Integrand size = 21, antiderivative size = 36 \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} d} \] Output:
arctan(a^(1/2)*sinh(d*x+c)/(a+b)^(1/2))/a^(1/2)/(a+b)^(1/2)/d
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} \sqrt {a+b} d} \] Input:
Integrate[Sech[c + d*x]/(a + b*Sech[c + d*x]^2),x]
Output:
ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 4635, 218}
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.
\(\displaystyle \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sec (i c+i d x)}{a+b \sec (i c+i d x)^2}dx\) |
\(\Big \downarrow \) 4635 |
\(\displaystyle \frac {\int \frac {1}{a \sinh ^2(c+d x)+a+b}d\sinh (c+d x)}{d}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} d \sqrt {a+b}}\) |
Input:
Int[Sech[c + d*x]/(a + b*Sech[c + d*x]^2),x]
Output:
ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(Sqrt[a]*Sqrt[a + b]*d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs. \(2(28)=56\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.22
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}}{d}\) | \(80\) |
default | \(\frac {\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{\sqrt {a +b}\, \sqrt {a}}}{d}\) | \(80\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, d}\) | \(106\) |
Input:
int(sech(d*x+c)/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(1/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)-2 *b^(1/2))/a^(1/2))+1/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/ 2*d*x+1/2*c)+2*b^(1/2))/a^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (28) = 56\).
Time = 0.22 (sec) , antiderivative size = 489, normalized size of antiderivative = 13.58 \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\left [-\frac {\sqrt {-a^{2} - a b} \log \left (\frac {a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} - 3 \, a - 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} - {\left (3 \, a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\left (\cosh \left (d x + c\right )^{3} + 3 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + \sinh \left (d x + c\right )^{3} + {\left (3 \, \cosh \left (d x + c\right )^{2} - 1\right )} \sinh \left (d x + c\right ) - \cosh \left (d x + c\right )\right )} \sqrt {-a^{2} - a b} + a}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right )}{2 \, {\left (a^{2} + a b\right )} d}, \frac {\sqrt {a^{2} + a b} \arctan \left (\frac {a \cosh \left (d x + c\right )^{3} + 3 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{3} + {\left (3 \, a + 4 \, b\right )} \cosh \left (d x + c\right ) + {\left (3 \, a \cosh \left (d x + c\right )^{2} + 3 \, a + 4 \, b\right )} \sinh \left (d x + c\right )}{2 \, \sqrt {a^{2} + a b}}\right ) - \sqrt {a^{2} + a b} \arctan \left (\frac {2 \, \sqrt {a^{2} + a b}}{a \cosh \left (d x + c\right ) + a \sinh \left (d x + c\right )}\right )}{{\left (a^{2} + a b\right )} d}\right ] \] Input:
integrate(sech(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
[-1/2*sqrt(-a^2 - a*b)*log((a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c)*sinh(d*x + c)^3 + a*sinh(d*x + c)^4 - 2*(3*a + 2*b)*cosh(d*x + c)^2 + 2*(3*a*cosh( d*x + c)^2 - 3*a - 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 - (3*a + 2* b)*cosh(d*x + c))*sinh(d*x + c) - 4*(cosh(d*x + c)^3 + 3*cosh(d*x + c)*sin h(d*x + c)^2 + sinh(d*x + c)^3 + (3*cosh(d*x + c)^2 - 1)*sinh(d*x + c) - c osh(d*x + c))*sqrt(-a^2 - a*b) + a)/(a*cosh(d*x + c)^4 + 4*a*cosh(d*x + c) *sinh(d*x + c)^3 + a*sinh(d*x + c)^4 + 2*(a + 2*b)*cosh(d*x + c)^2 + 2*(3* a*cosh(d*x + c)^2 + a + 2*b)*sinh(d*x + c)^2 + 4*(a*cosh(d*x + c)^3 + (a + 2*b)*cosh(d*x + c))*sinh(d*x + c) + a))/((a^2 + a*b)*d), (sqrt(a^2 + a*b) *arctan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sin h(d*x + c)^3 + (3*a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + 3*a + 4* b)*sinh(d*x + c))/sqrt(a^2 + a*b)) - sqrt(a^2 + a*b)*arctan(2*sqrt(a^2 + a *b)/(a*cosh(d*x + c) + a*sinh(d*x + c))))/((a^2 + a*b)*d)]
\[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\operatorname {sech}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(sech(d*x+c)/(a+b*sech(d*x+c)**2),x)
Output:
Integral(sech(c + d*x)/(a + b*sech(c + d*x)**2), x)
\[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \] Input:
integrate(sech(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="maxima")
Output:
integrate(sech(d*x + c)/(b*sech(d*x + c)^2 + a), x)
Exception generated. \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sech(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Time = 2.63 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.00 \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {\ln \left (-\frac {4\,\left (b-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,\left (a+b\right )}-\frac {8\,b\,{\mathrm {e}}^{c+d\,x}}{{\left (-a\right )}^{5/2}\,\sqrt {a+b}}\right )-\ln \left (\frac {8\,b\,{\mathrm {e}}^{c+d\,x}}{{\left (-a\right )}^{5/2}\,\sqrt {a+b}}-\frac {4\,\left (b-b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,\left (a+b\right )}\right )}{2\,\sqrt {-a}\,d\,\sqrt {a+b}} \] Input:
int(1/(cosh(c + d*x)*(a + b/cosh(c + d*x)^2)),x)
Output:
-(log(- (4*(b - b*exp(2*c + 2*d*x)))/(a^2*(a + b)) - (8*b*exp(c + d*x))/(( -a)^(5/2)*(a + b)^(1/2))) - log((8*b*exp(c + d*x))/((-a)^(5/2)*(a + b)^(1/ 2)) - (4*(b - b*exp(2*c + 2*d*x)))/(a^2*(a + b))))/(2*(-a)^(1/2)*d*(a + b) ^(1/2))
Time = 0.22 (sec) , antiderivative size = 457, normalized size of antiderivative = 12.69 \[ \int \frac {\text {sech}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\sqrt {a}\, \left (-2 \sqrt {b}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {e^{d x +c} a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right )+2 \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {e^{d x +c} a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) a +2 \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {e^{d x +c} a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) b -\sqrt {b}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )+\sqrt {b}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a -\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b +\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) a +\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b \right )}{2 a^{2} d \left (a +b \right )} \] Input:
int(sech(d*x+c)/(a+b*sech(d*x+c)^2),x)
Output:
(sqrt(a)*( - 2*sqrt(b)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*a tan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b))) + 2* sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt( 2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a + 2*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)) )*b - sqrt(b)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqr t(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a)) + sqrt(b)*sqrt( a + b)*sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a + b ) - a - 2*b) + e**(c + d*x)*sqrt(a)) - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2* b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log( - sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b + sqrt(2*sqrt(b)*sqrt(a + b) - a - 2* b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*a + s qrt(2*sqrt(b)*sqrt(a + b) - a - 2*b)*log(sqrt(2*sqrt(b)*sqrt(a + b) - a - 2*b) + e**(c + d*x)*sqrt(a))*b))/(2*a**2*d*(a + b))