Integrand size = 43, antiderivative size = 303 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {4 a^3 (15 A+17 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (105 A+121 B+143 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (15 A+17 B+21 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2 (6 A+11 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (105 A+143 B+99 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)} \] Output:
-4/15*a^3*(15*A+17*B+21*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/231*a ^3*(105*A+121*B+143*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/1155*a^3 *(210*A+253*B+264*C)*sin(d*x+c)/d/cos(d*x+c)^(5/2)+4/231*a^3*(105*A+121*B+ 143*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+4/15*a^3*(15*A+17*B+21*C)*sin(d*x+c)/ d/cos(d*x+c)^(1/2)+2/11*A*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^(11/2 )+2/99*(6*A+11*B)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/cos(d*x+c)^(9/2)+2 /693*(105*A+143*B+99*C)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(7/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.28 (sec) , antiderivative size = 1418, normalized size of antiderivative = 4.68 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Cos[c + d*x]^(13/2),x]
Output:
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((15*A + 1 7*B + 21*C)*Csc[c]*Sec[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44 *d) + (Sec[c]*Sec[c + d*x]^5*(9*A*Sin[c] + 33*A*Sin[d*x] + 11*B*Sin[d*x])) /(396*d) + (Sec[c]*Sec[c + d*x]^4*(231*A*Sin[c] + 77*B*Sin[c] + 378*A*Sin[ d*x] + 297*B*Sin[d*x] + 99*C*Sin[d*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]*(5 25*A*Sin[c] + 605*B*Sin[c] + 715*C*Sin[c] + 1155*A*Sin[d*x] + 1309*B*Sin[d *x] + 1617*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]^3*(1890*A*Sin[c] + 1485*B*Sin[c] + 495*C*Sin[c] + 2310*A*Sin[d*x] + 2618*B*Sin[d*x] + 2079*C *Sin[d*x]))/(13860*d) + (Sec[c]*Sec[c + d*x]^2*(2310*A*Sin[c] + 2618*B*Sin [c] + 2079*C*Sin[c] + 3150*A*Sin[d*x] + 3630*B*Sin[d*x] + 4290*C*Sin[d*x]) )/(13860*d)) - (5*A*(a + a*Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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]]]]) /(22*d*Sqrt[1 + Cot[c]^2]) - (11*B*(a + a*Cos[c + d*x])^3*Csc[c]*Hypergeom etricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2 ]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(S qrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - Ar cTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (13*C*(a + a*Cos[c + d*x])^3*C sc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]...
Time = 1.93 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.99, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3522, 27, 3042, 3454, 27, 3042, 3454, 27, 3042, 3447, 3042, 3500, 27, 3042, 3227, 3042, 3116, 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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{13/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 (a (6 A+11 B)+a (3 A+11 C) \cos (c+d x))}{2 \cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 (a (6 A+11 B)+a (3 A+11 C) \cos (c+d x))}{\cos ^{\frac {11}{2}}(c+d x)}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (6 A+11 B)+a (3 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{9} \int \frac {(\cos (c+d x) a+a)^2 \left ((105 A+143 B+99 C) a^2+3 (15 A+11 B+33 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {(\cos (c+d x) a+a)^2 \left ((105 A+143 B+99 C) a^2+3 (15 A+11 B+33 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((105 A+143 B+99 C) a^2+3 (15 A+11 B+33 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \frac {3 (\cos (c+d x) a+a) \left ((210 A+253 B+264 C) a^3+5 (21 A+22 B+33 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \frac {(\cos (c+d x) a+a) \left ((210 A+253 B+264 C) a^3+5 (21 A+22 B+33 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((210 A+253 B+264 C) a^3+5 (21 A+22 B+33 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \frac {5 (21 A+22 B+33 C) \cos ^2(c+d x) a^4+(210 A+253 B+264 C) a^4+\left (5 (21 A+22 B+33 C) a^4+(210 A+253 B+264 C) a^4\right ) \cos (c+d x)}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \frac {5 (21 A+22 B+33 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(210 A+253 B+264 C) a^4+\left (5 (21 A+22 B+33 C) a^4+(210 A+253 B+264 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {15 (105 A+121 B+143 C) a^4+77 (15 A+17 B+21 C) \cos (c+d x) a^4}{2 \cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {15 (105 A+121 B+143 C) a^4+77 (15 A+17 B+21 C) \cos (c+d x) a^4}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \frac {15 (105 A+121 B+143 C) a^4+77 (15 A+17 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)}dx+77 a^4 (15 A+17 B+21 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+77 a^4 (15 A+17 B+21 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (15 A+17 B+21 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (15 A+17 B+21 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (15 A+17 B+21 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{9} \left (\frac {2 (105 A+143 B+99 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {6}{7} \left (\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}\right )+77 a^4 (15 A+17 B+21 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )\right )+\frac {2 (6 A+11 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d \cos ^{\frac {9}{2}}(c+d x)}}{11 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac {11}{2}}(c+d x)}\) |
Input:
Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Cos[c + d*x]^(13/2),x]
Output:
(2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + ((2 *(6*A + 11*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^( 9/2)) + ((2*(105*A + 143*B + 99*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/ (7*d*Cos[c + d*x]^(7/2)) + (6*((2*a^4*(210*A + 253*B + 264*C)*Sin[c + d*x] )/(5*d*Cos[c + d*x]^(5/2)) + (15*a^4*(105*A + 121*B + 143*C)*((2*EllipticF [(c + d*x)/2, 2])/(3*d) + (2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))) + 77* a^4*(15*A + 17*B + 21*C)*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin[c + d* x])/(d*Sqrt[Cos[c + d*x]])))/5))/7)/9)/(11*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*(( b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[(n + 2)/(b^2*(n + 1)) I nt[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && IntegerQ[2*n]
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])
Leaf count of result is larger than twice the leaf count of optimal. \(1396\) vs. \(2(278)=556\).
Time = 20.45 (sec) , antiderivative size = 1397, normalized size of antiderivative = 4.61
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1397\) |
| parts | \(\text {Expression too large to display}\) | \(1739\) |
Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x ,method=_RETURNVERBOSE)
Output:
-16*(-(-2*cos(1/2*d*x+1/2*c)^2+1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(1/8*A*( -1/352*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)^6-9/616*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)^4-15/154* 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)/(c os(1/2*d*x+1/2*c)^2-1/2)^2+15/77*(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)))+1/8*C/sin(1/2*d*x+1/2*c)^2/(2*sin( 1/2*d*x+1/2*c)^2-1)*(-2*sin(1/2*d*x+1/2*c)^4+sin(1/2*d*x+1/2*c)^2)^(1/2)*( 2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-(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)))+(3/8* A+1/8*B)*(-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*s in(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) ^3-14/15*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...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.08 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{6} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (462 \, {\left (15 \, A + 17 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} + 30 \, {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 77 \, {\left (30 \, A + 34 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (42 \, A + 33 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 385 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 315 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d \cos \left (d x + c\right )^{6}} \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 3/2),x, algorithm="fricas")
Output:
-2/3465*(15*I*sqrt(2)*(105*A + 121*B + 143*C)*a^3*cos(d*x + c)^6*weierstra ssPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(105*A + 1 21*B + 143*C)*a^3*cos(d*x + c)^6*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*I*sqrt(2)*(15*A + 17*B + 21*C)*a^3*cos(d*x + c)^6*w eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 231*I*sqrt(2)*(15*A + 17*B + 21*C)*a^3*cos(d*x + c)^6*weierstrass Zeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - ( 462*(15*A + 17*B + 21*C)*a^3*cos(d*x + c)^5 + 30*(105*A + 121*B + 143*C)*a ^3*cos(d*x + c)^4 + 77*(30*A + 34*B + 27*C)*a^3*cos(d*x + c)^3 + 45*(42*A + 33*B + 11*C)*a^3*cos(d*x + c)^2 + 385*(3*A + B)*a^3*cos(d*x + c) + 315*A *a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^6)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Timed out} \] Input:
integrate((a+a*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/cos(d*x+c)* *(13/2),x)
Output:
Timed out
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 3/2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3/c os(d*x + c)^(13/2), x)
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {13}{2}}} \,d x } \] Input:
integrate((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(1 3/2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(a*cos(d*x + c) + a)^3/c os(d*x + c)^(13/2), x)
Time = 2.61 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.95 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=\text {Too large to display} \] Input:
int(((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(13/2),x)
Output:
(2*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2)*((120*A*a^3*sin(c + d*x))/( cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (45*A*a^3*sin(c + d*x))/( cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2)) + (15*A*a^3*sin(c + d*x))/( cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2)^(1/2)) + (136*B*a^3*sin(c + d*x))/ (cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (39*B*a^3*sin(c + d*x))/ (cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2)) + (5*B*a^3*sin(c + d*x))/( cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2)^(1/2)) + (153*C*a^3*sin(c + d*x))/ (cos(c + d*x)^(1/2)*(1 - cos(c + d*x)^2)^(1/2)) + (27*C*a^3*sin(c + d*x))/ (cos(c + d*x)^(5/2)*(1 - cos(c + d*x)^2)^(1/2))))/(45*d) + (8*hypergeom([- 3/4, 1/2], 5/4, cos(c + d*x)^2)*((42*A*a^3*sin(c + d*x))/(cos(c + d*x)^(3/ 2)*(1 - cos(c + d*x)^2)^(1/2)) + (7*A*a^3*sin(c + d*x))/(cos(c + d*x)^(7/2 )*(1 - cos(c + d*x)^2)^(1/2)) + (33*B*a^3*sin(c + d*x))/(cos(c + d*x)^(3/2 )*(1 - cos(c + d*x)^2)^(1/2)) + (11*C*a^3*sin(c + d*x))/(cos(c + d*x)^(3/2 )*(1 - cos(c + d*x)^2)^(1/2))))/(231*d) + (2*hypergeom([-3/4, 1/2], 1/4, c os(c + d*x)^2)*((168*A*a^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (119*A*a^3*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (21*A*a^3*sin(c + d*x))/(cos(c + d*x)^(11/2)*(1 - cos(c + d*x)^2)^(1/2)) + (209*B*a^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - cos(c + d*x)^2)^(1/2)) + (99*B*a^3*sin(c + d*x))/(cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (275*C*a^3*sin(c + d*x))/(cos(c + d*x)^(3/2)*(1 - c...
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{7}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) c \right ) \] Input:
int((a+a*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/cos(d*x+c)^(13/2),x )
Output:
a**3*(int(sqrt(cos(c + d*x))/cos(c + d*x)**7,x)*a + 3*int(sqrt(cos(c + d*x ))/cos(c + d*x)**6,x)*a + int(sqrt(cos(c + d*x))/cos(c + d*x)**6,x)*b + 3* int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a + 3*int(sqrt(cos(c + d*x))/cos (c + d*x)**5,x)*b + int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*c + int(sqrt (cos(c + d*x))/cos(c + d*x)**4,x)*a + 3*int(sqrt(cos(c + d*x))/cos(c + d*x )**4,x)*b + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x)*b + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**3,x )*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*c)