Integrand size = 43, antiderivative size = 231 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {4 a^3 (27 A+21 B+17 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (21 A+13 B+11 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^3 (42 A+41 B+32 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 C \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 (3 B+2 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 a d}+\frac {2 (63 A+99 B+73 C) \sqrt {\cos (c+d x)} \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{315 d} \] Output:
4/15*a^3*(27*A+21*B+17*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/21*a^3 *(21*A+13*B+11*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/105*a^3*(42*A +41*B+32*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+2/9*C*cos(d*x+c)^(1/2)*(a+a*cos( d*x+c))^3*sin(d*x+c)/d+2/21*(3*B+2*C)*cos(d*x+c)^(1/2)*(a^2+a^2*cos(d*x+c) )^2*sin(d*x+c)/a/d+2/315*(63*A+99*B+73*C)*cos(d*x+c)^(1/2)*(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 = 8.79 (sec) , antiderivative size = 1322, normalized size of antiderivative = 5.72 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx =\text {Too large to display} \] Input:
Integrate[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Sqrt[Cos[c + d*x]],x]
Output:
Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((27 *A + 21*B + 17*C)*Cot[c])/d + ((84*A + 107*B + 97*C)*Cos[d*x]*Sin[c])/(336 *d) + ((18*A + 54*B + 73*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((B + 3*C)*Cos[ 3*d*x]*Sin[3*c])/(112*d) + (C*Cos[4*d*x]*Sin[4*c])/(288*d) + ((84*A + 107* B + 97*C)*Cos[c]*Sin[d*x])/(336*d) + ((18*A + 54*B + 73*C)*Cos[2*c]*Sin[2* d*x])/(720*d) + ((B + 3*C)*Cos[3*c]*Sin[3*d*x])/(112*d) + (C*Cos[4*c]*Sin[ 4*d*x])/(288*d)) - (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 - 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]]] ])/(2*d*Sqrt[1 + Cot[c]^2]) - (13*B*(a + a*Cos[c + d*x])^3*Csc[c]*Hypergeo metricPFQ[{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 - A rcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (11*C*(a + a*Cos[c + d*x])^3* Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*S ec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Co t[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]) - (9*A*(a + a*Co s[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1...
Time = 1.61 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.04, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.442, Rules used = {3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 27, 3042, 3447, 3042, 3502, 27, 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.
| \(\displaystyle \int \frac {(a \cos (c+d x)+a)^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {2 \int \frac {(\cos (c+d x) a+a)^3 (a (9 A+C)+3 a (3 B+2 C) \cos (c+d x))}{2 \sqrt {\cos (c+d x)}}dx}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(\cos (c+d x) a+a)^3 (a (9 A+C)+3 a (3 B+2 C) \cos (c+d x))}{\sqrt {\cos (c+d x)}}dx}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (a (9 A+C)+3 a (3 B+2 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \cos (c+d x) a^2\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {(\cos (c+d x) a+a)^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \cos (c+d x) a^2\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left ((63 A+9 B+13 C) a^2+(63 A+99 B+73 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {3 (\cos (c+d x) a+a) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {(\cos (c+d x) a+a) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \cos (c+d x) a^3\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((63 A+24 B+23 C) a^3+3 (42 A+41 B+32 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {3 (42 A+41 B+32 C) \cos ^2(c+d x) a^4+(63 A+24 B+23 C) a^4+\left ((63 A+24 B+23 C) a^4+3 (42 A+41 B+32 C) a^4\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \int \frac {3 (42 A+41 B+32 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+(63 A+24 B+23 C) a^4+\left ((63 A+24 B+23 C) a^4+3 (42 A+41 B+32 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\frac {2}{3} \int \frac {3 \left (5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \cos (c+d x) a^4\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \cos (c+d x) a^4}{\sqrt {\cos (c+d x)}}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (\int \frac {5 (21 A+13 B+11 C) a^4+7 (27 A+21 B+17 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+7 a^4 (27 A+21 B+17 C) \int \sqrt {\cos (c+d x)}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+7 a^4 (27 A+21 B+17 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {6}{5} \left (5 a^4 (21 A+13 B+11 C) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 a^4 (27 A+21 B+17 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )+\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2 (63 A+99 B+73 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{5 d}+\frac {6}{5} \left (\frac {10 a^4 (21 A+13 B+11 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{d}+\frac {14 a^4 (27 A+21 B+17 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 (42 A+41 B+32 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{d}\right )\right )+\frac {6 (3 B+2 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{7 d}}{9 a}+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}\) |
Input:
Int[((a + a*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sqrt[ Cos[c + d*x]],x]
Output:
(2*C*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*( 3*B + 2*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7* d) + ((2*(63*A + 99*B + 73*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])* Sin[c + d*x])/(5*d) + (6*((14*a^4*(27*A + 21*B + 17*C)*EllipticE[(c + d*x) /2, 2])/d + (10*a^4*(21*A + 13*B + 11*C)*EllipticF[(c + d*x)/2, 2])/d + (2 *a^4*(42*A + 41*B + 32*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d))/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[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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(513\) vs. \(2(214)=428\).
Time = 20.24 (sec) , antiderivative size = 514, normalized size of antiderivative = 2.23
| method | result | size |
| default | \(-\frac {4 \sqrt {\left (-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (-560 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (360 B +2200 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-252 A -1296 B -3412 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (882 A +1806 B +2702 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-378 A -624 B -738 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+315 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 )-567 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 )+195 B \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 )-441 B \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 )+165 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 )-357 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 )}{315 \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 {-1+2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, d}\) | \(514\) |
| parts | \(\text {Expression too large to display}\) | \(1007\) |
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)^(1/2),x, method=_RETURNVERBOSE)
Output:
-4/315*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(-560* C*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^10+(360*B+2200*C)*sin(1/2*d*x+1/2* c)^8*cos(1/2*d*x+1/2*c)+(-252*A-1296*B-3412*C)*sin(1/2*d*x+1/2*c)^6*cos(1/ 2*d*x+1/2*c)+(882*A+1806*B+2702*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c) +(-378*A-624*B-738*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+315*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))-567*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))+195*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))-441*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))+165*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))- 357*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)*Ellipt icE(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)/(-1+2*cos(1/2*d*x+1/2*c)^2)^(1/2)/d
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.04 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (21 \, A + 13 \, B + 11 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (21 \, A + 13 \, B + 11 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 21 i \, \sqrt {2} {\left (27 \, A + 21 \, B + 17 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 i \, \sqrt {2} {\left (27 \, A + 21 \, B + 17 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, C a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 7 \, {\left (9 \, A + 27 \, B + 34 \, C\right )} a^{3} \cos \left (d x + c\right ) + 15 \, {\left (21 \, A + 26 \, B + 22 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d} \] 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 /2),x, algorithm="fricas")
Output:
-2/315*(15*I*sqrt(2)*(21*A + 13*B + 11*C)*a^3*weierstrassPInverse(-4, 0, c os(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(21*A + 13*B + 11*C)*a^3*weie rstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 21*I*sqrt(2)*(27*A + 21*B + 17*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( d*x + c) + I*sin(d*x + c))) + 21*I*sqrt(2)*(27*A + 21*B + 17*C)*a^3*weiers trassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) ) - (35*C*a^3*cos(d*x + c)^3 + 45*(B + 3*C)*a^3*cos(d*x + c)^2 + 7*(9*A + 27*B + 34*C)*a^3*cos(d*x + c) + 15*(21*A + 26*B + 22*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\text {Timed out} \] 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)* *(1/2),x)
Output:
Timed out
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] 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 /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/s qrt(cos(d*x + c)), x)
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (a \cos \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cos \left (d x + c\right )}} \,d x } \] 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 /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/s qrt(cos(d*x + c)), x)
Time = 0.97 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.86 \[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=\frac {2\,\left (B\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+B\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}+\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 {6\,A\,a^3\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {4\,A\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {2\,A\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{d}-\frac {2\,A\,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 {6\,B\,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\,B\,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\,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(((a + a*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c + d*x)^(1/2),x)
Output:
(2*(B*a^3*ellipticE(c/2 + (d*x)/2, 2) + B*a^3*ellipticF(c/2 + (d*x)/2, 2) + B*a^3*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (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 + (6*A*a^3*ellip ticE(c/2 + (d*x)/2, 2))/d + (4*A*a^3*ellipticF(c/2 + (d*x)/2, 2))/d + (2*A *a^3*cos(c + d*x)^(1/2)*sin(c + d*x))/d - (2*A*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)) - (6*B*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*B*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*(si n(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))
\[ \int \frac {(a+a \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx=a^{3} \left (\left (\int \frac {\sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) c \right ) \] 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)^(1/2),x)
Output:
a**3*(int(sqrt(cos(c + d*x))/cos(c + d*x),x)*a + 3*int(sqrt(cos(c + d*x)), x)*a + int(sqrt(cos(c + d*x)),x)*b + 3*int(sqrt(cos(c + d*x))*cos(c + d*x) ,x)*a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x),x)*b + int(sqrt(cos(c + d*x) )*cos(c + d*x),x)*c + int(sqrt(cos(c + d*x))*cos(c + d*x)**4,x)*c + int(sq rt(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)*a + 3*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)