Integrand size = 35, antiderivative size = 315 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (21 a^2 b B+63 b^3 B-6 a^3 C+2 a b^2 (70 A+41 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (35 A b^2+21 a b B-6 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 b^2 d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (35 A b^2+21 a b B-6 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 b d}+\frac {2 (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 b d}+\frac {2 C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d} \] Output:
2/105*(21*B*a^2*b+63*B*b^3-6*a^3*C+2*a*b^2*(70*A+41*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^2/d/((a+b*cos (d*x+c))/(a+b))^(1/2)-2/105*(a^2-b^2)*(35*A*b^2+21*B*a*b-6*C*a^2+25*C*b^2) *((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^2/d/(a+b*cos(d*x+c))^(1/2)+2/105*(35*A*b^2+21*B*a*b-6*C*a^2 +25*C*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d+2/35*(7*B*b-2*C*a)*(a+b*c os(d*x+c))^(3/2)*sin(d*x+c)/b/d+2/7*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/ d
Time = 3.07 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.82 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {4 \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left (b^2 \left (84 a b B+5 b^2 (7 A+5 C)+3 a^2 (35 A+17 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+\left (21 a^2 b B+63 b^3 B-6 a^3 C+2 a b^2 (70 A+41 C)\right ) \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )\right )+b (a+b \cos (c+d x)) \left (\left (140 A b^2+168 a b B+12 a^2 C+115 b^2 C\right ) \sin (c+d x)+3 b (2 (7 b B+8 a C) \sin (2 (c+d x))+5 b C \sin (3 (c+d x)))\right )}{210 b^2 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^ 2),x]
Output:
(4*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*(b^2*(84*a*b*B + 5*b^2*(7*A + 5*C) + 3*a^2*(35*A + 17*C))*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + (21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*((a + b)*EllipticE[(c + d*x) /2, (2*b)/(a + b)] - a*EllipticF[(c + d*x)/2, (2*b)/(a + b)])) + b*(a + b* Cos[c + d*x])*((140*A*b^2 + 168*a*b*B + 12*a^2*C + 115*b^2*C)*Sin[c + d*x] + 3*b*(2*(7*b*B + 8*a*C)*Sin[2*(c + d*x)] + 5*b*C*Sin[3*(c + d*x)])))/(21 0*b^2*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.62 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.03, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.514, Rules used = {3042, 3502, 27, 3042, 3232, 27, 3042, 3232, 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.
| \(\displaystyle \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {2 \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} (b (7 A+5 C)+(7 b B-2 a C) \cos (c+d x))dx}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{3/2} (b (7 A+5 C)+(7 b B-2 a C) \cos (c+d x))dx}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (b (7 A+5 C)+(7 b B-2 a C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (b (35 a A+21 b B+19 a C)+\left (-6 C a^2+21 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)\right )dx+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (b (35 a A+21 b B+19 a C)+\left (-6 C a^2+21 b B a+35 A b^2+25 b^2 C\right ) \cos (c+d x)\right )dx+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (b (35 a A+21 b B+19 a C)+\left (-6 C a^2+21 b B a+35 A b^2+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {b \left (3 (35 A+17 C) a^2+84 b B a+5 b^2 (7 A+5 C)\right )+\left (-6 C a^3+21 b B a^2+2 b^2 (70 A+41 C) a+63 b^3 B\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (3 (35 A+17 C) a^2+84 b B a+5 b^2 (7 A+5 C)\right )+\left (-6 C a^3+21 b B a^2+2 b^2 (70 A+41 C) a+63 b^3 B\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (3 (35 A+17 C) a^2+84 b B a+5 b^2 (7 A+5 C)\right )+\left (-6 C a^3+21 b B a^2+2 b^2 (70 A+41 C) a+63 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 \left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2 \sin (c+d x) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 \left (-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}\right )\right )+\frac {2 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{7 b}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 b d}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(2*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*b*d) + ((2*(7*b*B - 2*a*C )*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*(21*a^2*b*B + 63*b ^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*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)*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/(b*d*Sqrt[a + b*C os[c + d*x]]))/3 + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)/(7*b)
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_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1634\) vs. \(2(300)=600\).
Time = 39.31 (sec) , antiderivative size = 1635, normalized size of antiderivative = 5.19
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1635\) |
| parts | \(\text {Expression too large to display}\) | \(1942\) |
Input:
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETUR NVERBOSE)
Output:
-2/105*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(240*C* b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^8+(-168*B*b^4-312*C*a*b^3-360*C* b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(140*A*b^4+252*B*a*b^3+168*B* b^4+108*C*a^2*b^2+312*C*a*b^3+280*C*b^4)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+ 1/2*c)+(-70*A*a*b^3-70*A*b^4-84*B*a^2*b^2-126*B*a*b^3-42*B*b^4-6*C*a^3*b-5 4*C*a^2*b^2-128*C*a*b^3-80*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)- 35*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/( a-b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2+35*A* b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a -b))^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+140*A*(sin(1/2 *d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*E llipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b^2-140*A*(sin(1/2*d*x +1/2*c)^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*Ellip ticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a*b^3-21*B*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(co s(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+21*B*a*(sin(1/2*d*x+1/2*c)^2)^( 1/2)*(-2*b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticF(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))*b^3+21*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2* b/(a-b)*sin(1/2*d*x+1/2*c)^2+(a+b)/(a-b))^(1/2)*EllipticE(cos(1/2*d*x+1/2* c),(-2*b/(a-b))^(1/2))*a^3*b-21*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 604, normalized size of antiderivative = 1.92 \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="fricas")
Output:
-2/315*(sqrt(1/2)*(12*I*C*a^4 - 42*I*B*a^3*b + I*(35*A - 11*C)*a^2*b^2 + 1 26*I*B*a*b^3 + 15*I*(7*A + 5*C)*b^4)*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)*(-12*I*C*a^4 + 42*I*B*a^3*b - I*(35*A - 11*C)*a^2*b^2 - 126*I*B*a*b^3 - 15*I*(7*A + 5*C)*b^4)*sqrt(b)*weierstras sPInverse(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*c os(d*x + c) - 3*I*b*sin(d*x + c) + 2*a)/b) + 3*sqrt(1/2)*(6*I*C*a^3*b - 21 *I*B*a^2*b^2 - 2*I*(70*A + 41*C)*a*b^3 - 63*I*B*b^4)*sqrt(b)*weierstrassZe ta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInver se(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)*(-6*I*C*a^3*b + 21*I*B* a^2*b^2 + 2*I*(70*A + 41*C)*a*b^3 + 63*I*B*b^4)*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*(15*C*b^4*cos(d*x + c)^2 + 3*C*a^2*b^2 + 42*B*a*b^3 + 5*(7*A + 5*C)*b^4 + 3*(8*C*a*b^3 + 7*B*b^4)*cos(d*x + c))* sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b^3*d)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algori thm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
int((a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C*cos(c + d*x)^2), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{2}+2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) b^{2} \] Input:
int((a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
int(sqrt(cos(c + d*x)*b + a),x)*a**2 + 2*int(sqrt(cos(c + d*x)*b + a)*cos( c + d*x),x)*a*b + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**3,x)*b*c + in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*a*c + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**2,x)*b**2