 = 14.10 (sec) , antiderivative size = 1454, normalized size of antiderivative = 5.37 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \]
Integrate[Cos[c + d*x]^(3/2)*(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*((-20*A - 61*B - 64*C + 20*A*Cos[2*c ] + 5*B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(12*d) + (A*Cos[c ]*Sin[d*x])/(12*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*S ec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 28*C*Sin[d*x]))/(140*d) + (Sec[ c]*Sec[c + d*x]^2*(21*B*Sin[c] + 84*C*Sin[c] + 35*A*Sin[d*x] + 140*B*Sin[d *x] + 235*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 140*B *Sin[c] + 235*C*Sin[c] + 420*A*Sin[d*x] + 693*B*Sin[d*x] + 672*C*Sin[d*x]) )/(420*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (5*A*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*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]]]])/(3*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]*Hypergeometri cPFQ[{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 - A rcTan[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]...
Time = 2.07 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.08, 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, 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.
\(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (c+d x)^{3/2} (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\) |
\(\Big \downarrow \) 4600 |
\(\displaystyle \int \frac {(a \cos (c+d x)+a)^4 \left (A \cos ^2(c+d x)+B \cos (c+d x)+C\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+C\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx\) |
\(\Big \downarrow \) 3522 |
\(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^4 (a (7 B+8 C)+a (7 A-3 C) \cos (c+d x))}{2 \cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^4 (a (7 B+8 C)+a (7 A-3 C) \cos (c+d x))}{\cos ^{\frac {7}{2}}(c+d x)}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (7 B+8 C)+a (7 A-3 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \frac {\frac {2}{5} \int \frac {(\cos (c+d x) a+a)^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} \int \frac {(\cos (c+d x) a+a)^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((35 A+77 B+73 C) a^2+(35 A-21 B-39 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {(\cos (c+d x) a+a)^2 \left (a^3 (175 A+238 B+197 C)-21 a^3 (7 B+8 C) \cos (c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a^3 (175 A+238 B+197 C)-21 a^3 (7 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3454 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (2 \int \frac {3 (\cos (c+d x) a+a) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {(\cos (c+d x) a+a) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (175 A+189 B+141 C)-a^4 (175 A+287 B+253 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3447 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {-\left ((175 A+287 B+253 C) \cos ^2(c+d x) a^5\right )+(175 A+189 B+141 C) a^5+\left (a^5 (175 A+189 B+141 C)-a^5 (175 A+287 B+253 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \int \frac {-\left ((175 A+287 B+253 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5\right )+(175 A+189 B+141 C) a^5+\left (a^5 (175 A+189 B+141 C)-a^5 (175 A+287 B+253 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3502 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \int \frac {5 a^5 (35 A+28 B+17 C)-21 a^5 (7 B+8 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \int \frac {5 a^5 (35 A+28 B+17 C)-21 a^5 (7 B+8 C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3227 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx-21 a^5 (7 B+8 C) \int \sqrt {\cos (c+d x)}dx\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-21 a^5 (7 B+8 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \left (3 \left (\frac {2}{3} \left (5 a^5 (35 A+28 B+17 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {42 a^5 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
\(\Big \downarrow \) 3120 |
\(\displaystyle \frac {\frac {2 a^2 (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (\frac {2}{3} \left (\frac {2 (175 A+238 B+197 C) \sin (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d \sqrt {\cos (c+d x)}}+3 \left (\frac {2}{3} \left (\frac {10 a^5 (35 A+28 B+17 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}-\frac {42 a^5 (7 B+8 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )-\frac {2 a^5 (175 A+287 B+253 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^3 (35 A+77 B+73 C) \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )}{7 a}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac {7}{2}}(c+d x)}\) |
(2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + ((2*a ^2*(7*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2 )) + ((2*a^3*(35*A + 77*B + 73*C)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3* d*Cos[c + d*x]^(3/2)) + (2*((2*(175*A + 238*B + 197*C)*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + 3*((2*((-42*a^5*(7*B + 8*C)*E llipticE[(c + d*x)/2, 2])/d + (10*a^5*(35*A + 28*B + 17*C)*EllipticF[(c + d*x)/2, 2])/d))/3 - (2*a^5*(175*A + 287*B + 253*C)*Sqrt[Cos[c + d*x]]*Sin[ c + d*x])/(3*d))))/3)/5)/(7*a)
3.13.14.3.1 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[(-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. \(1534\) vs. \(2(299)=598\).
Time = 5.64 (sec) , antiderivative size = 1535, normalized size of antiderivative = 5.66
int(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, method=_RETURNVERBOSE)
8/105*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^4/sin(1/ 2*d*x+1/2*c)^3/(16*sin(1/2*d*x+1/2*c)^8-32*sin(1/2*d*x+1/2*c)^6+24*sin(1/2 *d*x+1/2*c)^4-8*sin(1/2*d*x+1/2*c)^2+1)*(882*B*(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))*s in(1/2*d*x+1/2*c)^2-2016*C*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^4+ 1008*C*(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))*(sin(1/2*d*x+1/2*c)^2)^(1/2)*sin(1/2*d*x+1/2*c)^2+3010*A*sin(1/2*d*x+1 /2*c)^6*cos(1/2*d*x+1/2*c)-140*B*(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))+1400*A*Elliptic F(cos(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)*sin(1/2*d*x+1/2*c)^6+1120*B*EllipticF(cos(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)*si n(1/2*d*x+1/2*c)^6+1176*B*EllipticE(cos(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)*sin(1/2*d*x+1/2*c)^6+6 80*C*EllipticF(cos(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)*sin(1/2*d*x+1/2*c)^6+1344*C*EllipticE(cos(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)*sin(1/2*d*x+1/2*c)^6-1470*A*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 2*c)+245*A*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-1764*B*(sin(1/2*d*x+...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.07 \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (10 i \, \sqrt {2} {\left (35 \, A + 28 \, B + 17 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 10 i \, \sqrt {2} {\left (35 \, A + 28 \, B + 17 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 42 i \, \sqrt {2} {\left (7 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 42 i \, \sqrt {2} {\left (7 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 21 \, {\left (20 \, A + 33 \, B + 32 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 5 \, {\left (7 \, A + 28 \, B + 47 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 21 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right ) + 15 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{105 \, d \cos \left (d x + c\right )^{4}} \]
integrate(cos(d*x+c)^(3/2)*(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)*(35*A + 28*B + 17*C)*a^4*cos(d*x + c)^4*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 10*I*sqrt(2)*(35*A + 28*B + 17*C)*a^4*cos(d*x + c)^4*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) + 42*I*sqrt(2)*(7*B + 8*C)*a^4*cos(d*x + c)^4*weierstrassZeta(-4 , 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 42*I*sqr t(2)*(7*B + 8*C)*a^4*cos(d*x + c)^4*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (35*A*a^4*cos(d*x + c)^4 + 2 1*(20*A + 33*B + 32*C)*a^4*cos(d*x + c)^3 + 5*(7*A + 28*B + 47*C)*a^4*cos( d*x + c)^2 + 21*(B + 4*C)*a^4*cos(d*x + c) + 15*C*a^4)*sqrt(cos(d*x + c))* sin(d*x + c))/(d*cos(d*x + c)^4)
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Timed out. \[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
integrate(cos(d*x+c)^(3/2)*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c) ^2),x, algorithm="maxima")
\[ \int \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
integrate(cos(d*x+c)^(3/2)*(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)^(3/2), x)
Time = 23.48 (sec) , antiderivative size = 560, normalized size of antiderivative = 2.07 \[ \int \cos ^{\frac {3}{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 (12\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+19\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+A\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {2\,\left (B\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{d}+\frac {2\,\left (\frac {34\,B\,a^4\,\sin \left (c+d\,x\right )}{\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {B\,a^4\,\sin \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{5/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d}+\frac {2\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,A\,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\,A\,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}}+\frac {8\,B\,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}}-\frac {8\,B\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {7}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\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 {4\,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 )}{d\,{\cos \left (c+d\,x\right )}^{3/2}\,\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 {5}{4},\frac {1}{2};\ -\frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{5\,d\,{\cos \left (c+d\,x\right )}^{5/2}\,\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 {7}{4},\frac {1}{2};\ -\frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,{\cos \left (c+d\,x\right )}^{7/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
(2*(12*A*a^4*ellipticE(c/2 + (d*x)/2, 2) + 19*A*a^4*ellipticF(c/2 + (d*x)/ 2, 2) + A*a^4*cos(c + d*x)^(1/2)*sin(c + d*x)))/(3*d) + (2*(B*a^4*elliptic E(c/2 + (d*x)/2, 2) + 4*B*a^4*ellipticF(c/2 + (d*x)/2, 2)))/d + (2*((34*B* a^4*sin(c + d*x))/(cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (B*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) + (2*C*a^4*ellipticF(c/2 + (d*x)/2, 2))/d + (8*A*a^4*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*c os(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (2*A*a^4*sin(c + d*x)*hypergeo m([-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*B*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*B*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*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)) + (4*C*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*C*a^4*sin(c + d*x)*hypergeom([-5/4, 1 /2], -1/4, cos(c + d*x)^2))/(5*d*cos(c + d*x)^(5/2)*(sin(c + d*x)^2)^(1/2) ) + (2*C*a^4*sin(c + d*x)*hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/(7 *d*cos(c + d*x)^(7/2)*(sin(c + d*x)^2)^(1/2))