Integrand size = 35, antiderivative size = 473 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^5 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (384 a^4 C+5 b^4 (7 A+5 C)+4 a^2 b^2 (70 A+29 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^5 d \sqrt {a+b \cos (c+d x)}}-\frac {2 \left (A b^2+a^2 C\right ) \cos ^3(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (2 a^2 b^2 (70 A-31 C)+192 a^4 C-5 b^4 (7 A+5 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b^4 \left (a^2-b^2\right ) d}-\frac {2 a \left (35 A b^2+48 a^2 C-13 b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{35 b^3 \left (a^2-b^2\right ) d}+\frac {2 \left (7 A b^2+8 a^2 C-b^2 C\right ) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{7 b^2 \left (a^2-b^2\right ) d} \] Output:
-2/105*a*(4*a^2*b^2*(70*A-43*C)+384*a^4*C-b^4*(175*A+107*C))*(a+b*cos(d*x+ c))^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/b^5/(a^2-b ^2)/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/105*(384*a^4*C+5*b^4*(7*A+5*C)+4*a^ 2*b^2*(70*A+29*C))*((a+b*cos(d*x+c))/(a+b))^(1/2)*InverseJacobiAM(1/2*d*x+ 1/2*c,2^(1/2)*(b/(a+b))^(1/2))/b^5/d/(a+b*cos(d*x+c))^(1/2)-2*(A*b^2+C*a^2 )*cos(d*x+c)^3*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+2/105*(2*a^ 2*b^2*(70*A-31*C)+192*a^4*C-5*b^4*(7*A+5*C))*(a+b*cos(d*x+c))^(1/2)*sin(d* x+c)/b^4/(a^2-b^2)/d-2/35*a*(35*A*b^2+48*C*a^2-13*C*b^2)*cos(d*x+c)*(a+b*c os(d*x+c))^(1/2)*sin(d*x+c)/b^3/(a^2-b^2)/d+2/7*(7*A*b^2+8*C*a^2-C*b^2)*co s(d*x+c)^2*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^2/(a^2-b^2)/d
Time = 2.78 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {-4 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b \left (2 a^2 b^3 (35 A-8 C)+96 a^4 b C+5 b^5 (7 A+5 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+a \left (4 a^2 b^2 (70 A-43 C)+384 a^4 C-b^4 (175 A+107 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+(a-b) b (a+b) \left (420 a^3 \left (A b^2+a^2 C\right ) \sin (c+d x)+\left (a^2-b^2\right ) \left (140 A b^2+348 a^2 C+115 b^2 C\right ) (a+b \cos (c+d x)) \sin (c+d x)-78 a b \left (a^2-b^2\right ) C (a+b \cos (c+d x)) \sin (2 (c+d x))+15 b^2 \left (a^2-b^2\right ) C (a+b \cos (c+d x)) \sin (3 (c+d x))\right )}{210 (a-b) b^5 (a+b) \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/ 2),x]
Output:
(-4*(a^2 - b^2)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b*(2*a^2*b^3*(35*A - 8 *C) + 96*a^4*b*C + 5*b^5*(7*A + 5*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b) ] + a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*((a + b) *EllipticE[(c + d*x)/2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + (a - b)*b*(a + b)*(420*a^3*(A*b^2 + a^2*C)*Sin[c + d*x] + (a^2 - b^2)*(140*A*b^2 + 348*a^2*C + 115*b^2*C)*(a + b*Cos[c + d*x])*Sin[c + d* x] - 78*a*b*(a^2 - b^2)*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)] + 15*b^2*( a^2 - b^2)*C*(a + b*Cos[c + d*x])*Sin[3*(c + d*x)]))/(210*(a - b)*b^5*(a + b)*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.79 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 3527, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3502, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle -\frac {2 \int \frac {\cos ^2(c+d x) \left (-\left (\left (7 A b^2+\left (8 a^2-b^2\right ) C\right ) \cos ^2(c+d x)\right )-a b (A+C) \cos (c+d x)+6 \left (C a^2+A b^2\right )\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (-\left (\left (8 C a^2+7 A b^2-b^2 C\right ) \cos ^2(c+d x)\right )-a b (A+C) \cos (c+d x)+6 \left (C a^2+A b^2\right )\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\left (-8 C a^2-7 A b^2+b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-a b (A+C) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (C a^2+A b^2\right )\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {2 \int -\frac {\cos (c+d x) \left (-a \left (48 C a^2+35 A b^2-13 b^2 C\right ) \cos ^2(c+d x)-b \left (2 C a^2+7 A b^2+5 b^2 C\right ) \cos (c+d x)+4 a \left (8 C a^2+7 A b^2-b^2 C\right )\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {\cos (c+d x) \left (-a \left (48 C a^2+35 A b^2-13 b^2 C\right ) \cos ^2(c+d x)-b \left (2 C a^2+7 A b^2+5 b^2 C\right ) \cos (c+d x)+4 a \left (8 C a^2+7 A b^2-b^2 C\right )\right )}{\sqrt {a+b \cos (c+d x)}}dx}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (-a \left (48 C a^2+35 A b^2-13 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (2 C a^2+7 A b^2+5 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a \left (8 C a^2+7 A b^2-b^2 C\right )\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {-\frac {\frac {2 \int -\frac {2 \left (48 C a^2+35 A b^2-13 b^2 C\right ) a^2-b \left (16 C a^2+35 A b^2+19 b^2 C\right ) \cos (c+d x) a-\left (192 C a^4+2 b^2 (70 A-31 C) a^2-5 b^4 (7 A+5 C)\right ) \cos ^2(c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {2 \left (48 C a^2+35 A b^2-13 b^2 C\right ) a^2-b \left (16 C a^2+35 A b^2+19 b^2 C\right ) \cos (c+d x) a-\left (192 C a^4+2 b^2 (70 A-31 C) a^2-5 b^4 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {2 \left (48 C a^2+35 A b^2-13 b^2 C\right ) a^2-b \left (16 C a^2+35 A b^2+19 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-192 C a^4-2 b^2 (70 A-31 C) a^2+5 b^4 (7 A+5 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {2 \int \frac {b \left (96 C a^4+2 b^2 (35 A-8 C) a^2+5 b^4 (7 A+5 C)\right )+a \left (384 C a^4+4 b^2 (70 A-43 C) a^2-b^4 (175 A+107 C)\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {b \left (96 C a^4+2 b^2 (35 A-8 C) a^2+5 b^4 (7 A+5 C)\right )+a \left (384 C a^4+4 b^2 (70 A-43 C) a^2-b^4 (175 A+107 C)\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\int \frac {b \left (96 C a^4+2 b^2 (35 A-8 C) a^2+5 b^4 (7 A+5 C)\right )+a \left (384 C a^4+4 b^2 (70 A-43 C) a^2-b^4 (175 A+107 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {\frac {2 a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{7 b}-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}}{b \left (a^2-b^2\right )}-\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {2 \left (a^2 C+A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {-\frac {2 \left (8 a^2 C+7 A b^2-b^2 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt {a+b \cos (c+d x)}}{7 b d}-\frac {-\frac {2 a \left (48 a^2 C+35 A b^2-13 b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 a \left (384 a^4 C+4 a^2 b^2 (70 A-43 C)-b^4 (175 A+107 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (384 a^4 C+4 a^2 b^2 (70 A+29 C)+5 b^4 (7 A+5 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (192 a^4 C+2 a^2 b^2 (70 A-31 C)-5 b^4 (7 A+5 C)\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}}{7 b}}{b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^3*(A + C*Cos[c + d*x]^2))/(a + b*Cos[c + d*x])^(3/2),x]
Output:
(-2*(A*b^2 + a^2*C)*Cos[c + d*x]^3*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Cos[c + d*x]]) - ((-2*(7*A*b^2 + 8*a^2*C - b^2*C)*Cos[c + d*x]^2*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(7*b*d) - ((-2*a*(35*A*b^2 + 48*a^2*C - 1 3*b^2*C)*Cos[c + d*x]*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) - ((( 2*a*(4*a^2*b^2*(70*A - 43*C) + 384*a^4*C - b^4*(175*A + 107*C))*Sqrt[a + b *Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[(a + b*Cos [c + d*x])/(a + b)]) - (2*(a^2 - b^2)*(384*a^4*C + 5*b^4*(7*A + 5*C) + 4*a ^2*b^2*(70*A + 29*C))*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d* x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*Cos[c + d*x]]))/(3*b) - (2*(2*a^2*b^ 2*(70*A - 31*C) + 192*a^4*C - 5*b^4*(7*A + 5*C))*Sqrt[a + b*Cos[c + d*x]]* Sin[c + d*x])/(3*b*d))/(5*b))/(7*b))/(b*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(1/2)*(c - Pi/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[ {a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 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^2*C + 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/(d*(n + 1)*(c^2 - d^ 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A* d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(m + n + 2) + C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Leaf count of result is larger than twice the leaf count of optimal. \(1791\) vs. \(2(456)=912\).
Time = 17.75 (sec) , antiderivative size = 1792, normalized size of antiderivative = 3.79
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1792\) |
| parts | \(\text {Expression too large to display}\) | \(2244\) |
Input:
int(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x,method=_RETUR NVERBOSE)
Output:
-(-(-2*b*cos(1/2*d*x+1/2*c)^2-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(A*a^2*b ^2+A*a*b^3+A*b^4+C*a^4+C*a^3*b+C*a^2*b^2+C*a*b^3+C*b^4)/b^5*(sin(1/2*d*x+1 /2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2* d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),(-2*b/(a-b))^(1/2))+32*C/b*(-1/14/b*cos(1/2*d*x+1/2*c)^5*(-2*b*sin(1/2*d *x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)-1/140/b^2*(-6*a+18*b)*cos(1/ 2*d*x+1/2*c)^3*(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2 )-1/420*(12*a^2-47*a*b+83*b^2)/b^3*cos(1/2*d*x+1/2*c)*(-2*b*sin(1/2*d*x+1/ 2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)+1/420*(12*a^2-47*a*b+83*b^2)/b^3* (a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)/(a-b))^ (1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*Ellipti cF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/210*(-6*a^3+28*a^2*b-58*a*b^2+ 84*b^3)/b^4*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c)^2+ a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^2)^( 1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1/2*d *x+1/2*c),(-2*b/(a-b))^(1/2))))+2/b^5*(A*a*b^2+2*A*b^3+C*a^3+2*C*a^2*b+3*C *a*b^2+4*C*b^3)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*b*cos(1/2*d*x+1/2*c )^2+a-b)/(a-b))^(1/2)/(-2*b*sin(1/2*d*x+1/2*c)^4+(a+b)*sin(1/2*d*x+1/2*c)^ 2)^(1/2)*(EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-EllipticE(cos(1 /2*d*x+1/2*c),(-2*b/(a-b))^(1/2)))+8/b^3*(A*b^2+C*a^2+3*C*a*b+6*C*b^2)*...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 917, normalized size of antiderivative = 1.94 \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x, algori thm="fricas")
Output:
2/315*(sqrt(1/2)*(-768*I*C*a^7 - 8*I*(70*A - 79*C)*a^5*b^2 + 2*I*(280*A + 83*C)*a^3*b^4 + 15*I*(7*A + 5*C)*a*b^6 + (-768*I*C*a^6*b - 8*I*(70*A - 79* C)*a^4*b^3 + 2*I*(280*A + 83*C)*a^2*b^5 + 15*I*(7*A + 5*C)*b^7)*cos(d*x + c))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9* a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b) + sqrt(1/ 2)*(768*I*C*a^7 + 8*I*(70*A - 79*C)*a^5*b^2 - 2*I*(280*A + 83*C)*a^3*b^4 - 15*I*(7*A + 5*C)*a*b^6 + (768*I*C*a^6*b + 8*I*(70*A - 79*C)*a^4*b^3 - 2*I *(280*A + 83*C)*a^2*b^5 - 15*I*(7*A + 5*C)*b^7)*cos(d*x + c))*sqrt(b)*weie rstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3* (3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(-384*I*C*a ^6*b - 4*I*(70*A - 43*C)*a^4*b^3 + I*(175*A + 107*C)*a^2*b^5 + (-384*I*C*a ^5*b^2 - 4*I*(70*A - 43*C)*a^3*b^4 + I*(175*A + 107*C)*a*b^6)*cos(d*x + c) )*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2) /b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2) /b^3, 1/3*(3*b*cos(d*x + c) + 3*I*b*sin(d*x + c) + 2*a)/b)) + 3*sqrt(1/2)* (384*I*C*a^6*b + 4*I*(70*A - 43*C)*a^4*b^3 - I*(175*A + 107*C)*a^2*b^5 + ( 384*I*C*a^5*b^2 + 4*I*(70*A - 43*C)*a^3*b^4 - I*(175*A + 107*C)*a*b^6)*cos (d*x + c))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cos(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b)) + ...
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3*(A+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**(3/2),x)
Output:
Timed out
\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^3/(b*cos(d*x + c) + a)^(3/2) , x)
\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*cos(d*x + c)^3/(b*cos(d*x + c) + a)^(3/2) , x)
Timed out. \[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int((cos(c + d*x)^3*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2),x)
Output:
int((cos(c + d*x)^3*(A + C*cos(c + d*x)^2))/(a + b*cos(c + d*x))^(3/2), x)
\[ \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) a \] Input:
int(cos(d*x+c)^3*(A+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x)
Output:
int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**5)/(cos(c + d*x)**2*b**2 + 2*c os(c + d*x)*a*b + a**2),x)*c + int((sqrt(cos(c + d*x)*b + a)*cos(c + d*x)* *3)/(cos(c + d*x)**2*b**2 + 2*cos(c + d*x)*a*b + a**2),x)*a