Integrand size = 50, antiderivative size = 221 \[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {2 \left (20 a b B-17 a^2 C+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 \left (a^2-b^2\right ) (5 b B-2 a C) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{15 d \sqrt {a+b \cos (c+d x)}}+\frac {2 b (5 b B-2 a C) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{15 d}+\frac {2 b C (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d} \] Output:
2/15*(20*B*a*b-17*C*a^2+9*C*b^2)*(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))/d/((a+b*cos(d*x+c))/(a+b))^(1/2)-2/15* (a^2-b^2)*(5*B*b-2*C*a)*((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)+2/15*b*(5*B*b -2*C*a)*(a+b*cos(d*x+c))^(1/2)*sin(d*x+c)/d+2/5*b*C*(a+b*cos(d*x+c))^(3/2) *sin(d*x+c)/d
Time = 2.35 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.81 \[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\frac {-2 (a+b) \left (-20 a b B+17 a^2 C-9 b^2 C\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+2 \left (a^2-b^2\right ) (-5 b B+2 a C) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )+2 b (a+b \cos (c+d x)) (5 b B+a C+3 b C \cos (c+d x)) \sin (c+d x)}{15 d \sqrt {a+b \cos (c+d x)}} \] Input:
Integrate[Sqrt[a + b*Cos[c + d*x]]*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b ^2*C*Cos[c + d*x]^2),x]
Output:
(-2*(a + b)*(-20*a*b*B + 17*a^2*C - 9*b^2*C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*b)/(a + b)] + 2*(a^2 - b^2)*(-5*b*B + 2*a* C)*Sqrt[(a + b*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*b)/(a + b) ] + 2*b*(a + b*Cos[c + d*x])*(5*b*B + a*C + 3*b*C*Cos[c + d*x])*Sin[c + d* x])/(15*d*Sqrt[a + b*Cos[c + d*x]])
Time = 1.36 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.340, Rules used = {3042, 3494, 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.
| \(\displaystyle \int \sqrt {a+b \cos (c+d x)} \left (a^2 (-C)+a b B+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left (a^2 (-C)+a b B+b^2 B \sin \left (c+d x+\frac {\pi }{2}\right )+b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3494 |
| \(\displaystyle \frac {\int (a+b \cos (c+d x))^{3/2} \left (C \cos (c+d x) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (C \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+(b B-a C) b^2\right )dx}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {2}{5} \int \frac {1}{2} \sqrt {a+b \cos (c+d x)} \left ((5 b B-2 a C) \cos (c+d x) b^3+\left (-5 C a^2+5 b B a+3 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {a+b \cos (c+d x)} \left ((5 b B-2 a C) \cos (c+d x) b^3+\left (-5 C a^2+5 b B a+3 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )} \left ((5 b B-2 a C) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-5 C a^2+5 b B a+3 b^2 C\right ) b^2\right )dx+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {2}{3} \int \frac {\left (-17 C a^2+20 b B a+9 b^2 C\right ) \cos (c+d x) b^3+\left (-15 C a^3+15 b B a^2+7 b^2 C a+5 b^3 B\right ) b^2}{2 \sqrt {a+b \cos (c+d x)}}dx+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (-17 C a^2+20 b B a+9 b^2 C\right ) \cos (c+d x) b^3+\left (-15 C a^3+15 b B a^2+7 b^2 C a+5 b^3 B\right ) b^2}{\sqrt {a+b \cos (c+d x)}}dx+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \int \frac {\left (-17 C a^2+20 b B a+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right ) b^3+\left (-15 C a^3+15 b B a^2+7 b^2 C a+5 b^3 B\right ) b^2}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \int \sqrt {a+b \cos (c+d x)}dx-b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \int \frac {1}{\sqrt {a+b \cos (c+d x)}}dx\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \int \sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}dx-b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}dx}{\sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \int \frac {1}{\sqrt {a+b \sin \left (c+d x+\frac {\pi }{2}\right )}}dx\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (c+d x+\frac {\pi }{2}\right )}{a+b}}}dx}{\sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {1}{5} \left (\frac {1}{3} \left (\frac {2 b^2 \left (-17 a^2 C+20 a b B+9 b^2 C\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {2 b^2 \left (a^2-b^2\right ) (5 b B-2 a C) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \cos (c+d x)}}\right )+\frac {2 b^3 (5 b B-2 a C) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3 d}\right )+\frac {2 b^3 C \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{5 d}}{b^2}\) |
Input:
Int[Sqrt[a + b*Cos[c + d*x]]*(a*b*B - a^2*C + b^2*B*Cos[c + d*x] + b^2*C*C os[c + d*x]^2),x]
Output:
((2*b^3*C*(a + b*Cos[c + d*x])^(3/2)*Sin[c + d*x])/(5*d) + (((2*b^2*(20*a* b*B - 17*a^2*C + 9*b^2*C)*Sqrt[a + b*Cos[c + d*x]]*EllipticE[(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 )*(5*b*B - 2*a*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*(5*b*B - 2*a*C) *Sqrt[a + b*Cos[c + d*x]]*Sin[c + d*x])/(3*d))/5)/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. \(989\) vs. \(2(210)=420\).
Time = 20.37 (sec) , antiderivative size = 990, normalized size of antiderivative = 4.48
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(990\) |
| parts | \(\text {Expression too large to display}\) | \(1292\) |
Input:
int((a+b*cos(d*x+c))^(1/2)*(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:
-2/15*((2*b*cos(1/2*d*x+1/2*c)^2+a-b)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-24*C*b ^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^6+(20*B*b^3+16*C*a*b^2+24*C*b^3)* sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-10*B*a*b^2-10*B*b^3-2*C*a^2*b-8* C*a*b^2-6*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-5*B*a^2*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))+5*B*b^3*(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))+20*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^2*b-20*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^2+2*a^3*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))-2*C*a*b^2*(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))-17*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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^3+17 *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)*EllipticE(cos(1/2*d*x+1/2*c),(-2*b/(a-b))^(1/2))*a^2*b+9*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))^(...
Result contains complex when optimal does not.
Time = 0.11 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.23 \[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(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:
-2/45*(sqrt(1/2)*(-11*I*C*a^3 + 5*I*B*a^2*b + 3*I*C*a*b^2 + 15*I*B*b^3)*sq rt(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)*(11 *I*C*a^3 - 5*I*B*a^2*b - 3*I*C*a*b^2 - 15*I*B*b^3)*sqrt(b)*weierstrassPInv erse(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)*(17*I*C*a^2*b - 20*I*B *a*b^2 - 9*I*C*b^3)*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)*(-17*I*C*a^2*b + 20*I*B*a*b^2 + 9*I*C*b^3)*sqrt(b)*weier strassZeta(4/3*(4*a^2 - 3*b^2)/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstra ssPInverse(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*(3*C*b^3*cos(d*x + c) + C *a*b^2 + 5*B*b^3)*sqrt(b*cos(d*x + c) + a)*sin(d*x + c))/(b*d)
\[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=- \int C a^{2} \sqrt {a + b \cos {\left (c + d x \right )}}\, dx - \int \left (- B a b \sqrt {a + b \cos {\left (c + d x \right )}}\right )\, dx - \int \left (- B b^{2} \sqrt {a + b \cos {\left (c + d x \right )}} \cos {\left (c + d x \right )}\right )\, dx - \int \left (- C b^{2} \sqrt {a + b \cos {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )}\right )\, dx \] Input:
integrate((a+b*cos(d*x+c))**(1/2)*(B*a*b-a**2*C+b**2*B*cos(d*x+c)+b**2*C*c os(d*x+c)**2),x)
Output:
-Integral(C*a**2*sqrt(a + b*cos(c + d*x)), x) - Integral(-B*a*b*sqrt(a + b *cos(c + d*x)), x) - Integral(-B*b**2*sqrt(a + b*cos(c + d*x))*cos(c + d*x ), x) - Integral(-C*b**2*sqrt(a + b*cos(c + d*x))*cos(c + d*x)**2, x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C b^{2} \cos \left (d x + c\right )^{2} + B b^{2} \cos \left (d x + c\right ) - C a^{2} + B a b\right )} \sqrt {b \cos \left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(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)*sqrt (b*cos(d*x + c) + a), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C b^{2} \cos \left (d x + c\right )^{2} + B b^{2} \cos \left (d x + c\right ) - C a^{2} + B a b\right )} \sqrt {b \cos \left (d x + c\right ) + a} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^(1/2)*(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)*sqrt (b*cos(d*x + c) + a), x)
Timed out. \[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (-C\,a^2+B\,a\,b+C\,b^2\,{\cos \left (c+d\,x\right )}^2+B\,b^2\,\cos \left (c+d\,x\right )\right ) \,d x \] Input:
int((a + b*cos(c + d*x))^(1/2)*(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))^(1/2)*(C*b^2*cos(c + d*x)^2 - C*a^2 + B*a*b + B*b ^2*cos(c + d*x)), x)
\[ \int \sqrt {a+b \cos (c+d x)} \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx=-\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a^{2} c +\left (\int \sqrt {\cos \left (d x +c \right ) b +a}d x \right ) a \,b^{2}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )d x \right ) b^{3}+\left (\int \sqrt {\cos \left (d x +c \right ) b +a}\, \cos \left (d x +c \right )^{2}d x \right ) b^{2} c \] Input:
int((a+b*cos(d*x+c))^(1/2)*(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**2*c + int(sqrt(cos(c + d*x)*b + a),x )*a*b**2 + int(sqrt(cos(c + d*x)*b + a)*cos(c + d*x),x)*b**3 + int(sqrt(co s(c + d*x)*b + a)*cos(c + d*x)**2,x)*b**2*c