Integrand size = 37, antiderivative size = 638 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (80 A b^2-3 a^2 C+52 b^2 C\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d}-\frac {\sqrt {a+b} \left (3 a^3 C-2 a^2 b C-8 b^3 (4 A+3 C)-4 a b^2 (20 A+13 C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^2 d}-\frac {\sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{64 b^3 d}+\frac {a \left (80 A b^2-3 a^2 C+52 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{64 b^2 d \sqrt {\cos (c+d x)}}-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{32 b d}-\frac {a C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{8 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{4 b d} \] Output:
-1/64*(a-b)*(a+b)^(1/2)*(80*A*b^2-3*C*a^2+52*C*b^2)*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)/b^2/d-1/64*(a +b)^(1/2)*(3*a^3*C-2*a^2*b*C-8*b^3*(4*A+3*C)-4*a*b^2*(20*A+13*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)*(3*a^4*C+24*a^2*b^2*(2*A+C)+16*b^4*(4*A+3*C))*cot( d*x+c)*EllipticPi((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+1/64*a*(80*A*b^2-3*C*a^2+52*C*b^2)*(a+b*cos(d*x+c))^(1 /2)*sin(d*x+c)/b^2/d/cos(d*x+c)^(1/2)-1/32*(3*a^2*C-4*b^2*(4*A+3*C))*cos(d *x+c)^(1/2)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/b/d-1/8*a*C*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 = 7.74 (sec) , antiderivative size = 1270, normalized size of antiderivative = 1.99 \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + C*Cos[c + d*x ]^2),x]
Output:
-1/128*((-4*a*(-112*a*A*b^2 + a^3*C - 76*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)]*S qrt[((a + b*Cos[c + d*x])*Csc[(c + d*x)/2]^2)/a]*Csc[c + d*x]*EllipticF[Ar cSin[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*(-128*a^2*A*b - 64*A*b^3 - 76*a^2*b*C - 48*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]/Sqr t[2]], (-2*a)/(-a + b)]*Sin[(c + d*x)/2]^4)/((a + b)*Sqrt[Cos[c + d*x]]*Sq rt[a + b*Cos[c + d*x]]) - (Sqrt[((a + b)*Cot[(c + d*x)/2]^2)/(-a + b)]*Sqr t[-(((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]*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*(-80*a* A*b^2 + 3*a^3*C - 52*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)/ 2]^2)/(-a + b)]*Sqrt[-(((a + b)*Cos[c + d*x]*Csc[(c + d*x)/2]^2)/a)]*Sq...
Time = 3.48 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.541, Rules used = {3042, 3529, 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.
| \(\displaystyle \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3529 |
| \(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left (-3 a C \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{2 \sqrt {\cos (c+d x)}}dx}{4 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \cos (c+d x))^{3/2} \left (-3 a C \cos ^2(c+d x)+2 b (4 A+3 C) \cos (c+d x)+a C\right )}{\sqrt {\cos (c+d x)}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-3 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b (4 A+3 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a C\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {3 \sqrt {a+b \cos (c+d x)} \left (C a^2+2 b (8 A+5 C) \cos (c+d x) a-\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {a+b \cos (c+d x)} \left (C a^2+2 b (8 A+5 C) \cos (c+d x) a-\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}}dx-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \int \frac {\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (C a^2+2 b (8 A+5 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (4 b^2 (4 A+3 C)-3 a^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{2} \int \frac {a \left (-3 C a^2+80 A b^2+52 b^2 C\right ) \cos ^2(c+d x)+2 b \left ((32 A+19 C) a^2+4 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (C a^2+4 b^2 (4 A+3 C)\right )}{2 \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \int \frac {a \left (-3 C a^2+80 A b^2+52 b^2 C\right ) \cos ^2(c+d x)+2 b \left ((32 A+19 C) a^2+4 b^2 (4 A+3 C)\right ) \cos (c+d x)+a \left (C a^2+4 b^2 (4 A+3 C)\right )}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \int \frac {a \left (-3 C a^2+80 A b^2+52 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 b \left ((32 A+19 C) a^2+4 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left (C a^2+4 b^2 (4 A+3 C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3540 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {\int -\frac {\left (-3 C a^2+80 A b^2+52 b^2 C\right ) a^2-2 b \left (C a^2+4 b^2 (4 A+3 C)\right ) \cos (c+d x) a-\left (3 C a^4+24 b^2 (2 A+C) a^2+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}+\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {\left (-3 C a^2+80 A b^2+52 b^2 C\right ) a^2-2 b \left (C a^2+4 b^2 (4 A+3 C)\right ) \cos (c+d x) a-\left (3 C a^4+24 b^2 (2 A+C) a^2+16 b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {\left (-3 C a^2+80 A b^2+52 b^2 C\right ) a^2-2 b \left (C a^2+4 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (-3 C a^4-24 b^2 (2 A+C) a^2-16 b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3532 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a^2 \left (-3 C a^2+80 A b^2+52 b^2 C\right )-2 a b \left (C a^2+4 b^2 (4 A+3 C)\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx-\left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}}dx}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a^2 \left (-3 C a^2+80 A b^2+52 b^2 C\right )-2 a b \left (C a^2+4 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx-\left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3288 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\int \frac {a^2 \left (-3 C a^2+80 A b^2+52 b^2 C\right )-2 a b \left (C a^2+4 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3477 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a^2 \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \int \frac {\cos (c+d x)+1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}}dx+a \left (3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}dx+\frac {2 \sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a^2 \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+a \left (3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3295 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {a^2 \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )+1}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}+\frac {2 \sqrt {a+b} \left (3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
| \(\Big \downarrow \) 3473 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {a \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{b d \sqrt {\cos (c+d x)}}-\frac {\frac {2 (a-b) \sqrt {a+b} \left (-3 a^2 C+80 A b^2+52 b^2 C\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{d}+\frac {2 \sqrt {a+b} \left (3 a^4 C+24 a^2 b^2 (2 A+C)+16 b^4 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{b d}+\frac {2 \sqrt {a+b} \left (3 a^3 C-2 a^2 b C-4 a b^2 (20 A+13 C)-8 b^3 (4 A+3 C)\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{d}}{2 b}\right )-\frac {\left (3 a^2 C-4 b^2 (4 A+3 C)\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{2 d}\right )-\frac {a C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{d}}{8 b}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{5/2}}{4 b d}\) |
Input:
Int[Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*(A + 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) + ( -((a*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/d) + (- 1/2*((3*a^2*C - 4*b^2*(4*A + 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Cos[c + d *x]]*Sin[c + d*x])/d + (-1/2*((2*(a - b)*Sqrt[a + b]*(80*A*b^2 - 3*a^2*C + 52*b^2*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)])/d + (2*Sqrt[a + b]*(3*a^3* C - 2*a^2*b*C - 8*b^3*(4*A + 3*C) - 4*a*b^2*(20*A + 13*C))*Cot[c + d*x]*El lipticF[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 + (2*Sqrt[a + 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 + (a*(80*A*b^2 - 3*a^2*C + 52*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]]))/4)/2)/(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_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* (a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , A, 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. \(2185\) vs. \(2(575)=1150\).
Time = 28.11 (sec) , antiderivative size = 2186, normalized size of antiderivative = 3.43
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2186\) |
| parts | \(\text {Expression too large to display}\) | \(2201\) |
Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x,method=_R ETURNVERBOSE)
Output:
1/64/d*((-80*cos(d*x+c)^2-160*cos(d*x+c)-80)*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*EllipticE(c ot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-80*cos(d*x+c)^2-160*cos(d*x+c )-80)*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) )+(3*cos(d*x+c)^2+6*cos(d*x+c)+3)*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))+(-52*cos(d*x+c)^2-104*cos(d*x+c)-52)*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^2*b^2*EllipticE(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(-52*cos( d*x+c)^2-104*cos(d*x+c)-52)*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))+(128*cos(d*x+c)^2+256*cos(d*x+c)+128)*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),(-(a-b)/(a+b))^(1/2))+(-32*cos(d*x+c )^2-64*cos(d*x+c)-32)*A*(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*b^3*EllipticF(cot(d*x+c)-csc(d*x+c),(-(a -b)/(a+b))^(1/2))+(-2*cos(d*x+c)^2-4*cos(d*x+c)-2)*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*Ellipti cF(cot(d*x+c)-csc(d*x+c),(-(a-b)/(a+b))^(1/2))+(76*cos(d*x+c)^2+152*cos...
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, al gorithm="fricas")
Output:
Timed out
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a+b*cos(d*x+c))**(3/2)*(A+C*cos(d*x+c)**2),x)
Output:
Timed out
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, al gorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c)), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x, al gorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^(3/2)*sqrt(cos(d*x + c)), x)
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2} \,d x \] Input:
int(cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2),x )
Output:
int(cos(c + d*x)^(1/2)*(A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^(3/2), x)
\[ \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \left (A+C \cos ^2(c+d x)\right ) \, dx=\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}d x \right ) b c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}d x \right ) a c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \sqrt {\cos \left (d x +c \right )}d x \right ) a^{2} \] Input:
int(cos(d*x+c)^(1/2)*(a+b*cos(d*x+c))^(3/2)*(A+C*cos(d*x+c)^2),x)
Output:
int(sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x),x)*a*b + int( sqrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**3,x)*b*c + int(s qrt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x))*cos(c + d*x)**2,x)*a*c + int(sq rt(cos(c + d*x)*b + a)*sqrt(cos(c + d*x)),x)*a**2