Integrand size = 29, antiderivative size = 123 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)}}\right )}{f} \]
2*arctan(a^(1/2)*c^(1/2)*tan(f*x+e)/(a+a*sec(f*x+e))^(1/2)/(c+d*sec(f*x+e) )^(1/2))*a^(1/2)*c^(1/2)/f+2*arctanh(a^(1/2)*d^(1/2)*tan(f*x+e)/(a+a*sec(f *x+e))^(1/2)/(c+d*sec(f*x+e))^(1/2))*a^(1/2)*d^(1/2)/f
Time = 14.89 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.95 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=-\frac {2 \cot (e+f x) \sqrt {a (1+\sec (e+f x))} \sqrt {c+d \sec (e+f x)} \left (-2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+c \cos (e+f x)}}{\sqrt {d} \sqrt {c-c \cos (e+f x)}}\right ) \sqrt {c (1+\cos (e+f x))} \sin ^2\left (\frac {1}{2} (e+f x)\right )+\arctan \left (\frac {\sqrt {c (1+\cos (e+f x))} \sqrt {d+c \cos (e+f x)}}{\sqrt {c^2 \sin ^2(e+f x)}}\right ) \sqrt {c-c \cos (e+f x)} \sqrt {c^2 \sin ^2(e+f x)}\right )}{f \sqrt {c (1+\cos (e+f x))} \sqrt {c-c \cos (e+f x)} \sqrt {d+c \cos (e+f x)}} \]
(-2*Cot[e + f*x]*Sqrt[a*(1 + Sec[e + f*x])]*Sqrt[c + d*Sec[e + f*x]]*(-2*S qrt[c]*Sqrt[d]*ArcTanh[(Sqrt[c]*Sqrt[d + c*Cos[e + f*x]])/(Sqrt[d]*Sqrt[c - c*Cos[e + f*x]])]*Sqrt[c*(1 + Cos[e + f*x])]*Sin[(e + f*x)/2]^2 + ArcTan [(Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[d + c*Cos[e + f*x]])/Sqrt[c^2*Sin[e + f* x]^2]]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[c^2*Sin[e + f*x]^2]))/(f*Sqrt[c*(1 + Cos[e + f*x])]*Sqrt[c - c*Cos[e + f*x]]*Sqrt[d + c*Cos[e + f*x]])
Time = 0.71 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {3042, 4420, 3042, 4422, 216, 4468, 219}
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.
| \(\displaystyle \int \sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \csc \left (e+f x+\frac {\pi }{2}\right )+a} \sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 4420 |
| \(\displaystyle c \int \frac {\sqrt {\sec (e+f x) a+a}}{\sqrt {c+d \sec (e+f x)}}dx+d \int \frac {\sec (e+f x) \sqrt {\sec (e+f x) a+a}}{\sqrt {c+d \sec (e+f x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \int \frac {\sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4422 |
| \(\displaystyle d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx-\frac {2 a c \int \frac {1}{\frac {a c \tan ^2(e+f x)}{(\sec (e+f x) a+a) (c+d \sec (e+f x))}+1}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle d \int \frac {\csc \left (e+f x+\frac {\pi }{2}\right ) \sqrt {\csc \left (e+f x+\frac {\pi }{2}\right ) a+a}}{\sqrt {c+d \csc \left (e+f x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 4468 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}-\frac {2 a d \int \frac {1}{1-\frac {a d \tan ^2(e+f x)}{(\sec (e+f x) a+a) (c+d \sec (e+f x))}}d\left (-\frac {\tan (e+f x)}{\sqrt {\sec (e+f x) a+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {a} \sqrt {c} \arctan \left (\frac {\sqrt {a} \sqrt {c} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}+\frac {2 \sqrt {a} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {c+d \sec (e+f x)}}\right )}{f}\) |
(2*Sqrt[a]*Sqrt[c]*ArcTan[(Sqrt[a]*Sqrt[c]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x]])])/f + (2*Sqrt[a]*Sqrt[d]*ArcTanh[(Sqrt[ a]*Sqrt[d]*Tan[e + f*x])/(Sqrt[a + a*Sec[e + f*x]]*Sqrt[c + d*Sec[e + f*x] ])])/f
3.2.84.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.) + (c_)], x_Symbol] :> Simp[c Int[Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]], x], x] + Simp[d Int[Csc[e + f*x]*(Sqrt[a + b*Csc[e + f* x]]/Sqrt[c + d*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && Ne Q[b*c - a*d, 0]
Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)] *(d_.) + (c_)], x_Symbol] :> Simp[-2*(a/f) Subst[Int[1/(1 + a*c*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[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[(csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)])/Sq rt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[-2*(b/f) Subs t[Int[1/(1 - b*d*x^2), x], x, Cot[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*Sqrt[c + d*Csc[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1495\) vs. \(2(99)=198\).
Time = 5.00 (sec) , antiderivative size = 1496, normalized size of antiderivative = 12.16
-1/f/(c-d)^(1/2)*2^(1/2)/(-d)^(1/2)/(c^2-2*c*d+d^2)*(a*(sec(f*x+e)+1))^(1/ 2)*(c+d*sec(f*x+e))^(1/2)*(2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e)-d*cot(f*x+ e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*(c -d)^(1/2))/(c-d)^(1/2))*c^3-3*2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e)-d*cot(f *x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2) *(c-d)^(1/2))/(c-d)^(1/2))*c^2*d+3*2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e)-d* cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^ (1/2)*(c-d)^(1/2))/(c-d)^(1/2))*c*d^2-2^(1/2)*(-d)^(1/2)*ln(-(c*cot(f*x+e) -d*cot(f*x+e)-c*csc(f*x+e)+d*csc(f*x+e)-(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1 ))^(1/2)*(c-d)^(1/2))/(c-d)^(1/2))*d^3-2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos( f*x+e))/(cos(f*x+e)+1))^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e ))*c^3+3*2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)- (c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*c^2*d-3*2^(1/2)*(-d)^(1/2)* ln((-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d) ^(1/2)*csc(f*x+e))*c*d^2+2^(1/2)*(-d)^(1/2)*ln((-2*(d+c*cos(f*x+e))/(cos(f *x+e)+1))^(1/2)-(c-d)^(1/2)*cot(f*x+e)+(c-d)^(1/2)*csc(f*x+e))*d^3+(c-d)^( 1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)* sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*d-c*cos(f*x+e)+d*cos(f*x+e)+c-d)/(cos(f *x+e)-1+sin(f*x+e)))*c^2*d-2*(c-d)^(1/2)*ln(-2*(2^(1/2)*(-d)^(1/2)*(-2*(d+ c*cos(f*x+e))/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)-sin(f*x+e)*c-sin(f*x+e)*...
Time = 0.50 (sec) , antiderivative size = 806, normalized size of antiderivative = 6.55 \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\left [\frac {\sqrt {a d} \log \left (\frac {2 \, \sqrt {a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c - a d\right )} \cos \left (f x + e\right )^{2} + 2 \, a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )}\right ) + \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right ) - \sqrt {a d} \log \left (\frac {2 \, \sqrt {a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + {\left (a c - a d\right )} \cos \left (f x + e\right )^{2} + 2 \, a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )}\right )}{f}, -\frac {2 \, \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a d \sin \left (f x + e\right )}\right ) - \sqrt {-a c} \log \left (\frac {2 \, a c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c + a d + {\left (a c + a d\right )} \cos \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, {\left (\sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a c \sin \left (f x + e\right )}\right ) + \sqrt {-a d} \arctan \left (\frac {\sqrt {-a d} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) + d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a d \sin \left (f x + e\right )}\right )\right )}}{f}\right ] \]
[(sqrt(a*d)*log((2*sqrt(a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt( (c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + (a*c - a*d) *cos(f*x + e)^2 + 2*a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e)^2 + cos( f*x + e))) + sqrt(-a*c)*log((2*a*c*cos(f*x + e)^2 - 2*sqrt(-a*c)*sqrt((a*c os(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos (f*x + e)*sin(f*x + e) - a*c + a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e) + 1)))/f, -(2*sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e) + a)/cos( f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(a*c*sin(f* x + e))) - sqrt(a*d)*log((2*sqrt(a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)*sin(f*x + e) + (a *c - a*d)*cos(f*x + e)^2 + 2*a*d + (a*c + a*d)*cos(f*x + e))/(cos(f*x + e) ^2 + cos(f*x + e))))/f, -(2*sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e )/(a*d*sin(f*x + e))) - sqrt(-a*c)*log((2*a*c*cos(f*x + e)^2 - 2*sqrt(-a*c )*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f* x + e))*cos(f*x + e)*sin(f*x + e) - a*c + a*d + (a*c + a*d)*cos(f*x + e))/ (cos(f*x + e) + 1)))/f, -2*(sqrt(a*c)*arctan(sqrt(a*c)*sqrt((a*cos(f*x + e ) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/ (a*c*sin(f*x + e))) + sqrt(-a*d)*arctan(sqrt(-a*d)*sqrt((a*cos(f*x + e) + a)/cos(f*x + e))*sqrt((c*cos(f*x + e) + d)/cos(f*x + e))*cos(f*x + e)/(...
\[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \sqrt {c + d \sec {\left (e + f x \right )}}\, dx \]
\[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c} \,d x } \]
\[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} \sqrt {d \sec \left (f x + e\right ) + c} \,d x } \]
Timed out. \[ \int \sqrt {a+a \sec (e+f x)} \sqrt {c+d \sec (e+f x)} \, dx=\int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,\sqrt {c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]