Integrand size = 25, antiderivative size = 246 \[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\cos ^2(e+f x) \sin (e+f x) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{3 f}+\frac {(2 a+b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}{3 a f \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right ) \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{3 a f \left (a+b-a \sin ^2(e+f x)\right )} \]
1/3*cos(f*x+e)^2*sin(f*x+e)*(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/f+1/ 3*(2*a+b)*EllipticE(sin(f*x+e),(a/(a+b))^(1/2))*(cos(f*x+e)^2)^(1/2)*(sec( f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)/a/f/(1-a*sin(f*x+e)^2/(a+b))^(1/2)-1/ 3*b*(a+b)*EllipticF(sin(f*x+e),(a/(a+b))^(1/2))*(cos(f*x+e)^2)^(1/2)*(sec( f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/a/f/(a +b-a*sin(f*x+e)^2)
Result contains complex when optimal does not.
Time = 13.40 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.63 \[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\frac {\cos (e+f x) \sqrt {a+b \sec ^2(e+f x)} \left (\frac {6 \sqrt {2} \sqrt {a+2 b+a \cos (2 (e+f x))} E\left (e+f x\left |\frac {a}{a+b}\right .\right )}{\sqrt {\frac {a+2 b+a \cos (2 (e+f x))}{a+b}}}+\frac {\sqrt {-\frac {1}{b}} b \csc ^2(2 (e+f x)) \sec (2 (e+f x)) \left (4 i \sqrt {2} \left (a^2+3 a b+2 b^2\right ) \sqrt {-\frac {a \cos ^2(e+f x)}{b}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))}}{\sqrt {2}}\right )|\frac {b}{a+b}\right ) \sqrt {\frac {a \sin ^2(e+f x)}{a+b}}+a \left (a \sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))} (-1+\cos (4 (e+f x)))-4 i \sqrt {2} (a+b) \sqrt {-\frac {a \cos ^2(e+f x)}{b}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {1}{b}} \sqrt {a+2 b+a \cos (2 (e+f x))}}{\sqrt {2}}\right ),\frac {b}{a+b}\right ) \sqrt {\frac {a \sin ^2(e+f x)}{a+b}}\right )\right ) \sin (4 (e+f x))}{2 a^2}\right )}{12 f \sqrt {a+2 b+a \cos (2 (e+f x))}} \]
(Cos[e + f*x]*Sqrt[a + b*Sec[e + f*x]^2]*((6*Sqrt[2]*Sqrt[a + 2*b + a*Cos[ 2*(e + f*x)]]*EllipticE[e + f*x, a/(a + b)])/Sqrt[(a + 2*b + a*Cos[2*(e + f*x)])/(a + b)] + (Sqrt[-b^(-1)]*b*Csc[2*(e + f*x)]^2*Sec[2*(e + f*x)]*((4 *I)*Sqrt[2]*(a^2 + 3*a*b + 2*b^2)*Sqrt[-((a*Cos[e + f*x]^2)/b)]*EllipticE[ I*ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], b/( a + b)]*Sqrt[(a*Sin[e + f*x]^2)/(a + b)] + a*(a*Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]]*(-1 + Cos[4*(e + f*x)]) - (4*I)*Sqrt[2]*(a + b)*Sqr t[-((a*Cos[e + f*x]^2)/b)]*EllipticF[I*ArcSinh[(Sqrt[-b^(-1)]*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]])/Sqrt[2]], b/(a + b)]*Sqrt[(a*Sin[e + f*x]^2)/(a + b)]))*Sin[4*(e + f*x)])/(2*a^2)))/(12*f*Sqrt[a + 2*b + a*Cos[2*(e + f*x)]] )
Time = 0.49 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 4636, 2057, 2058, 319, 27, 399, 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.
| \(\displaystyle \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {a+b \sec (e+f x)^2}}{\sec (e+f x)^3}dx\) |
| \(\Big \downarrow \) 4636 |
| \(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right ) \sqrt {a+\frac {b}{1-\sin ^2(e+f x)}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \frac {\int \left (1-\sin ^2(e+f x)\right ) \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}d\sin (e+f x)}{f}\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \int \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}d\sin (e+f x)}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 319 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {2}{3} \int \frac {-\left ((2 a+b) \sin ^2(e+f x)\right )+2 a+2 b}{2 \sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \int \frac {2 (a+b)-(2 a+b) \sin ^2(e+f x)}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \left (\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{a}\right )+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \left (\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {1}{\sqrt {1-\sin ^2(e+f x)} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \left (\frac {(2 a+b) \int \frac {\sqrt {-a \sin ^2(e+f x)+a+b}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \left (\frac {(2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{\sqrt {1-\sin ^2(e+f x)}}d\sin (e+f x)}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}} \left (\frac {1}{3} \left (\frac {(2 a+b) \sqrt {-a \sin ^2(e+f x)+a+b} E\left (\arcsin (\sin (e+f x))\left |\frac {a}{a+b}\right .\right )}{a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}-\frac {b (a+b) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),\frac {a}{a+b}\right )}{a \sqrt {-a \sin ^2(e+f x)+a+b}}\right )+\frac {1}{3} \sqrt {1-\sin ^2(e+f x)} \sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}\right )}{f \sqrt {-a \sin ^2(e+f x)+a+b}}\) |
(Sqrt[1 - Sin[e + f*x]^2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x ]^2)]*((Sin[e + f*x]*Sqrt[1 - Sin[e + f*x]^2]*Sqrt[a + b - a*Sin[e + f*x]^ 2])/3 + (((2*a + b)*EllipticE[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[a + b - a*Sin[e + f*x]^2])/(a*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)]) - (b*(a + b) *EllipticF[ArcSin[Sin[e + f*x]], a/(a + b)]*Sqrt[1 - (a*Sin[e + f*x]^2)/(a + b)])/(a*Sqrt[a + b - a*Sin[e + f*x]^2]))/3))/(f*Sqrt[a + b - a*Sin[e + f*x]^2])
3.3.32.3.1 Defintions of rubi rules used
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)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[x*(a + b*x^2)^p*((c + d*x^2)^q/(2*(p + q) + 1)), x] + Simp[2/(2*(p + q) + 1) Int[(a + b*x^2)^(p - 1)*(c + d*x^2)^(q - 1)*Simp[a*c*(p + q) + (q*(b* c - a*d) + a*d*(p + q))*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b* c - a*d, 0] && GtQ[q, 0] && GtQ[p, 0] && 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[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b/(1 - ff^2*x^2)^(n/2))^p/(1 - ff^2*x^2)^((m + 1)/2), x], x , Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 7.23 (sec) , antiderivative size = 6300, normalized size of antiderivative = 25.61
\[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\text {Timed out} \]
\[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{3} \,d x } \]
\[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int { \sqrt {b \sec \left (f x + e\right )^{2} + a} \cos \left (f x + e\right )^{3} \,d x } \]
Timed out. \[ \int \cos ^3(e+f x) \sqrt {a+b \sec ^2(e+f x)} \, dx=\int {\cos \left (e+f\,x\right )}^3\,\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}} \,d x \]