Integrand size = 50, antiderivative size = 285 \[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{105 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{105 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b \left (56 a b B-41 a^2 C+25 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 b (7 b B-2 a C) (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{35 d}+\frac {2 b C (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{7 d} \] Output:
2/105*(161*B*a^2*b+63*B*b^3-146*C*a^3+82*C*a*b^2)*(a+b*cos(d*x+c))^(1/2)*E llipticE(sin(1/2*d*x+1/2*c),2^(1/2)*(b/(a+b))^(1/2))/d/((a+b*cos(d*x+c))/( a+b))^(1/2)-2/105*(a^2-b^2)*(56*B*a*b-41*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))/d/(a+ b*cos(d*x+c))^(1/2)+2/105*b*(56*B*a*b-41*C*a^2+25*C*b^2)*(a+b*cos(d*x+c))^ (1/2)*sin(d*x+c)/d+2/35*b*(7*B*b-2*C*a)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/ d+2/7*b*C*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/d
Time = 2.64 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (105 a^3 b B+119 a b^3 B-105 a^4 C+16 a^2 b^2 C+25 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+2 \left (161 a^2 b B+63 b^3 B-146 a^3 C+82 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )+b (a+b \cos (c+d x)) \left (154 a b B-64 a^2 C+65 b^2 C+6 b (7 b B+8 a C) \cos (c+d x)+15 b^2 C \cos (2 (c+d x))\right ) \sin (c+d x)}{105 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[(a + b*Cos[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*Cos[c + d*x]^2),x]
Output:
(2*(105*a^3*b*B + 119*a*b^3*B - 105*a^4*C + 16*a^2*b^2*C + 25*b^4*C)*Sqrt[ (a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)] + 2*(1 61*a^2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/ (a + b)]*((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])*(154*a*b*B - 64*a^2*C + 65*b^2*C + 6*b*(7*b*B + 8*a*C)*Cos[c + d*x] + 15*b^2*C*Cos[2*(c + d*x)])*S in[c + d*x])/(105*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.76 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.09, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 3494, 3042, 3232, 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.
| \(\displaystyle \int (a+b \cos (c+d x))^{3/2} \left (a^2 (-C)+a b B+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a^2 (-C)+a b B+b^2 B \sin \left (c+d x+\frac {\pi }{2}\right )+b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3494 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{5/2} \left (C \cos (c+d x) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{7} \int \frac {1}{2} (a+b \cos (c+d x))^{3/2} \left ((7 b B-2 a C) \cos (c+d x) b^3+\left (-7 C a^2+7 b B a+5 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \int (a+b \cos (c+d x))^{3/2} \left ((7 b B-2 a C) \cos (c+d x) b^3+\left (-7 C a^2+7 b B a+5 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((7 b B-2 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-7 C a^2+7 b B a+5 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left (\left (-41 C a^2+56 b B a+25 b^2 C\right ) \cos (c+d x) b^3+\left (-35 C a^3+35 b B a^2+19 b^2 C a+21 b^3 B\right ) b^2\right )dx+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left (\left (-41 C a^2+56 b B a+25 b^2 C\right ) \cos (c+d x) b^3+\left (-35 C a^3+35 b B a^2+19 b^2 C a+21 b^3 B\right ) b^2\right )dx+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (\left (-41 C a^2+56 b B a+25 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-35 C a^3+35 b B a^2+19 b^2 C a+21 b^3 B\right ) b^2\right )dx+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (-146 C a^3+161 b B a^2+82 b^2 C a+63 b^3 B\right ) \cos (c+d x) b^3+\left (-105 C a^4+105 b B a^3+16 b^2 C a^2+119 b^3 B a+25 b^4 C\right ) b^2}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (-146 C a^3+161 b B a^2+82 b^2 C a+63 b^3 B\right ) \cos (c+d x) b^3+\left (-105 C a^4+105 b B a^3+16 b^2 C a^2+119 b^3 B a+25 b^4 C\right ) b^2}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (-146 C a^3+161 b B a^2+82 b^2 C a+63 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-105 C a^4+105 b B a^3+16 b^2 C a^2+119 b^3 B a+25 b^4 C\right ) b^2}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \int \sqrt {a+b \cos (c+d x)}dx-b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {2 b^3 \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}+\frac {1}{3} \left (\frac {2 b^2 \left (-146 a^3 C+161 a^2 b B+82 a b^2 C+63 b^3 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 b^2 \left (a^2-b^2\right ) \left (-41 a^2 C+56 a b B+25 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}\right )\right )+\frac {2 b^3 (7 b B-2 a C) \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{7 d}}{b^2}\) |
Input:
Int[(a + b*Cos[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C *Cos[c + d*x]^2),x]
Output:
((2*b^3*C*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(7*d) + ((2*b^3*(7*b*B - 2*a*C)*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*b^2*(161*a^ 2*b*B + 63*b^3*B - 146*a^3*C + 82*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Ellipt icE[(c + d*x)/2, (2*b)/(a + b)])/(d*Sqrt[(a + b*Cos[c + d*x])/(a + b)]) - (2*b^2*(a^2 - b^2)*(56*a*b*B - 41*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)])/(d*Sqrt[a + b*Cos[c + d*x]]))/3 + (2*b^3*(56*a*b*B - 41*a^2*C + 25*b^2*C)*Sqrt[a + b*Cos[c + d*x ]]*Sin[c + d*x])/(3*d))/5)/7)/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_)*((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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1301\) vs. \(2(270)=540\).
Time = 28.60 (sec) , antiderivative size = 1302, normalized size of antiderivative = 4.57
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1302\) |
| parts | \(\text {Expression too large to display}\) | \(1945\) |
Input:
int((a+b*cos(d*x+c))^(3/2)*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^ 2),x,method=_RETURNVERBOSE)
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)+(392*B*a*b^3+168*B*b^4-32*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)+(-15 4*B*a^2*b^2-196*B*a*b^3-42*B*b^4+64*C*a^3*b+16*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)-56*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(cos(1/2 *d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3*b+56*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+161*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-161*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*b/(a-b)*si n(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^2*b^2+63*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*b^3-63*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)) *b^4+41*C*(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^4-66*C *(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...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.97 \[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d *x+c)^2),x, algorithm="fricas")
Output:
-2/315*(sqrt(1/2)*(-23*I*C*a^4 - 7*I*B*a^3*b - 116*I*C*a^2*b^2 + 231*I*B*a *b^3 + 75*I*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)*(23*I*C*a^4 + 7*I*B*a^3*b + 116*I*C*a^2*b^2 - 231*I*B*a* b^3 - 75*I*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) + 3*sqrt(1/2)*(146*I*C*a^3*b - 161*I*B*a^2*b^2 - 82*I*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*sqrt( 1/2)*(-146*I*C*a^3*b + 161*I*B*a^2*b^2 + 82*I*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, we ierstrassPInverse(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 - 32*C*a^2*b^2 + 77*B*a*b^3 + 25*C*b^4 + 3*(8*C*a*b^3 + 7*B*b^4)*co s(d*x + c))*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b*d)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate((a+b*cos(d*x+c))**(3/2)*(B*a*b-a**2*C+b**2*B*cos(d*x+c)+b**2*C*c os(d*x+c)**2),x)
Output:
Timed out
\[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C b^{2} \cos \left (d x + c\right )^{2} + B b^{2} \cos \left (d x + c\right ) - C a^{2} + B a b\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d *x+c)^2),x, algorithm="maxima")
Output:
integrate((C*b^2*cos(d*x + c)^2 + B*b^2*cos(d*x + c) - C*a^2 + B*a*b)*(b*c os(d*x + c) + a)^(3/2), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C b^{2} \cos \left (d x + c\right )^{2} + B b^{2} \cos \left (d x + c\right ) - C a^{2} + B a b\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(3/2)*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d *x+c)^2),x, algorithm="giac")
Output:
integrate((C*b^2*cos(d*x + c)^2 + B*b^2*cos(d*x + c) - C*a^2 + B*a*b)*(b*c os(d*x + c) + a)^(3/2), x)
Timed out. \[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int {\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (-C\,a^2+B\,a\,b+C\,b^2\,{\cos \left (c+d\,x\right )}^2+B\,b^2\,\cos \left (c+d\,x\right )\right ) \,d x \] Input:
int((a + b*cos(c + d*x))^(3/2)*(C*b^2*cos(c + d*x)^2 - C*a^2 + B*a*b + B*b ^2*cos(c + d*x)),x)
Output:
int((a + b*cos(c + d*x))^(3/2)*(C*b^2*cos(c + d*x)^2 - C*a^2 + B*a*b + B*b ^2*cos(c + d*x)), x)
\[ \int (a+b \cos (c+d x))^{3/2} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=-\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{3} c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{2} b^{2}-\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a^{2} b c +2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) a \,b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}d x \right ) b^{3} c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) a \,b^{2} c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) b^{4} \] Input:
int((a+b*cos(d*x+c))^(3/2)*(B*a*b-a^2*C+b^2*B*cos(d*x+c)+b^2*C*cos(d*x+c)^ 2),x)
Output:
- int(sqrt(cos(c + d*x)*b + a),x)*a**3*c + int(sqrt(cos(c + d*x)*b + a),x )*a**2*b**2 - int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x),x)*a**2*b*c + 2*in t(sqrt(cos(c + d*x)*b + a)*cos(c + d*x),x)*a*b**3 + int(sqrt(cos(c + d*x)* b + a)*cos(c + d*x)**3,x)*b**3*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d* x)**2,x)*a*b**2*c + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x)**2,x)*b**4