Integrand size = 45, antiderivative size = 704 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 a b^2 d}-\frac {\sqrt {a+b} \left (9 a^3 C-6 a^2 b (4 B+C)-8 b^3 (12 A+16 B+9 C)-4 a b^2 (60 A+28 B+39 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{192 b^2 d}+\frac {\sqrt {a+b} \left (8 a^3 b B-96 a b^3 B-3 a^4 C-24 a^2 b^2 (2 A+C)-16 b^4 (4 A+3 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^3 d}+\frac {\left (24 a^2 b B+128 b^3 B-9 a^3 C+12 a b^2 (20 A+13 C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{192 b^2 d \sqrt {\cos (c+d x)}}+\frac {\left (4 b^2 (4 A+3 C)+a (8 b B-3 a C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}+\frac {(8 b B-3 a C) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{24 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d} \] Output:
-1/192*(a-b)*(a+b)^(1/2)*(24*B*a^2*b+128*B*b^3-9*a^3*C+12*a*b^2*(20*A+13*C ))*cot(d*x+c)*EllipticE((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2 ),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/( a-b))^(1/2)/a/b^2/d-1/192*(a+b)^(1/2)*(9*a^3*C-6*a^2*b*(4*B+C)-8*b^3*(12*A +16*B+9*C)-4*a*b^2*(60*A+28*B+39*C))*cot(d*x+c)*EllipticF((a+b*cos(d*x+c)) ^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(-(a+b)/(a-b))^(1/2))*(a*(1-sec(d*x+c) )/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^(1/2)/b^2/d+1/64*(a+b)^(1/2)*(8*B* a^3*b-96*B*a*b^3-3*a^4*C-24*a^2*b^2*(2*A+C)-16*b^4*(4*A+3*C))*cot(d*x+c)*E llipticPi((a+b*cos(d*x+c))^(1/2)/(a+b)^(1/2)/cos(d*x+c)^(1/2),(a+b)/b,(-(a +b)/(a-b))^(1/2))*(a*(1-sec(d*x+c))/(a+b))^(1/2)*(a*(1+sec(d*x+c))/(a-b))^ (1/2)/b^3/d+1/192*(24*B*a^2*b+128*B*b^3-9*a^3*C+12*a*b^2*(20*A+13*C))*(a+b *cos(d*x+c))^(1/2)*sin(d*x+c)/b^2/d/cos(d*x+c)^(1/2)+1/32*(4*b^2*(4*A+3*C) +a*(8*B*b-3*C*a))*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d+1 /24*(8*B*b-3*C*a)*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*sin(d*x+c)/b/d+1 /4*C*cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(5/2)*sin(d*x+c)/b/d
Result contains complex when optimal does not.
Time = 7.92 (sec) , antiderivative size = 1317, normalized size of antiderivative = 1.87 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x ] + C*Cos[c + d*x]^2),x]
Output:
-1/384*((-4*a*(-336*a*A*b^2 - 136*a^2*b*B - 128*b^3*B + 3*a^3*C - 228*a*b^ 2*C)*Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d *x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2) /a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x) /2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos [c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - 4*a*(-384*a^2*A*b - 192*A*b^3 - 416 *a*b^2*B - 228*a^2*b*C - 144*b^3*C)*((Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(- a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sqrt[((a + b* Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[ArcSin[Sqrt[(( a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[ (c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]]) - (S qrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Cs c[(c + d*x)/2]^2)/a)]*Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Cs c[c + d*x]*EllipticPi[-(a/b), ArcSin[Sqrt[((a + b*Cos[c + d*x])*Csc[(c + d *x)/2]^2)/a]/Sqrt[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/(b*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d*x]])) + 2*(-240*a*A*b^2 - 24*a^2*b*B - 128*b^ 3*B + 9*a^3*C - 156*a*b^2*C)*((I*Cos[(c + d*x)/2]*Sqrt[a + b*Cos[c + d*x]] *EllipticE[I*ArcSinh[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]], (-2*a)/(-a - b) ]*Sec[c + d*x])/(b*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[((a + b*Cos[ c + d*x])*Sec[c + d*x])/(a + b)]) + (2*a*((a*Sqrt[((a + b)*Cot[(c + d*x...
Time = 3.82 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3540, 25, 3042, 3532, 3042, 3288, 3477, 3042, 3295, 3473}
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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B-3 a C) \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{2 \sqrt {\cos (c+d x)}}dx}{4 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left ((8 b B-3 a C) \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{\sqrt {\cos (c+d x)}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left ((8 b B-3 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (4 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \cos ^2(c+d x)+2 b (24 a A+16 b B+15 a C) \cos (c+d x)+a (8 b B+3 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\sqrt {a+b \cos (c+d x)} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \cos ^2(c+d x)+2 b (24 a A+16 b B+15 a C) \cos (c+d x)+a (8 b B+3 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (3 \left (4 (4 A+3 C) b^2+a (8 b B-3 a C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (24 a A+16 b B+15 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (8 b B+3 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{2} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \cos ^2(c+d x)+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \int \frac {\left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((96 A+57 C) a^2+104 b B a+12 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\int -\frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {3 \left (-3 C a^4+8 b B a^3-24 b^2 (2 A+C) a^2-96 b^3 B a-16 b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+3 \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+3 \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a \left (-9 C a^3+24 b B a^2+12 b^2 (20 A+13 C) a+128 b^3 B\right )-2 a b \left (3 C a^2+56 b B a+48 A b^2+36 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \cot (c+d x) \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {\frac {1}{6} \left (\frac {1}{4} \left (\frac {\sin (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 \sqrt {a+b} \cot (c+d x) \left (9 a^3 C-6 a^2 b (4 B+C)-4 a b^2 (60 A+28 B+39 C)-8 b^3 (12 A+16 B+9 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}+\frac {2 (a-b) \sqrt {a+b} \cot (c+d x) \left (-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d}-\frac {6 \sqrt {a+b} \cot (c+d x) \left (-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )+\frac {3 \sin (c+d x) \sqrt {\cos (c+d x)} \left (a (8 b B-3 a C)+4 b^2 (4 A+3 C)\right ) \sqrt {a+b \cos (c+d x)}}{2 d}\right )+\frac {(8 b B-3 a C) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + B*Cos[c + d*x] + C* Cos[c + d*x]^2),x]
Output:
(C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(5/2)*Sin[c + d*x])/(4*b*d) + ( ((8*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x ])/(3*d) + ((3*(4*b^2*(4*A + 3*C) + a*(8*b*B - 3*a*C))*Sqrt[Cos[c + d*x]]* Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(2*d) + (-1/2*((2*(a - b)*Sqrt[a + b]*(24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Cot[c + d*x ]*EllipticE[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x] ])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(a*d) + (2*Sqrt[a + b]*(9*a^3*C - 6*a^2*b*(4*B + C) - 8*b^3*(12*A + 16*B + 9*C) - 4*a*b^2*(60*A + 28*B + 39*C))*Cot[c + d* x]*EllipticF[ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x ]])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/d - (6*Sqrt[a + b]*(8*a^3*b*B - 96*a*b^3*B - 3* a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Cot[c + d*x]*EllipticPi [(a + b)/b, ArcSin[Sqrt[a + b*Cos[c + d*x]]/(Sqrt[a + b]*Sqrt[Cos[c + d*x] ])], -((a + b)/(a - b))]*Sqrt[(a*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[(a*(1 + Sec[c + d*x]))/(a - b)])/(b*d))/b + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[C os[c + d*x]]))/4)/6)/(8*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[(b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[2*b*(Tan[e + f*x]/(d*f))*Rt[(c + d)/b, 2]*Sqrt[c *((1 + Csc[e + f*x])/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*Ellipti cPi[(c + d)/d, ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2, 0] && PosQ[(c + d)/b]
Int[1/(Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f _.)*(x_)]]), x_Symbol] :> Simp[-2*(Tan[e + f*x]/(a*f))*Rt[(a + b)/d, 2]*Sqr t[a*((1 - Csc[e + f*x])/(a + b))]*Sqrt[a*((1 + Csc[e + f*x])/(a - b))]*Elli pticF[ArcSin[Sqrt[a + b*Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]]/Rt[(a + b)/d, 2] ], -(a + b)/(a - b)], x] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0] && PosQ[(a + b)/d]
Int[((A_) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((b_.)*sin[(e_.) + (f_.)*(x_)]) ^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*A* (c - d)*(Tan[e + f*x]/(f*b*c^2))*Rt[(c + d)/b, 2]*Sqrt[c*((1 + Csc[e + f*x] )/(c - d))]*Sqrt[c*((1 - Csc[e + f*x])/(c + d))]*EllipticE[ArcSin[Sqrt[c + d*Sin[e + f*x]]/Sqrt[b*Sin[e + f*x]]/Rt[(c + d)/b, 2]], -(c + d)/(c - d)], x] /; FreeQ[{b, c, d, e, f, A, B}, x] && NeQ[c^2 - d^2, 0] && EqQ[A, B] && PosQ[(c + d)/b]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])^(3/2)*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> S imp[(A - B)/(a - b) Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f* x]]), x], x] - Simp[(A*b - a*B)/(a - b) Int[(1 + Sin[e + f*x])/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*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] && NeQ[A, B]
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])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(3/2)*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[C/b^2 Int[Sqrt[a + b*Sin[e + f*x]] /Sqrt[c + d*Sin[e + f*x]], x], x] + Simp[1/b^2 Int[(A*b^2 - a^2*C + b*(b* B - 2*a*C)*Sin[e + f*x])/((a + b*Sin[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x ]]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] & & NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(Sqrt[c + d*Sin[e + f *x]]/(d*f*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[1/(2*d) Int[(1/((a + b*Si n[e + f*x])^(3/2)*Sqrt[c + d*Sin[e + f*x]]))*Simp[2*a*A*d - C*(b*c - a*d) - 2*(a*c*C - d*(A*b + a*B))*Sin[e + f*x] + (2*b*B*d - C*(b*c + a*d))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a *d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(3140\) vs. \(2(641)=1282\).
Time = 39.63 (sec) , antiderivative size = 3141, normalized size of antiderivative = 4.46
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3141\) |
| parts | \(\text {Expression too large to display}\) | \(3195\) |
Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x,method=_RETURNVERBOSE)
Output:
1/192/d*(-9*C*a^4*cos(d*x+c)*sin(d*x+c)+(-240*cos(d*x+c)^2-480*cos(d*x+c)- 240)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos( d*x+c)))^(1/2)*a*b^3*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2)) +(-24*cos(d*x+c)^2-48*cos(d*x+c)-24)*B*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d* x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3*b*EllipticE(cot(d*x+c)- csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-24*cos(d*x+c)^2-48*cos(d*x+c)-24)*B*(1/ (a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^( 1/2)*a^2*b^2*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-128*c os(d*x+c)^2-256*cos(d*x+c)-128)*B*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)) )^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^3*EllipticE(cot(d*x+c)-csc(d *x+c),(-(a-b)/(a+b))^(1/2))+(9*cos(d*x+c)^2+18*cos(d*x+c)+9)*C*(1/(a+b)*(a +b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^3 *b*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-156*cos(d*x+c)^ 2-312*cos(d*x+c)-156)*C*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(c os(d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^2*EllipticE(cot(d*x+c)-csc(d*x+c),(- (a-b)/(a+b))^(1/2))+(-156*cos(d*x+c)^2-312*cos(d*x+c)-156)*C*(1/(a+b)*(a+b *cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*a*b^3 *EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(384*cos(d*x+c)^2+7 68*cos(d*x+c)+384)*A*(1/(a+b)*(a+b*cos(d*x+c))/(1+cos(d*x+c)))^(1/2)*(cos( d*x+c)/(1+cos(d*x+c)))^(1/2)*a^2*b^2*EllipticF(cot(d*x+c)-csc(d*x+c),(-...
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="fricas")
Output:
integral((C*b*cos(d*x + c)^3 + (C*a + B*b)*cos(d*x + c)^2 + A*a + (B*a + A *b)*cos(d*x + c))*sqrt(b*cos(d*x + c) + a)*sqrt(cos(d*x + c)), x)
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**(3/2)*(A+B*cos(d*x+c)+C*cos( d*x+c)**2),x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d* x+c)^2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^(3/ 2)*sqrt(cos(d*x + c)), x)
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \] Input:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C* cos(c + d*x)^2),x)
Output:
int(cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2)*(A + B*cos(c + d*x) + C* cos(c + d*x)^2), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=2 \left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+B*cos(d*x+c)+C*cos(d*x+c)^2 ),x)
Output:
2*int(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x),x)*a*b + in t(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b*c + int (sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*c + int( sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*b**2 + int( sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)),x)*a**2