Integrand size = 10, antiderivative size = 24 \[ \int \sqrt {1+\sec ^2(x)} \, dx=\text {arcsinh}\left (\frac {\tan (x)}{\sqrt {2}}\right )+\arctan \left (\frac {\tan (x)}{\sqrt {2+\tan ^2(x)}}\right ) \] Output:
arcsinh(1/2*tan(x)*2^(1/2))+arctan(tan(x)/(2+tan(x)^2)^(1/2))
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(24)=48\).
Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.38 \[ \int \sqrt {1+\sec ^2(x)} \, dx=\frac {\sqrt {2} \left (\arcsin \left (\frac {\sin (x)}{\sqrt {2}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sin (x)}{\sqrt {3+\cos (2 x)}}\right )\right ) \cos (x) \sqrt {1+\sec ^2(x)}}{\sqrt {3+\cos (2 x)}} \] Input:
Integrate[Sqrt[1 + Sec[x]^2],x]
Output:
(Sqrt[2]*(ArcSin[Sin[x]/Sqrt[2]] + ArcTanh[(Sqrt[2]*Sin[x])/Sqrt[3 + Cos[2 *x]]])*Cos[x]*Sqrt[1 + Sec[x]^2])/Sqrt[3 + Cos[2*x]]
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4616, 301, 222, 291, 216}
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.
\(\displaystyle \int \sqrt {\sec ^2(x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {\sec (x)^2+1}dx\) |
\(\Big \downarrow \) 4616 |
\(\displaystyle \int \frac {\sqrt {\tan ^2(x)+2}}{\tan ^2(x)+1}d\tan (x)\) |
\(\Big \downarrow \) 301 |
\(\displaystyle \int \frac {1}{\sqrt {\tan ^2(x)+2}}d\tan (x)+\int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^2(x)+2}}d\tan (x)\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \int \frac {1}{\left (\tan ^2(x)+1\right ) \sqrt {\tan ^2(x)+2}}d\tan (x)+\text {arcsinh}\left (\frac {\tan (x)}{\sqrt {2}}\right )\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \int \frac {1}{\frac {\tan ^2(x)}{\tan ^2(x)+2}+1}d\frac {\tan (x)}{\sqrt {\tan ^2(x)+2}}+\text {arcsinh}\left (\frac {\tan (x)}{\sqrt {2}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \text {arcsinh}\left (\frac {\tan (x)}{\sqrt {2}}\right )+\arctan \left (\frac {\tan (x)}{\sqrt {\tan ^2(x)+2}}\right )\) |
Input:
Int[Sqrt[1 + Sec[x]^2],x]
Output:
ArcSinh[Tan[x]/Sqrt[2]] + ArcTan[Tan[x]/Sqrt[2 + Tan[x]^2]]
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ d Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b*x^2)^ (p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E qQ[b*c + 3*a*d, 0]))
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(21)=42\).
Time = 7.09 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.62
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {2+2 \sec \left (x \right )^{2}}\, \left (\left (1-\cos \left (x \right )\right )^{2} \csc \left (x \right )^{2}-1\right ) \left (2 \arctan \left (\frac {2 \csc \left (x \right )-2 \cot \left (x \right )}{\sqrt {2 \left (1-\cos \left (x \right )\right )^{4} \csc \left (x \right )^{4}+2}}\right )-\operatorname {arctanh}\left (\frac {\sin \left (x \right )-2}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {\cos \left (x \right )^{2}+1}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-\operatorname {arctanh}\left (\frac {\sin \left (x \right )+2}{\left (\cos \left (x \right )+1\right ) \sqrt {\frac {\cos \left (x \right )^{2}+1}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )\right )}{8 \sqrt {\frac {\cos \left (x \right )^{2}+1}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(135\) |
Input:
int((1+sec(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/8*2^(1/2)*(2+2*sec(x)^2)^(1/2)*((1-cos(x))^2*csc(x)^2-1)*(2*arctan(2/(2 *(1-cos(x))^4*csc(x)^4+2)^(1/2)*(csc(x)-cot(x)))-arctanh((sin(x)-2)/(cos(x )+1)/((cos(x)^2+1)/(cos(x)+1)^2)^(1/2))-arctanh((sin(x)+2)/(cos(x)+1)/((co s(x)^2+1)/(cos(x)+1)^2)^(1/2)))/((cos(x)^2+1)/(cos(x)+1)^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (21) = 42\).
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 5.46 \[ \int \sqrt {1+\sec ^2(x)} \, dx=\frac {1}{2} \, \arctan \left (\frac {\sqrt {\frac {\cos \left (x\right )^{2} + 1}{\cos \left (x\right )^{2}}} \cos \left (x\right )^{3} \sin \left (x\right ) + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} + \cos \left (x\right )^{2} - 1}\right ) - \frac {1}{2} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) + \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right ) + {\left (\cos \left (x\right )^{2} + \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {\cos \left (x\right )^{2} + 1}{\cos \left (x\right )^{2}}} + 1\right ) - \frac {1}{2} \, \log \left (\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right ) + {\left (\cos \left (x\right )^{2} - \cos \left (x\right ) \sin \left (x\right )\right )} \sqrt {\frac {\cos \left (x\right )^{2} + 1}{\cos \left (x\right )^{2}}} + 1\right ) \] Input:
integrate((1+sec(x)^2)^(1/2),x, algorithm="fricas")
Output:
1/2*arctan((sqrt((cos(x)^2 + 1)/cos(x)^2)*cos(x)^3*sin(x) + cos(x)*sin(x)) /(cos(x)^4 + cos(x)^2 - 1)) - 1/2*arctan(sin(x)/cos(x)) + 1/2*log(cos(x)^2 + cos(x)*sin(x) + (cos(x)^2 + cos(x)*sin(x))*sqrt((cos(x)^2 + 1)/cos(x)^2 ) + 1) - 1/2*log(cos(x)^2 - cos(x)*sin(x) + (cos(x)^2 - cos(x)*sin(x))*sqr t((cos(x)^2 + 1)/cos(x)^2) + 1)
\[ \int \sqrt {1+\sec ^2(x)} \, dx=\int \sqrt {\sec ^{2}{\left (x \right )} + 1}\, dx \] Input:
integrate((1+sec(x)**2)**(1/2),x)
Output:
Integral(sqrt(sec(x)**2 + 1), x)
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 1391, normalized size of antiderivative = 57.96 \[ \int \sqrt {1+\sec ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate((1+sec(x)^2)^(1/2),x, algorithm="maxima")
Output:
-1/2*arctan2(2*(2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4 )*sin(1/2*arctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)), 2*(2 *(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12* sin(4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2 (sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 8) + 3/2*arctan2(2*( 2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12 *sin(4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*sin(1/2*arctan 2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 2*sin(2*x), 2*(2*(6 *cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin (4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(si n(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + 2*cos(2*x) + 6) - arcta n2((2*(6*cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin(4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*sin(1/2*ar ctan2(sin(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + sin(2*x), (2*(6 *cos(2*x) + 1)*cos(4*x) + cos(4*x)^2 + 36*cos(2*x)^2 + sin(4*x)^2 + 12*sin (4*x)*sin(2*x) + 36*sin(2*x)^2 + 12*cos(2*x) + 1)^(1/4)*cos(1/2*arctan2(si n(4*x) + 6*sin(2*x), cos(4*x) + 6*cos(2*x) + 1)) + cos(2*x) + 3) - 1/2*log (-(2*sqrt(2)*(abs(2*e^(2*I*x) + 2)^4 + 4*cos(2*x)^4 + 4*sin(2*x)^4 - 4*(co s(2*x)^2 - sin(2*x)^2 - 2*cos(2*x) + 1)*abs(2*e^(2*I*x) + 2)^2 - 16*cos...
\[ \int \sqrt {1+\sec ^2(x)} \, dx=\int { \sqrt {\sec \left (x\right )^{2} + 1} \,d x } \] Input:
integrate((1+sec(x)^2)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(sec(x)^2 + 1), x)
Timed out. \[ \int \sqrt {1+\sec ^2(x)} \, dx=\int \sqrt {\frac {1}{{\cos \left (x\right )}^2}+1} \,d x \] Input:
int((1/cos(x)^2 + 1)^(1/2),x)
Output:
int((1/cos(x)^2 + 1)^(1/2), x)
\[ \int \sqrt {1+\sec ^2(x)} \, dx=\int \sqrt {\sec \left (x \right )^{2}+1}d x \] Input:
int((1+sec(x)^2)^(1/2),x)
Output:
int(sqrt(sec(x)**2 + 1),x)