Integrand size = 15, antiderivative size = 38 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d} \]
Time = 1.41 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.84 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {\sqrt {1+e^{2 (c+d x)}} \left (\text {arcsinh}\left (e^{c+d x}\right )+\text {arctanh}\left (\sqrt {1+e^{2 (c+d x)}}\right )\right ) \sqrt {a-a \text {sech}(c+d x)}}{d \left (-1+e^{c+d x}\right )} \]
(Sqrt[1 + E^(2*(c + d*x))]*(ArcSinh[E^(c + d*x)] + ArcTanh[Sqrt[1 + E^(2*( c + d*x))]])*Sqrt[a - a*Sech[c + d*x]])/(d*(-1 + E^(c + d*x)))
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4261, 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.
\(\displaystyle \int \sqrt {a-a \text {sech}(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {a-a \csc \left (i c+i d x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4261 |
\(\displaystyle -\frac {2 i a \int \frac {1}{a-\frac {a^2 \tanh ^2(c+d x)}{a-a \text {sech}(c+d x)}}d\frac {i a \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}}{d}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tanh (c+d x)}{\sqrt {a-a \text {sech}(c+d x)}}\right )}{d}\) |
3.1.83.3.1 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[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(a + x^2), x], x, b*(Cot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
\[\int \sqrt {a -\operatorname {sech}\left (d x +c \right ) a}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 642 vs. \(2 (32) = 64\).
Time = 0.28 (sec) , antiderivative size = 642, normalized size of antiderivative = 16.89 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\frac {\sqrt {a} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + a \sinh \left (d x + c\right )^{4} + 3 \, a \cosh \left (d x + c\right )^{3} + {\left (4 \, a \cosh \left (d x + c\right ) + 3 \, a\right )} \sinh \left (d x + c\right )^{3} + 5 \, a \cosh \left (d x + c\right )^{2} + {\left (6 \, a \cosh \left (d x + c\right )^{2} + 9 \, a \cosh \left (d x + c\right ) + 5 \, a\right )} \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{5} + {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{4} + \sinh \left (d x + c\right )^{5} + 3 \, \cosh \left (d x + c\right )^{4} + {\left (10 \, \cosh \left (d x + c\right )^{2} + 12 \, \cosh \left (d x + c\right ) + 5\right )} \sinh \left (d x + c\right )^{3} + 5 \, \cosh \left (d x + c\right )^{3} + {\left (10 \, \cosh \left (d x + c\right )^{3} + 18 \, \cosh \left (d x + c\right )^{2} + 15 \, \cosh \left (d x + c\right ) + 7\right )} \sinh \left (d x + c\right )^{2} + 7 \, \cosh \left (d x + c\right )^{2} + {\left (5 \, \cosh \left (d x + c\right )^{4} + 12 \, \cosh \left (d x + c\right )^{3} + 15 \, \cosh \left (d x + c\right )^{2} + 14 \, \cosh \left (d x + c\right ) + 4\right )} \sinh \left (d x + c\right ) + 4 \, \cosh \left (d x + c\right ) + 4\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} + 4 \, a \cosh \left (d x + c\right ) + {\left (4 \, a \cosh \left (d x + c\right )^{3} + 9 \, a \cosh \left (d x + c\right )^{2} + 10 \, a \cosh \left (d x + c\right ) + 4 \, a\right )} \sinh \left (d x + c\right ) + 4 \, a}{\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right ) + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3}}\right ) + \sqrt {a} \log \left (-\frac {a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + {\left (\cosh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right ) - 1\right )} \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} - \cosh \left (d x + c\right )^{2} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) + 1\right )} \sinh \left (d x + c\right ) + \cosh \left (d x + c\right ) - 1\right )} \sqrt {a} \sqrt {\frac {a}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}} - a \cosh \left (d x + c\right ) + {\left (2 \, a \cosh \left (d x + c\right ) - a\right )} \sinh \left (d x + c\right ) + a}{\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )}\right )}{2 \, d} \]
1/2*(sqrt(a)*log((a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 + 3*a*cosh(d*x + c )^3 + (4*a*cosh(d*x + c) + 3*a)*sinh(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6 *a*cosh(d*x + c)^2 + 9*a*cosh(d*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + c) + 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 + 3*cosh( d*x + c)^4 + (10*cosh(d*x + c)^2 + 12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (10*cosh(d*x + c)^3 + 18*cosh(d*x + c)^2 + 15*cosh(d* x + c) + 7)*sinh(d*x + c)^2 + 7*cosh(d*x + c)^2 + (5*cosh(d*x + c)^4 + 12* cosh(d*x + c)^3 + 15*cosh(d*x + c)^2 + 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) + 4)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)* sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + 4*a*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 9*a*cosh(d*x + c)^2 + 10*a*cosh(d*x + c) + 4*a)*sinh(d*x + c) + 4 *a)/(cosh(d*x + c)^3 + 3*cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)*s inh(d*x + c)^2 + sinh(d*x + c)^3)) + sqrt(a)*log(-(a*cosh(d*x + c)^2 + a*s inh(d*x + c)^2 + (cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - a*cosh(d*x + c) + (2*a*cosh(d*x + c) - a)*sinh(d*x + c) + a)/(cosh(d*x + c) + sinh(d*x + c) )))/d
\[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int \sqrt {- a \operatorname {sech}{\left (c + d x \right )} + a}\, dx \]
\[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int { \sqrt {-a \operatorname {sech}\left (d x + c\right ) + a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (32) = 64\).
Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66 \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=-\frac {\frac {2 \, a \arctan \left (-\frac {\sqrt {a} e^{\left (d x + c\right )} - \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{\sqrt {-a}} + \sqrt {a} \log \left ({\left | -\sqrt {a} e^{\left (d x + c\right )} + \sqrt {a e^{\left (2 \, d x + 2 \, c\right )} + a} \right |}\right ) \mathrm {sgn}\left (e^{\left (d x + c\right )} - 1\right )}{d} \]
-(2*a*arctan(-(sqrt(a)*e^(d*x + c) - sqrt(a*e^(2*d*x + 2*c) + a))/sqrt(-a) )*sgn(e^(d*x + c) - 1)/sqrt(-a) + sqrt(a)*log(abs(-sqrt(a)*e^(d*x + c) + s qrt(a*e^(2*d*x + 2*c) + a)))*sgn(e^(d*x + c) - 1))/d
Timed out. \[ \int \sqrt {a-a \text {sech}(c+d x)} \, dx=\int \sqrt {a-\frac {a}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \]