Integrand size = 25, antiderivative size = 170 \[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=-\frac {\sqrt {a} \sqrt {a+b} E\left (\arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )|\frac {a+b}{a}\right ) \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}{b f \sqrt {\cos ^2(e+f x)} \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}}+\frac {\sec (e+f x) \left (a+b-a \sin ^2(e+f x)\right ) \tan (e+f x)}{b f \sqrt {\sec ^2(e+f x) \left (a+b-a \sin ^2(e+f x)\right )}} \]
-EllipticE(sin(f*x+e)*a^(1/2)/(a+b)^(1/2),((a+b)/a)^(1/2))*a^(1/2)*(a+b)^( 1/2)*(1-a*sin(f*x+e)^2/(a+b))^(1/2)/b/f/(cos(f*x+e)^2)^(1/2)/(sec(f*x+e)^2 *(a+b-a*sin(f*x+e)^2))^(1/2)+sec(f*x+e)*(a+b-a*sin(f*x+e)^2)*tan(f*x+e)/b/ f/(sec(f*x+e)^2*(a+b-a*sin(f*x+e)^2))^(1/2)
\[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx \]
Time = 0.43 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4636, 2057, 2058, 316, 25, 27, 331, 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.
\(\displaystyle \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (e+f x)^3}{\sqrt {a+b \sec (e+f x)^2}}dx\) |
\(\Big \downarrow \) 4636 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^2 \sqrt {a+\frac {b}{1-\sin ^2(e+f x)}}}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \frac {\int \frac {1}{\left (1-\sin ^2(e+f x)\right )^2 \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}d\sin (e+f x)}{f}\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \int \frac {1}{\left (1-\sin ^2(e+f x)\right )^{3/2} \sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 316 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\int -\frac {a \sqrt {1-\sin ^2(e+f x)}}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}+\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\int \frac {a \sqrt {1-\sin ^2(e+f x)}}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\sqrt {-a \sin ^2(e+f x)+a+b}}d\sin (e+f x)}{b}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 331 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {a \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} \int \frac {\sqrt {1-\sin ^2(e+f x)}}{\sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}}}d\sin (e+f x)}{b \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {-a \sin ^2(e+f x)+a+b} \left (\frac {\sin (e+f x) \sqrt {-a \sin ^2(e+f x)+a+b}}{b \sqrt {1-\sin ^2(e+f x)}}-\frac {\sqrt {a} \sqrt {a+b} \sqrt {1-\frac {a \sin ^2(e+f x)}{a+b}} E\left (\arcsin \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b}}\right )|\frac {a+b}{a}\right )}{b \sqrt {-a \sin ^2(e+f x)+a+b}}\right )}{f \sqrt {1-\sin ^2(e+f x)} \sqrt {\frac {-a \sin ^2(e+f x)+a+b}{1-\sin ^2(e+f x)}}}\) |
(Sqrt[a + b - a*Sin[e + f*x]^2]*((Sin[e + f*x]*Sqrt[a + b - a*Sin[e + f*x] ^2])/(b*Sqrt[1 - Sin[e + f*x]^2]) - (Sqrt[a]*Sqrt[a + b]*EllipticE[ArcSin[ (Sqrt[a]*Sin[e + f*x])/Sqrt[a + b]], (a + b)/a]*Sqrt[1 - (a*Sin[e + f*x]^2 )/(a + b)])/(b*Sqrt[a + b - a*Sin[e + f*x]^2])))/(f*Sqrt[1 - Sin[e + f*x]^ 2]*Sqrt[(a + b - a*Sin[e + f*x]^2)/(1 - Sin[e + f*x]^2)])
3.3.58.3.1 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[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
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[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
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 = 7.89 (sec) , antiderivative size = 4103, normalized size of antiderivative = 24.14
1/f/((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)/b/(2*I*a^(1/2)*b^(1/2)-a+b)*(2 *I*a^(3/2)*b^(1/2)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot (f*x+e))+2*I*a^(1/2)*b^(3/2)*((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc( f*x+e)-cot(f*x+e))-4*(-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+ a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b) )^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e ))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*Ellipt icF(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I *a^(3/2)*b^(1/2)-4*I*a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*a*b+2* (-(2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*cs c(f*x+e)^2-b*(1-cos(f*x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2) *b^(1/2)*(1-cos(f*x+e))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*( 1-cos(f*x+e))^2*csc(f*x+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^( 1/2)+a-b)/(a+b))^(1/2)*(csc(f*x+e)-cot(f*x+e)),(-(4*I*a^(3/2)*b^(1/2)-4*I* a^(1/2)*b^(3/2)-a^2+6*a*b-b^2)/(a+b)^2)^(1/2))*a*b-4*((2*I*a^(1/2)*b^(1/2) +a-b)/(a+b))^(1/2)*a*b*(1-cos(f*x+e))^3*csc(f*x+e)^3-(-(2*I*a^(1/2)*b^(1/2 )*(1-cos(f*x+e))^2*csc(f*x+e)^2+a*(1-cos(f*x+e))^2*csc(f*x+e)^2-b*(1-cos(f *x+e))^2*csc(f*x+e)^2-a-b)/(a+b))^(1/2)*((2*I*a^(1/2)*b^(1/2)*(1-cos(f*x+e ))^2*csc(f*x+e)^2-a*(1-cos(f*x+e))^2*csc(f*x+e)^2+b*(1-cos(f*x+e))^2*csc(f *x+e)^2+a+b)/(a+b))^(1/2)*EllipticE(((2*I*a^(1/2)*b^(1/2)+a-b)/(a+b))^(...
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 594, normalized size of antiderivative = 3.49 \[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=-\frac {2 \, \sqrt {a} {\left (-i \, a - 2 i \, b\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + 2 \, \sqrt {a} {\left (i \, a + 2 i \, b\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} F(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (-2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (i \, a + 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) + {\left (2 i \, a^{\frac {3}{2}} \sqrt {\frac {a b + b^{2}}{a^{2}}} + \sqrt {a} {\left (-i \, a - 2 i \, b\right )}\right )} \sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} E(\arcsin \left (\sqrt {\frac {2 \, a \sqrt {\frac {a b + b^{2}}{a^{2}}} - a - 2 \, b}{a}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} + 2 \, a b\right )} \sqrt {\frac {a b + b^{2}}{a^{2}}}}{a^{2}}) - 2 \, a \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right )}{2 \, a b f} \]
-1/2*(2*sqrt(a)*(-I*a - 2*I*b)*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/ a)*elliptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f* x + e) + I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a* b + b^2)/a^2))/a^2) + 2*sqrt(a)*(I*a + 2*I*b)*sqrt((2*a*sqrt((a*b + b^2)/a ^2) - a - 2*b)/a)*elliptic_f(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) + (-2*I*a^(3/2)*sqrt((a*b + b^2)/a^2) + sqrt(a)*(I*a + 2*I*b))*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elli ptic_e(arcsin(sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) + I*sin(f*x + e))), (a^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2 )/a^2))/a^2) + (2*I*a^(3/2)*sqrt((a*b + b^2)/a^2) + sqrt(a)*(-I*a - 2*I*b) )*sqrt((2*a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*elliptic_e(arcsin(sqrt((2* a*sqrt((a*b + b^2)/a^2) - a - 2*b)/a)*(cos(f*x + e) - I*sin(f*x + e))), (a ^2 + 8*a*b + 8*b^2 + 4*(a^2 + 2*a*b)*sqrt((a*b + b^2)/a^2))/a^2) - 2*a*sqr t((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(a*b*f)
\[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {\sec ^{3}{\left (e + f x \right )}}{\sqrt {a + b \sec ^{2}{\left (e + f x \right )}}}\, dx \]
\[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )^{3}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
\[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int { \frac {\sec \left (f x + e\right )^{3}}{\sqrt {b \sec \left (f x + e\right )^{2} + a}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(e+f x)}{\sqrt {a+b \sec ^2(e+f x)}} \, dx=\int \frac {1}{{\cos \left (e+f\,x\right )}^3\,\sqrt {a+\frac {b}{{\cos \left (e+f\,x\right )}^2}}} \,d x \]