Integrand size = 25, antiderivative size = 188 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\cos (e+f x) \sin (e+f x)}{a f \sqrt {a+b \sin ^2(e+f x)}}+\frac {\sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{a b f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{b f \sqrt {a+b \sin ^2(e+f x)}} \]
cos(f*x+e)*sin(f*x+e)/a/f/(a+b*sin(f*x+e)^2)^(1/2)+EllipticE(sin(f*x+e),(- b/a)^(1/2))*sec(f*x+e)*(cos(f*x+e)^2)^(1/2)*(a+b*sin(f*x+e)^2)^(1/2)/a/b/f /(1+b*sin(f*x+e)^2/a)^(1/2)-EllipticF(sin(f*x+e),(-b/a)^(1/2))*sec(f*x+e)* (cos(f*x+e)^2)^(1/2)*(1+b*sin(f*x+e)^2/a)^(1/2)/b/f/(a+b*sin(f*x+e)^2)^(1/ 2)
Time = 0.54 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.71 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\frac {\sqrt {2} a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )-\sqrt {2} a \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+b \sin (2 (e+f x))}{\sqrt {2} a b f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
(Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticE[e + f*x, -(b/a )] - Sqrt[2]*a*Sqrt[(2*a + b - b*Cos[2*(e + f*x)])/a]*EllipticF[e + f*x, - (b/a)] + b*Sin[2*(e + f*x)])/(Sqrt[2]*a*b*f*Sqrt[2*a + b - b*Cos[2*(e + f* x)]])
Time = 0.37 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.95, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3671, 314, 25, 389, 323, 321, 330, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (e+f x)^2}{\left (a+b \sin (e+f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3671 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\left (b \sin ^2(e+f x)+a\right )^{3/2}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\sin (e+f x) \sqrt {1-\sin ^2(e+f x)}}{a \sqrt {a+b \sin ^2(e+f x)}}-\frac {\int -\frac {\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}\right )}{f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\int \frac {\sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 389 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {b \sin ^2(e+f x)+a}}d\sin (e+f x)}{b}}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}d\sin (e+f x)}{b \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\int \frac {\sqrt {b \sin ^2(e+f x)+a}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\sqrt {a+b \sin ^2(e+f x)} \int \frac {\sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {\cos ^2(e+f x)} \sec (e+f x) \left (\frac {\frac {\sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{b \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {a \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{b \sqrt {a+b \sin ^2(e+f x)}}}{a}+\frac {\sqrt {1-\sin ^2(e+f x)} \sin (e+f x)}{a \sqrt {a+b \sin ^2(e+f x)}}\right )}{f}\) |
(Sqrt[Cos[e + f*x]^2]*Sec[e + f*x]*((Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2] )/(a*Sqrt[a + b*Sin[e + f*x]^2]) + ((EllipticE[ArcSin[Sin[e + f*x]], -(b/a )]*Sqrt[a + b*Sin[e + f*x]^2])/(b*Sqrt[1 + (b*Sin[e + f*x]^2)/a]) - (a*Ell ipticF[ArcSin[Sin[e + f*x]], -(b/a)]*Sqrt[1 + (b*Sin[e + f*x]^2)/a])/(b*Sq rt[a + b*Sin[e + f*x]^2]))/a))/f
3.4.60.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* (p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d *(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[1/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b Int [1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N eQ[b*c - a*d, 0] && !SimplerSqrtQ[-b/a, -d/c]
Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^( p_.), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff*(Sqrt[ Cos[e + f*x]^2]/(f*Cos[e + f*x])) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*ff^2*x^2)^p, x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && !IntegerQ[p]
Time = 2.07 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )-a \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {\frac {a +b \left (\sin ^{2}\left (f x +e \right )\right )}{a}}\, E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right )+\left (\sin ^{3}\left (f x +e \right )\right ) b -b \sin \left (f x +e \right )}{a b \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(145\) |
-(a*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*EllipticF(sin(f*x+e) ,(-1/a*b)^(1/2))-a*(cos(f*x+e)^2)^(1/2)*((a+b*sin(f*x+e)^2)/a)^(1/2)*Ellip ticE(sin(f*x+e),(-1/a*b)^(1/2))+sin(f*x+e)^3*b-b*sin(f*x+e))/a/b/cos(f*x+e )/(a+b*sin(f*x+e)^2)^(1/2)/f
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.13 \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=-\frac {2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b^{2} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 4 \, {\left (i \, b^{2} \cos \left (f x + e\right )^{2} - i \, a b - i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} \sqrt {\frac {a^{2} + a b}{b^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + 4 \, {\left (-i \, b^{2} \cos \left (f x + e\right )^{2} + i \, a b + i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} \sqrt {\frac {a^{2} + a b}{b^{2}}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) - {\left (2 \, {\left (i \, b^{2} \cos \left (f x + e\right )^{2} - i \, a b - i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (-2 i \, a b - i \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 2 i \, a^{2} + 3 i \, a b + i \, b^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) - {\left (2 \, {\left (-i \, b^{2} \cos \left (f x + e\right )^{2} + i \, a b + i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} - {\left ({\left (2 i \, a b + i \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 i \, a^{2} - 3 i \, a b - i \, b^{2}\right )} \sqrt {-b}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}})}{2 \, {\left (a b^{3} f \cos \left (f x + e\right )^{2} - {\left (a^{2} b^{2} + a b^{3}\right )} f\right )}} \]
-1/2*(2*sqrt(-b*cos(f*x + e)^2 + a + b)*b^2*cos(f*x + e)*sin(f*x + e) + 4* (I*b^2*cos(f*x + e)^2 - I*a*b - I*b^2)*sqrt(-b)*sqrt((2*b*sqrt((a^2 + a*b) /b^2) + 2*a + b)/b)*sqrt((a^2 + a*b)/b^2)*elliptic_f(arcsin(sqrt((2*b*sqrt ((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) + 4*(-I*b^2*cos (f*x + e)^2 + I*a*b + I*b^2)*sqrt(-b)*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2* a + b)/b)*sqrt((a^2 + a*b)/b^2)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 + a* b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b ^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - (2*(I*b^2*cos(f*x + e)^ 2 - I*a*b - I*b^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((-2*I*a*b - I*b^2)*co s(f*x + e)^2 + 2*I*a^2 + 3*I*a*b + I*b^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) + I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*( 2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2) - (2*(-I*b^2*cos(f*x + e)^2 + I*a *b + I*b^2)*sqrt(-b)*sqrt((a^2 + a*b)/b^2) - ((2*I*a*b + I*b^2)*cos(f*x + e)^2 - 2*I*a^2 - 3*I*a*b - I*b^2)*sqrt(-b))*sqrt((2*b*sqrt((a^2 + a*b)/b^2 ) + 2*a + b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 + a*b)/b^2) + 2*a + b)/b)*(cos(f*x + e) - I*sin(f*x + e))), (8*a^2 + 8*a*b + b^2 - 4*(2*a*b + b^2)*sqrt((a^2 + a*b)/b^2))/b^2))/(a*b^3*f*cos(f*x + e)^2 - (a^2*b^2 + a*b ^3)*f)
Timed out. \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int { \frac {\cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^2(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]