Integrand size = 42, antiderivative size = 387 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=-\frac {2 \left (40 a^3 b B-25 a b^3 B-48 a^4 C+24 a^2 b^2 C+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 b^4 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {2 \left (40 a^2 b B+5 b^3 B-48 a^3 C-12 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 b^4 d \sqrt {a+b \cos (c+d x)}}+\frac {2 a (b B-a C) \cos ^2(c+d x) \sin (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (20 a^2 b B-5 b^3 B-24 a^3 C+9 a b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 b^3 \left (a^2-b^2\right ) d}-\frac {2 \left (5 a b B-6 a^2 C+b^2 C\right ) \cos (c+d x) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d} \] Output:
-2/15*(40*B*a^3*b-25*B*a*b^3-48*C*a^4+24*C*a^2*b^2+9*C*b^4)*(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^4/(a^2-b^ 2)/d/((a+b*cos(d*x+c))/(a+b))^(1/2)+2/15*(40*B*a^2*b+5*B*b^3-48*C*a^3-12*C *a*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^4/d/(a+b*cos(d*x+c))^(1/2)+2*a*(B*b-C*a)*cos(d*x+c)^ 2*sin(d*x+c)/b/(a^2-b^2)/d/(a+b*cos(d*x+c))^(1/2)+2/15*(20*B*a^2*b-5*B*b^3 -24*C*a^3+9*C*a*b^2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^3/(a^2-b^2)/d-2/5 *(5*B*a*b-6*C*a^2+C*b^2)*cos(d*x+c)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b^2/ (a^2-b^2)/d
Time = 2.75 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.79 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\frac {\frac {2 b^2 \left (-10 a^2 b B-5 b^3 B+12 a^3 C+3 a b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{(a-b) (a+b)}+\frac {2 \left (-40 a^3 b B+25 a b^3 B+48 a^4 C-24 a^2 b^2 C-9 b^4 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \left ((a+b) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )\right )}{(a-b) (a+b)}+\frac {30 a^3 b (-b B+a C) \sin (c+d x)}{-a^2+b^2}+2 b (5 b B-9 a C) (a+b \cos (c+d x)) \sin (c+d x)+3 b^2 C (a+b \cos (c+d x)) \sin (2 (c+d x))}{15 b^4 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[ c + d*x])^(3/2),x]
Output:
((2*b^2*(-10*a^2*b*B - 5*b^3*B + 12*a^3*C + 3*a*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b)])/((a - b)*(a + b)) + (2*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*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)]))/((a - b)*(a + b)) + (30*a^3*b *(-(b*B) + a*C)*Sin[c + d*x])/(-a^2 + b^2) + 2*b*(5*b*B - 9*a*C)*(a + b*Co s[c + d*x])*Sin[c + d*x] + 3*b^2*C*(a + b*Cos[c + d*x])*Sin[2*(c + d*x)])/ (15*b^4*d*Sqrt[a + b*Cos[c + d*x]])
Time = 2.23 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 3508, 3042, 3468, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3508 |
| \(\displaystyle \int \frac {\cos ^3(c+d x) (B+C \cos (c+d x))}{(a+b \cos (c+d x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \left (B+C \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3468 |
| \(\displaystyle \frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {2 \int -\frac {\cos (c+d x) \left (-\left (\left (-6 C a^2+5 b B a+b^2 C\right ) \cos ^2(c+d x)\right )-b (b B-a C) \cos (c+d x)+4 a (b B-a C)\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\cos (c+d x) \left (-\left (\left (-6 C a^2+5 b B a+b^2 C\right ) \cos ^2(c+d x)\right )-b (b B-a C) \cos (c+d x)+4 a (b B-a C)\right )}{\sqrt {a+b \cos (c+d x)}}dx}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \left (\left (6 C a^2-5 b B a-b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b (b B-a C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 a (b B-a C)\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {2 \int -\frac {-\left (\left (-24 C a^3+20 b B a^2+9 b^2 C a-5 b^3 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^2+5 b B a-3 b^2 C\right ) \cos (c+d x)+2 a \left (-6 C a^2+5 b B a+b^2 C\right )}{2 \sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\int \frac {-\left (\left (-24 C a^3+20 b B a^2+9 b^2 C a-5 b^3 B\right ) \cos ^2(c+d x)\right )-b \left (-2 C a^2+5 b B a-3 b^2 C\right ) \cos (c+d x)+2 a \left (-6 C a^2+5 b B a+b^2 C\right )}{\sqrt {a+b \cos (c+d x)}}dx}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\int \frac {\left (24 C a^3-20 b B a^2-9 b^2 C a+5 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-b \left (-2 C a^2+5 b B a-3 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left (-6 C a^2+5 b B a+b^2 C\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {-\frac {\frac {2 \int \frac {b \left (-12 C a^3+10 b B a^2-3 b^2 C a+5 b^3 B\right )+\left (-48 C a^4+40 b B a^3+24 b^2 C a^2-25 b^3 B a+9 b^4 C\right ) \cos (c+d x)}{2 \sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {b \left (-12 C a^3+10 b B a^2-3 b^2 C a+5 b^3 B\right )+\left (-48 C a^4+40 b B a^3+24 b^2 C a^2-25 b^3 B a+9 b^4 C\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}}dx}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\int \frac {b \left (-12 C a^3+10 b B a^2-3 b^2 C a+5 b^3 B\right )+\left (-48 C a^4+40 b B a^3+24 b^2 C a^2-25 b^3 B a+9 b^4 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \int \sqrt {a+b \cos (c+d x)}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx}{b}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx}{b}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {\left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{b}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\frac {\frac {2 \left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}}{b \left (a^2-b^2\right )}+\frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 a (b B-a C) \sin (c+d x) \cos ^2(c+d x)}{b d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {-\frac {2 \left (-6 a^2 C+5 a b B+b^2 C\right ) \sin (c+d x) \cos (c+d x) \sqrt {a+b \cos (c+d x)}}{5 b d}-\frac {\frac {\frac {2 \left (-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{b d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) \left (-48 a^3 C+40 a^2 b B-12 a b^2 C+5 b^3 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{b d \sqrt {a+b \cos (c+d x)}}}{3 b}-\frac {2 \left (-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 b d}}{5 b}}{b \left (a^2-b^2\right )}\) |
Input:
Int[(Cos[c + d*x]^2*(B*Cos[c + d*x] + C*Cos[c + d*x]^2))/(a + b*Cos[c + d* x])^(3/2),x]
Output:
(2*a*(b*B - a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b* Cos[c + d*x]]) + ((-2*(5*a*b*B - 6*a^2*C + b^2*C)*Cos[c + d*x]*Sqrt[a + b* Cos[c + d*x]]*Sin[c + d*x])/(5*b*d) - (((2*(40*a^3*b*B - 25*a*b^3*B - 48*a ^4*C + 24*a^2*b^2*C + 9*b^4*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)*(40*a^2*b*B + 5*b^3*B - 48*a^3*C - 12*a*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*Cos[c + d*x]]))/(3*b) - (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*b*d))/(5*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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si mp[(-(b*c - a*d))*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(b*c - a*d)*(B*c - A*d)*(m - 1) + a*d*(a*A*c + b*B*c - (A*b + a *B)*d)*(n + 1) + (b*(b*d*(B*c - A*d) + a*(A*c*d + B*(c^2 - 2*d^2)))*(n + 1) - a*(b*c - a*d)*(B*c - A*d)*(n + 2))*Sin[e + f*x] + b*(d*(A*b*c + a*B*c - a*A*d)*(m + n + 1) - b*B*(c^2*m + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2 , 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] && LtQ[n, -1]
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[1/b^2 Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*(b*B - a*C + b*C*Sin[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*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. \(1311\) vs. \(2(374)=748\).
Time = 8.16 (sec) , antiderivative size = 1312, normalized size of antiderivative = 3.39
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1312\) |
| parts | \(\text {Expression too large to display}\) | \(2030\) |
Input:
int(cos(d*x+c)^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2),x,me thod=_RETURNVERBOSE)
Output:
-(-(-2*cos(1/2*d*x+1/2*c)^2*b-a+b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*(B*a^2*b +B*a*b^2+B*b^3-C*a^3-C*a^2*b-C*a*b^2-C*b^3)/b^4*(sin(1/2*d*x+1/2*c)^2)^(1/ 2)*((2*cos(1/2*d*x+1/2*c)^2*b+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))+16*C/b*(-1/10/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)-1/60/b^2*(-4*a+12*b)*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)+1/60/b^2*(-4* a+12*b)*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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))-1/60*(4*a^2-15*a*b+27*b^ 2)/b^3*(a-b)*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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)* (-EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))+EllipticF(cos(1/2*d*x+1 /2*c),(-2*b/(a-b))^(1/2))))+8*(B*b-C*a-3*C*b)/b^2*(-1/6/b*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)+1/6*(a-b)/ b*(sin(1/2*d*x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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(c os(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))-1/12/b^2*(-2*a+6*b)*(a-b)*(sin(1/2*d *x+1/2*c)^2)^(1/2)*((2*cos(1/2*d*x+1/2*c)^2*b+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)*(-EllipticE(cos(1/2*...
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 890, normalized size of antiderivative = 2.30 \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(cos(d*x+c)^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2 ),x, algorithm="fricas")
Output:
2/45*(sqrt(1/2)*(96*I*C*a^6 - 80*I*B*a^5*b - 84*I*C*a^4*b^2 + 80*I*B*a^3*b ^3 - 27*I*C*a^2*b^4 + 15*I*B*a*b^5 + (96*I*C*a^5*b - 80*I*B*a^4*b^2 - 84*I *C*a^3*b^3 + 80*I*B*a^2*b^4 - 27*I*C*a*b^5 + 15*I*B*b^6)*cos(d*x + c))*sqr t(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)*(-96 *I*C*a^6 + 80*I*B*a^5*b + 84*I*C*a^4*b^2 - 80*I*B*a^3*b^3 + 27*I*C*a^2*b^4 - 15*I*B*a*b^5 + (-96*I*C*a^5*b + 80*I*B*a^4*b^2 + 84*I*C*a^3*b^3 - 80*I* B*a^2*b^4 + 27*I*C*a*b^5 - 15*I*B*b^6)*cos(d*x + c))*sqrt(b)*weierstrassPI nverse(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)*(48*I*C*a^5*b - 40*I *B*a^4*b^2 - 24*I*C*a^3*b^3 + 25*I*B*a^2*b^4 - 9*I*C*a*b^5 + (48*I*C*a^4*b ^2 - 40*I*B*a^3*b^3 - 24*I*C*a^2*b^4 + 25*I*B*a*b^5 - 9*I*C*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)*(-48*I*C*a^5*b + 40*I*B*a^4*b^2 + 24*I*C*a^3*b^3 - 25*I*B*a^2*b^4 + 9*I *C*a*b^5 + (-48*I*C*a^4*b^2 + 40*I*B*a^3*b^3 + 24*I*C*a^2*b^4 - 25*I*B*a*b ^5 + 9*I*C*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...
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**2*(B*cos(d*x+c)+C*cos(d*x+c)**2)/(a+b*cos(d*x+c))**( 3/2),x)
Output:
Timed out
\[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2 ),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)^2/(b*cos(d*x + c) + a)^(3/2), x)
\[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right )\right )} \cos \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(cos(d*x+c)^2*(B*cos(d*x+c)+C*cos(d*x+c)^2)/(a+b*cos(d*x+c))^(3/2 ),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c))*cos(d*x + c)^2/(b*cos(d*x + c) + a)^(3/2), x)
Timed out. \[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \] Input:
int((cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d* x))^(3/2),x)
Output:
int((cos(c + d*x)^2*(B*cos(c + d*x) + C*cos(c + d*x)^2))/(a + b*cos(c + d* x))^(3/2), x)
\[ \int \frac {\cos ^2(c+d x) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx=\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) c +\left (\int \frac {\sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{3}}{\cos \left (d x +c \right )^{2} b^{2}+2 \cos \left (d x +c \right ) a b +a^{2}}d x \right ) b \] Input:
int(cos(d*x+c)^2*(B*cos(d*x+c)+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)**4)/(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)*b