Integrand size = 35, antiderivative size = 279 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a^3 (175 A+221 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {4 a^3 (95 A+121 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (95 A+121 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (175 A+221 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{585 d}+\frac {40 a^3 (118 A+143 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{9009 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{13 d}+\frac {12 A \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{143 a d}+\frac {2 (145 A+143 C) \cos ^{\frac {5}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{1287 d} \] Output:
4/195*a^3*(175*A+221*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/231*a^3* (95*A+121*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/231*a^3*(95*A+121* C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+4/585*a^3*(175*A+221*C)*cos(d*x+c)^(3/2)* sin(d*x+c)/d+40/9009*a^3*(118*A+143*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/13* A*cos(d*x+c)^(5/2)*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+12/143*A*cos(d*x+c)^(5/ 2)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d+2/1287*(145*A+143*C)*cos(d*x+c)^( 5/2)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.75 (sec) , antiderivative size = 1022, normalized size of antiderivative = 3.66 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2 ),x]
Output:
a^3*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390* ((175*A + 221*C)*Cot[c])/d + ((1811*A + 2134*C)*Cos[d*x]*Sin[c])/(7392*d) + ((7825*A + 7592*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((215*A + 132*C)*Cos [3*d*x]*Sin[3*c])/(4928*d) + ((59*A + 13*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + (3*A*Cos[5*d*x]*Sin[5*c])/(704*d) + (A*Cos[6*d*x]*Sin[6*c])/(1664*d) + ( (1811*A + 2134*C)*Cos[c]*Sin[d*x])/(7392*d) + ((7825*A + 7592*C)*Cos[2*c]* Sin[2*d*x])/(74880*d) + ((215*A + 132*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ( (59*A + 13*C)*Cos[4*c]*Sin[4*d*x])/(3744*d) + (3*A*Cos[5*c]*Sin[5*d*x])/(7 04*d) + (A*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (95*A*(1 + 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 - 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]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (11*C*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Co t[c]]]^2]*Sec[c/2 + (d*x)/2]^6*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]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (35 *A*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{- 1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]* Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[...
Time = 1.75 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.02, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.629, Rules used = {3042, 4602, 3042, 3525, 27, 3042, 3455, 27, 3042, 3455, 3042, 3447, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \cos ^{\frac {13}{2}}(c+d x) (a \sec (c+d x)+a)^3 \left (A+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^{13/2} (a \sec (c+d x)+a)^3 \left (A+C \sec (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4602 |
| \(\displaystyle \int \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3 \left (A \cos ^2(c+d x)+C\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A \sin \left (c+d x+\frac {\pi }{2}\right )^2+C\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^3 (a (5 A+13 C)+6 a A \cos (c+d x))dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^3 (a (5 A+13 C)+6 a A \cos (c+d x))dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (5 A+13 C)+6 a A \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{11} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 \left ((85 A+143 C) a^2+(145 A+143 C) \cos (c+d x) a^2\right )dx+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 \left ((85 A+143 C) a^2+(145 A+143 C) \cos (c+d x) a^2\right )dx+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((85 A+143 C) a^2+(145 A+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a) \left ((745 A+1001 C) a^3+10 (118 A+143 C) \cos (c+d x) a^3\right )dx+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((745 A+1001 C) a^3+10 (118 A+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (10 (118 A+143 C) \cos ^2(c+d x) a^4+(745 A+1001 C) a^4+\left (10 (118 A+143 C) a^4+(745 A+1001 C) a^4\right ) \cos (c+d x)\right )dx+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (10 (118 A+143 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(745 A+1001 C) a^4+\left (10 (118 A+143 C) a^4+(745 A+1001 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (117 (95 A+121 C) a^4+77 (175 A+221 C) \cos (c+d x) a^4\right )dx+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (117 (95 A+121 C) a^4+77 (175 A+221 C) \cos (c+d x) a^4\right )dx+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (117 (95 A+121 C) a^4+77 (175 A+221 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (95 A+121 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^4 (175 A+221 C) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (95 A+121 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 a^4 (175 A+221 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+221 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (95 A+121 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (77 a^4 (175 A+221 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (95 A+121 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2}{9} \left (\frac {1}{7} \left (117 a^4 (95 A+121 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 a^4 (175 A+221 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{11} \left (\frac {2 (145 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{9 d}+\frac {2}{9} \left (\frac {20 a^4 (118 A+143 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (77 a^4 (175 A+221 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a^4 (95 A+121 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {12 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{11 d}}{13 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}\) |
Input:
Int[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]
Output:
(2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + ((12 *A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (( 2*(145*A + 143*C)*Cos[c + d*x]^(5/2)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x] )/(9*d) + (2*((20*a^4*(118*A + 143*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7* d) + (117*a^4*(95*A + 121*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqr t[Cos[c + d*x]]*Sin[c + d*x])/(3*d)) + 77*a^4*(175*A + 221*C)*((6*Elliptic E[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)))/7)) /9)/11)/(13*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[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[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)*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_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x _)])^(m_.)*((A_.) + (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 + A*Cos [e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] && !IntegerQ[n] && IntegerQ[m]
Time = 163.72 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.66
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-221760 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+1058400 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-2122400 A -80080 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (2331040 A +314600 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1535860 A -487916 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (633710 A +386386 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-121230 A -105534 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+18525 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-40425 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+23595 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-51051 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45045 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(464\) |
Input:
int(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x,method=_RETU RNVERBOSE)
Output:
-4/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-221 760*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+1058400*A*cos(1/2*d*x+1/2*c )*sin(1/2*d*x+1/2*c)^12+(-2122400*A-80080*C)*sin(1/2*d*x+1/2*c)^10*cos(1/2 *d*x+1/2*c)+(2331040*A+314600*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+( -1535860*A-487916*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(633710*A+386 386*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-121230*A-105534*C)*sin(1/ 2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+18525*A*(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))-40425* A*(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))+23595*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)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-51051*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)))/(-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*cos(1/2*d*x+1/2*c)^2-1)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.93 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (195 i \, \sqrt {2} {\left (95 \, A + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 195 i \, \sqrt {2} {\left (95 \, A + 121 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (175 \, A + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (175 \, A + 221 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (3465 \, A a^{3} \cos \left (d x + c\right )^{5} + 12285 \, A a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (50 \, A + 13 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 585 \, {\left (38 \, A + 33 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 154 \, {\left (175 \, A + 221 \, C\right )} a^{3} \cos \left (d x + c\right ) + 390 \, {\left (95 \, A + 121 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{45045 \, d} \] Input:
integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algor ithm="fricas")
Output:
-2/45045*(195*I*sqrt(2)*(95*A + 121*C)*a^3*weierstrassPInverse(-4, 0, cos( d*x + c) + I*sin(d*x + c)) - 195*I*sqrt(2)*(95*A + 121*C)*a^3*weierstrassP Inverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(175*A + 221 *C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I *sin(d*x + c))) + 231*I*sqrt(2)*(175*A + 221*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (3465*A*a^3* cos(d*x + c)^5 + 12285*A*a^3*cos(d*x + c)^4 + 385*(50*A + 13*C)*a^3*cos(d* x + c)^3 + 585*(38*A + 33*C)*a^3*cos(d*x + c)^2 + 154*(175*A + 221*C)*a^3* cos(d*x + c) + 390*(95*A + 121*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(13/2)*(a+a*sec(d*x+c))**3*(A+C*sec(d*x+c)**2),x)
Output:
Timed out
Timed out. \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algor ithm="maxima")
Output:
Timed out
\[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {13}{2}} \,d x } \] Input:
integrate(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x, algor ithm="giac")
Output:
integrate((C*sec(d*x + c)^2 + A)*(a*sec(d*x + c) + a)^3*cos(d*x + c)^(13/2 ), x)
Time = 14.43 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.29 \[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C\,a^3\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,A\,a^3\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^3\,{\cos \left (c+d\,x\right )}^{15/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {15}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{15\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,C\,a^3\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:
int(cos(c + d*x)^(13/2)*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^3,x)
Output:
(C*a^3*((2*cos(c + d*x)^(1/2)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2 , 2))/3))/d - (2*A*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4 ], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^2)^(1/2)) - (6*A*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11 *d*(sin(c + d*x)^2)^(1/2)) - (6*A*a^3*cos(c + d*x)^(13/2)*sin(c + d*x)*hyp ergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x)^(15/2)*sin(c + d*x)*hypergeom([1/2, 15/4], 19/4, co s(c + d*x)^2))/(15*d*(sin(c + d*x)^2)^(1/2)) - (6*C*a^3*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*C*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9 /4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/( 11*d*(sin(c + d*x)^2)^(1/2))
\[ \int \cos ^{\frac {13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{5}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{3}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{3}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )^{2}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6} \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{6}d x \right ) a \right ) \] Input:
int(cos(d*x+c)^(13/2)*(a+a*sec(d*x+c))^3*(A+C*sec(d*x+c)^2),x)
Output:
a**3*(int(sqrt(cos(c + d*x))*cos(c + d*x)**6*sec(c + d*x)**5,x)*c + 3*int( sqrt(cos(c + d*x))*cos(c + d*x)**6*sec(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x))*cos(c + d*x)**6*sec(c + d*x)**3,x)*a + 3*int(sqrt(cos(c + d*x))*cos (c + d*x)**6*sec(c + d*x)**3,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)* *6*sec(c + d*x)**2,x)*a + int(sqrt(cos(c + d*x))*cos(c + d*x)**6*sec(c + d *x)**2,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**6*sec(c + d*x),x)*a + int(sqrt(cos(c + d*x))*cos(c + d*x)**6,x)*a)