Integrand size = 43, antiderivative size = 267 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {4 a^3 (17 A+21 B+27 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (11 A+13 B+21 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^3 (17 A+21 B+27 C) \sin (c+d x)}{15 d \sqrt {\cos (c+d x)}}+\frac {2 A (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 (2 A+3 B) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (73 A+99 B+63 C) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d \cos ^{\frac {5}{2}}(c+d x)} \] Output:
-4/15*a^3*(17*A+21*B+27*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/21*a^ 3*(11*A+13*B+21*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/105*a^3*(32* A+41*B+42*C)*sin(d*x+c)/d/cos(d*x+c)^(3/2)+4/15*a^3*(17*A+21*B+27*C)*sin(d *x+c)/d/cos(d*x+c)^(1/2)+2/9*A*(a+a*cos(d*x+c))^3*sin(d*x+c)/d/cos(d*x+c)^ (9/2)+2/21*(2*A+3*B)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/a/d/cos(d*x+c)^(7/2 )+2/315*(73*A+99*B+63*C)*(a^3+a^3*cos(d*x+c))*sin(d*x+c)/d/cos(d*x+c)^(5/2 )
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 13.05 (sec) , antiderivative size = 1364, normalized size of antiderivative = 5.11 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] 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]^(11/2),x]
Output:
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(((17*A + 2 1*B + 27*C)*Csc[c]*Sec[c])/(30*d) + (A*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36 *d) + (Sec[c]*Sec[c + d*x]^4*(7*A*Sin[c] + 27*A*Sin[d*x] + 9*B*Sin[d*x]))/ (252*d) + (Sec[c]*Sec[c + d*x]^3*(135*A*Sin[c] + 45*B*Sin[c] + 238*A*Sin[d *x] + 189*B*Sin[d*x] + 63*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]^2*( 238*A*Sin[c] + 189*B*Sin[c] + 63*C*Sin[c] + 330*A*Sin[d*x] + 390*B*Sin[d*x ] + 315*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(110*A*Sin[c] + 130*B *Sin[c] + 105*C*Sin[c] + 238*A*Sin[d*x] + 294*B*Sin[d*x] + 378*C*Sin[d*x]) )/(420*d)) - (11*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 - Ar cTan[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]) - (13*B*(a + a*Cos[c + d*x])^3*Csc[c]*Hypergeome tricPFQ[{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[-(Sq rt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - Arc Tan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (C*(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 - 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 ...
Time = 1.80 (sec) , antiderivative size = 264, 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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{11/2}}dx\) |
| \(\Big \downarrow \) 3522 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 (3 a (2 A+3 B)+a (A+9 C) \cos (c+d x))}{2 \cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 (3 a (2 A+3 B)+a (A+9 C) \cos (c+d x))}{\cos ^{\frac {9}{2}}(c+d x)}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a (2 A+3 B)+a (A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((73 A+99 B+63 C) a^2+(13 A+9 B+63 C) \cos (c+d x) a^2\right )}{2 \cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((73 A+99 B+63 C) a^2+(13 A+9 B+63 C) \cos (c+d x) a^2\right )}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((73 A+99 B+63 C) a^2+(13 A+9 B+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3454 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left (3 (32 A+41 B+42 C) a^3+(23 A+24 B+63 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(\cos (c+d x) a+a) \left (3 (32 A+41 B+42 C) a^3+(23 A+24 B+63 C) \cos (c+d x) a^3\right )}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (3 (32 A+41 B+42 C) a^3+(23 A+24 B+63 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(23 A+24 B+63 C) \cos ^2(c+d x) a^4+3 (32 A+41 B+42 C) a^4+\left (3 (32 A+41 B+42 C) a^4+(23 A+24 B+63 C) a^4\right ) \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(23 A+24 B+63 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+3 (32 A+41 B+42 C) a^4+\left (3 (32 A+41 B+42 C) a^4+(23 A+24 B+63 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3 \left (7 (17 A+21 B+27 C) a^4+5 (11 A+13 B+21 C) \cos (c+d x) a^4\right )}{2 \cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+21 B+27 C) a^4+5 (11 A+13 B+21 C) \cos (c+d x) a^4}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {7 (17 A+21 B+27 C) a^4+5 (11 A+13 B+21 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+21 B+27 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)}dx+5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (7 a^4 (17 A+21 B+27 C) \int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3116 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+21 B+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (17 A+21 B+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (11 A+13 B+21 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+7 a^4 (17 A+21 B+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )+\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (73 A+99 B+63 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{5 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {6}{5} \left (\frac {10 a^4 (11 A+13 B+21 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {2 a^4 (32 A+41 B+42 C) \sin (c+d x)}{d \cos ^{\frac {3}{2}}(c+d x)}+7 a^4 (17 A+21 B+27 C) \left (\frac {2 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )\right )\right )+\frac {6 (2 A+3 B) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d \cos ^{\frac {7}{2}}(c+d x)}}{9 a}+\frac {2 A \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac {9}{2}}(c+d x)}\) |
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]^(11/2),x]
Output:
(2*A*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + ((6*( 2*A + 3*B)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2 )) + ((2*(73*A + 99*B + 63*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(5*d* Cos[c + d*x]^(5/2)) + (6*((10*a^4*(11*A + 13*B + 21*C)*EllipticF[(c + d*x) /2, 2])/d + (2*a^4*(32*A + 41*B + 42*C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2 )) + 7*a^4*(17*A + 21*B + 27*C)*((-2*EllipticE[(c + d*x)/2, 2])/d + (2*Sin [c + d*x])/(d*Sqrt[Cos[c + d*x]]))))/5)/7)/(9*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. \(1234\) vs. \(2(246)=492\).
Time = 16.62 (sec) , antiderivative size = 1235, normalized size of antiderivative = 4.63
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1235\) |
| parts | \(\text {Expression too large to display}\) | \(1461\) |
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)^(11/2),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*C*( 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*A*(-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)^3-14/15*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*c os(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))))+(3/8*A+1/8*B)*(-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*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)^2+5/21*(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*B+3/8*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)...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.14 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (11 \, A + 13 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 21 i \, \sqrt {2} {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{5} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (42 \, {\left (17 \, A + 21 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right )^{4} + 15 \, {\left (22 \, A + 26 \, B + 21 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 7 \, {\left (34 \, A + 27 \, B + 9 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 45 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + 35 \, A a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )^{5}} \] 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 1/2),x, algorithm="fricas")
Output:
-2/315*(15*I*sqrt(2)*(11*A + 13*B + 21*C)*a^3*cos(d*x + c)^5*weierstrassPI nverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(11*A + 13*B + 21*C)*a^3*cos(d*x + c)^5*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( d*x + c)) + 21*I*sqrt(2)*(17*A + 21*B + 27*C)*a^3*cos(d*x + c)^5*weierstra ssZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*I*sqrt(2)*(17*A + 21*B + 27*C)*a^3*cos(d*x + c)^5*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (42*(17*A + 21*B + 27*C)*a^3*cos(d*x + c)^4 + 15*(22*A + 26*B + 21*C)*a^3*cos(d*x + c)^3 + 7*(34*A + 27*B + 9*C)*a^3*cos(d*x + c)^2 + 45*(3*A + B)*a^3*cos(d*x + c) + 35*A*a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c)^5)
Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\text {Timed out} \] 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)* *(11/2),x)
Output:
Timed out
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] 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 1/2),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)^(11/2), x)
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\cos \left (d x + c\right )^{\frac {11}{2}}} \,d x } \] 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 1/2),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)^(11/2), x)
Time = 2.33 (sec) , antiderivative size = 457, normalized size of antiderivative = 1.71 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx =\text {Too large to display} \] 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)^(11/2),x)
Output:
(2*C*a^3*ellipticF(c/2 + (d*x)/2, 2))/d + (70*A*a^3*sin(c + d*x)*hypergeom ([-9/4, 1/2], -5/4, cos(c + d*x)^2) + 270*A*a^3*cos(c + d*x)*sin(c + d*x)* hypergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2) + 210*A*a^3*cos(c + d*x)^3*si n(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2) + 378*A*a^3*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/(315*d* cos(c + d*x)^(9/2)*(1 - cos(c + d*x)^2)^(1/2)) + ((2*B*a^3*sin(c + d*x)*hy pergeom([-7/4, 1/2], -3/4, cos(c + d*x)^2))/7 + (6*B*a^3*cos(c + d*x)*sin( c + d*x)*hypergeom([-5/4, 1/2], -1/4, cos(c + d*x)^2))/5 + 2*B*a^3*cos(c + d*x)^2*sin(c + d*x)*hypergeom([-3/4, 1/2], 1/4, cos(c + d*x)^2) + 2*B*a^3 *cos(c + d*x)^3*sin(c + d*x)*hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/ (d*cos(c + d*x)^(7/2)*(1 - cos(c + d*x)^2)^(1/2)) + (6*C*a^3*sin(c + d*x)* hypergeom([-1/4, 1/2], 3/4, cos(c + d*x)^2))/(d*cos(c + d*x)^(1/2)*(sin(c + d*x)^2)^(1/2)) + (2*C*a^3*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)) + (2*C*a^3*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))
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{6}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) a +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{5}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{4}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) a +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{3}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) b +3 \left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )^{2}}d x \right ) c \right ) \] 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)^(11/2),x )
Output:
a**3*(int(sqrt(cos(c + d*x))/cos(c + d*x),x)*c + int(sqrt(cos(c + d*x))/co s(c + d*x)**6,x)*a + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**5,x)*a + int(s qrt(cos(c + d*x))/cos(c + d*x)**5,x)*b + 3*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 + int(sqrt(co s(c + d*x))/cos(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x))/cos(c + d*x)**3, x)*a + 3*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)*b + 3*int(sqrt(cos(c + d*x))/cos(c + d*x)**2,x)*c)