Integrand size = 54, antiderivative size = 213 \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {(8 a A-4 A b-4 a B+7 b B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 \sqrt {a} d}-\frac {\sqrt {2} (a-b) (A-B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}+\frac {(4 A b+4 a B-b B) \sqrt {\cos (c+d x)} \sin (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {b B \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}} \] Output:
1/4*(8*A*a-4*A*b-4*B*a+7*B*b)*arcsin(a^(1/2)*sin(d*x+c)/(a+a*cos(d*x+c))^( 1/2))/a^(1/2)/d-2^(1/2)*(a-b)*(A-B)*arctan(1/2*a^(1/2)*sin(d*x+c)*2^(1/2)/ cos(d*x+c)^(1/2)/(a+a*cos(d*x+c))^(1/2))/a^(1/2)/d+1/4*(4*A*b+4*B*a-B*b)*c os(d*x+c)^(1/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+1/2*b*B*cos(d*x+c)^(3/ 2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)
Time = 0.89 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.61 \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\left ((4 A b+4 a B-b B) \arcsin \left (\sqrt {1-\cos (c+d x)}\right )-8 (a-b) (A-B) \arcsin \left (\sqrt {\cos (c+d x)}\right )+4 \sqrt {2} a A \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )-4 \sqrt {2} A b \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )-4 \sqrt {2} a B \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )+4 \sqrt {2} b B \arctan \left (\frac {\sqrt {\cos (c+d x)}}{\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}}\right )+2 b B \sqrt {1-\cos (c+d x)} \cos ^{\frac {3}{2}}(c+d x)+4 A b \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}+4 a B \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}-b B \sqrt {-((-1+\cos (c+d x)) \cos (c+d x))}\right ) \sin (c+d x)}{4 d \sqrt {1-\cos (c+d x)} \sqrt {a (1+\cos (c+d x))}} \] Input:
Integrate[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x]^2))/Sqrt[a + a*Cos[c + d*x]],x]
Output:
(((4*A*b + 4*a*B - b*B)*ArcSin[Sqrt[1 - Cos[c + d*x]]] - 8*(a - b)*(A - B) *ArcSin[Sqrt[Cos[c + d*x]]] + 4*Sqrt[2]*a*A*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt [Sin[(c + d*x)/2]^2]] - 4*Sqrt[2]*A*b*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[( c + d*x)/2]^2]] - 4*Sqrt[2]*a*B*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[(c + d* x)/2]^2]] + 4*Sqrt[2]*b*B*ArcTan[Sqrt[Cos[c + d*x]]/Sqrt[Sin[(c + d*x)/2]^ 2]] + 2*b*B*Sqrt[1 - Cos[c + d*x]]*Cos[c + d*x]^(3/2) + 4*A*b*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])] + 4*a*B*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d *x])] - b*B*Sqrt[-((-1 + Cos[c + d*x])*Cos[c + d*x])])*Sin[c + d*x])/(4*d* Sqrt[1 - Cos[c + d*x]]*Sqrt[a*(1 + Cos[c + d*x])])
Time = 1.29 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {3042, 3524, 27, 3042, 3462, 27, 3042, 3461, 3042, 3253, 223, 3261, 218}
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.
| \(\displaystyle \int \frac {\sqrt {\cos (c+d x)} \left ((a B+A b) \cos (c+d x)+a A+b B \cos ^2(c+d x)\right )}{\sqrt {a \cos (c+d x)+a}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left ((a B+A b) \sin \left (c+d x+\frac {\pi }{2}\right )+a A+b B \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sqrt {a \sin \left (c+d x+\frac {\pi }{2}\right )+a}}dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int \frac {\sqrt {\cos (c+d x)} (a (4 a A+3 b B)+a (4 A b-B b+4 a B) \cos (c+d x))}{2 \sqrt {\cos (c+d x) a+a}}dx}{2 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {\cos (c+d x)} (a (4 a A+3 b B)+a (4 A b-B b+4 a B) \cos (c+d x))}{\sqrt {\cos (c+d x) a+a}}dx}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a (4 a A+3 b B)+a (4 A b-B b+4 a B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3462 |
| \(\displaystyle \frac {\frac {\int \frac {(4 A b-B b+4 a B) a^2+(8 a A-4 b A-4 a B+7 b B) \cos (c+d x) a^2}{2 \sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {(4 A b-B b+4 a B) a^2+(8 a A-4 b A-4 a B+7 b B) \cos (c+d x) a^2}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {(4 A b-B b+4 a B) a^2+(8 a A-4 b A-4 a B+7 b B) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3461 |
| \(\displaystyle \frac {\frac {a (8 a A-4 a B-4 A b+7 b B) \int \frac {\sqrt {\cos (c+d x) a+a}}{\sqrt {\cos (c+d x)}}dx-8 a^2 (a-b) (A-B) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}dx}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a (8 a A-4 a B-4 A b+7 b B) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx-8 a^2 (a-b) (A-B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3253 |
| \(\displaystyle \frac {\frac {-8 a^2 (a-b) (A-B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx-\frac {2 a (8 a A-4 a B-4 A b+7 b B) \int \frac {1}{\sqrt {1-\frac {a \sin ^2(c+d x)}{\cos (c+d x) a+a}}}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x) a+a}}\right )}{d}}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 223 |
| \(\displaystyle \frac {\frac {\frac {2 a^{3/2} (8 a A-4 a B-4 A b+7 b B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}-8 a^2 (a-b) (A-B) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}dx}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 3261 |
| \(\displaystyle \frac {\frac {\frac {16 a^3 (a-b) (A-B) \int \frac {1}{\frac {\sin (c+d x) \tan (c+d x) a^3}{\cos (c+d x) a+a}+2 a^2}d\left (-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {\cos (c+d x) a+a}}\right )}{d}+\frac {2 a^{3/2} (8 a A-4 a B-4 A b+7 b B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\frac {2 a^{3/2} (8 a A-4 a B-4 A b+7 b B) \arcsin \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{d}-\frac {8 \sqrt {2} a^{3/2} (a-b) (A-B) \arctan \left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{d}}{2 a}+\frac {a (4 a B+4 A b-b B) \sin (c+d x) \sqrt {\cos (c+d x)}}{d \sqrt {a \cos (c+d x)+a}}}{4 a}+\frac {b B \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{2 d \sqrt {a \cos (c+d x)+a}}\) |
Input:
Int[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Cos[c + d*x] + b*B*Cos[c + d*x] ^2))/Sqrt[a + a*Cos[c + d*x]],x]
Output:
(b*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Cos[c + d*x]]) + ((( 2*a^(3/2)*(8*a*A - 4*A*b - 4*a*B + 7*b*B)*ArcSin[(Sqrt[a]*Sin[c + d*x])/Sq rt[a + a*Cos[c + d*x]]])/d - (8*Sqrt[2]*a^(3/2)*(a - b)*(A - B)*ArcTan[(Sq rt[a]*Sin[c + d*x])/(Sqrt[2]*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])] )/d)/(2*a) + (a*(4*A*b + 4*a*B - b*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d* Sqrt[a + a*Cos[c + d*x]]))/(4*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(d_.)*sin[(e_.) + (f_.) *(x_)]], x_Symbol] :> Simp[-2/f Subst[Int[1/Sqrt[1 - x^2/a], x], x, b*(Co s[e + f*x]/Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, d, e, f}, x] && E qQ[a^2 - b^2, 0] && EqQ[d, a/b]
Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e _.) + (f_.)*(x_)]]), x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*S in[e + f*x]]))], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Sim p[(A*b - a*B)/b Int[1/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]) , x], x] + Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]]/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] && EqQ[ a^2 - b^2, 0] && NeQ[c^2 - d^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] :> Sim p[(-B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/(f*(m + n + 1))), x] + Simp[1/(b*(m + n + 1)) Int[(a + b*Sin[e + f*x])^m*(c + d*S in[e + f*x])^(n - 1)*Simp[A*b*c*(m + n + 1) + B*(a*c*m + b*d*n) + (A*b*d*(m + n + 1) + B*(a*d*m + b*c*n))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 0] && (IntegerQ[n] || EqQ[m + 1/2, 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/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(382\) vs. \(2(182)=364\).
Time = 0.98 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.80
| method | result | size |
| default | \(\frac {\left (8 A \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \sqrt {2}\, a -4 A \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \sqrt {2}\, b -4 B \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \sqrt {2}\, a +7 B \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right ) \sqrt {2}\, b +4 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, b \sin \left (d x +c \right )+4 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}\, a \sin \left (d x +c \right )+\sin \left (d x +c \right ) \left (-1+2 \cos \left (d x +c \right )\right ) \sqrt {2}\, B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, b +8 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) a -8 A \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) b -8 B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) a +8 B \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ) b \right ) \sqrt {2}\, \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {\cos \left (d x +c \right )}}{8 d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, a}\) | \(383\) |
| parts | \(\frac {\left (A b +B a \right ) \left (\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}-\sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )-2 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{2 d a \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {A \sqrt {\cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+\arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right )}{d \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}+\frac {B b \left (7 \sqrt {2}\, \arctan \left (\sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \tan \left (d x +c \right )\right )+\sin \left (d x +c \right ) \left (-1+2 \cos \left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+8 \arcsin \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )\right ) \sqrt {\cos \left (d x +c \right )}\, \sqrt {2}\, \sqrt {a \left (1+\cos \left (d x +c \right )\right )}}{8 d a \left (1+\cos \left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\) | \(406\) |
Input:
int(cos(d*x+c)^(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)^2)/(a+a*cos( d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/8/d*(8*A*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*a- 4*A*arctan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*b-4*B*arc tan((cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*a+7*B*arctan((co s(d*x+c)/(1+cos(d*x+c)))^(1/2)*tan(d*x+c))*2^(1/2)*b+4*A*(cos(d*x+c)/(1+co s(d*x+c)))^(1/2)*2^(1/2)*b*sin(d*x+c)+4*B*(cos(d*x+c)/(1+cos(d*x+c)))^(1/2 )*2^(1/2)*a*sin(d*x+c)+sin(d*x+c)*(-1+2*cos(d*x+c))*2^(1/2)*B*(cos(d*x+c)/ (1+cos(d*x+c)))^(1/2)*b+8*A*arcsin(cot(d*x+c)-csc(d*x+c))*a-8*A*arcsin(cot (d*x+c)-csc(d*x+c))*b-8*B*arcsin(cot(d*x+c)-csc(d*x+c))*a+8*B*arcsin(cot(d *x+c)-csc(d*x+c))*b)*2^(1/2)*(a*(1+cos(d*x+c)))^(1/2)*cos(d*x+c)^(1/2)/(1+ cos(d*x+c))/(cos(d*x+c)/(1+cos(d*x+c)))^(1/2)/a
Time = 12.85 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {{\left (2 \, B b \cos \left (d x + c\right ) + 4 \, B a + {\left (4 \, A - B\right )} b\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + {\left (4 \, {\left (2 \, A - B\right )} a - {\left (4 \, A - 7 \, B\right )} b + {\left (4 \, {\left (2 \, A - B\right )} a - {\left (4 \, A - 7 \, B\right )} b\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )}\right ) - \frac {4 \, \sqrt {2} {\left ({\left (A - B\right )} a^{2} - {\left (A - B\right )} a b + {\left ({\left (A - B\right )} a^{2} - {\left (A - B\right )} a b\right )} \cos \left (d x + c\right )\right )} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a}}\right )}{\sqrt {a}}}{4 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \] Input:
integrate(cos(d*x+c)^(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)^2)/(a+ a*cos(d*x+c))^(1/2),x, algorithm="fricas")
Output:
1/4*((2*B*b*cos(d*x + c) + 4*B*a + (4*A - B)*b)*sqrt(a*cos(d*x + c) + a)*s qrt(cos(d*x + c))*sin(d*x + c) + (4*(2*A - B)*a - (4*A - 7*B)*b + (4*(2*A - B)*a - (4*A - 7*B)*b)*cos(d*x + c))*sqrt(a)*arctan(sqrt(a*cos(d*x + c) + a)*sqrt(a)*sqrt(cos(d*x + c))*sin(d*x + c)/(a*cos(d*x + c)^2 + a*cos(d*x + c))) - 4*sqrt(2)*((A - B)*a^2 - (A - B)*a*b + ((A - B)*a^2 - (A - B)*a*b )*cos(d*x + c))*arctan(1/2*sqrt(2)*sqrt(a*cos(d*x + c) + a)*sqrt(cos(d*x + c))*sin(d*x + c)/((cos(d*x + c)^2 + cos(d*x + c))*sqrt(a)))/sqrt(a))/(a*d *cos(d*x + c) + a*d)
Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)**2)/( a+a*cos(d*x+c))**(1/2),x)
Output:
Timed out
\[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\int { \frac {{\left (B b \cos \left (d x + c\right )^{2} + A a + {\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a \cos \left (d x + c\right ) + a}} \,d x } \] Input:
integrate(cos(d*x+c)^(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)^2)/(a+ a*cos(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate((B*b*cos(d*x + c)^2 + A*a + (B*a + A*b)*cos(d*x + c))*sqrt(cos(d *x + c))/sqrt(a*cos(d*x + c) + a), x)
Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)^(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)^2)/(a+ a*cos(d*x+c))^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\int \frac {\sqrt {\cos \left (c+d\,x\right )}\,\left (B\,b\,{\cos \left (c+d\,x\right )}^2+\left (A\,b+B\,a\right )\,\cos \left (c+d\,x\right )+A\,a\right )}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \] Input:
int((cos(c + d*x)^(1/2)*(A*a + cos(c + d*x)*(A*b + B*a) + B*b*cos(c + d*x) ^2))/(a + a*cos(c + d*x))^(1/2),x)
Output:
int((cos(c + d*x)^(1/2)*(A*a + cos(c + d*x)*(A*b + B*a) + B*b*cos(c + d*x) ^2))/(a + a*cos(c + d*x))^(1/2), x)
\[ \int \frac {\sqrt {\cos (c+d x)} \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx=\frac {\sqrt {a}\, \left (2 \left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}d x \right ) a b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\cos \left (d x +c \right )+1}d x \right ) b^{2}+\left (\int \frac {\sqrt {\cos \left (d x +c \right )+1}\, \sqrt {\cos \left (d x +c \right )}}{\cos \left (d x +c \right )+1}d x \right ) a^{2}\right )}{a} \] Input:
int(cos(d*x+c)^(1/2)*(a*A+(A*b+B*a)*cos(d*x+c)+b*B*cos(d*x+c)^2)/(a+a*cos( d*x+c))^(1/2),x)
Output:
(sqrt(a)*(2*int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))*cos(c + d*x))/( cos(c + d*x) + 1),x)*a*b + int((sqrt(cos(c + d*x) + 1)*sqrt(cos(c + d*x))* cos(c + d*x)**2)/(cos(c + d*x) + 1),x)*b**2 + int((sqrt(cos(c + d*x) + 1)* sqrt(cos(c + d*x)))/(cos(c + d*x) + 1),x)*a**2))/a