Integrand size = 13, antiderivative size = 64 \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=-\frac {a \arctan (\sinh (x))}{b^2}+\frac {2 a^2 \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^2 \sqrt {a+b}}+\frac {\tanh (x)}{b} \]
-a*arctan(sinh(x))/b^2+2*a^2*arctan((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/b ^2/(a-b)^(1/2)/(a+b)^(1/2)+tanh(x)/b
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {-2 a \arctan \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {2 a^2 \arctan \left (\frac {(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+b \tanh (x)}{b^2} \]
(-2*a*ArcTan[Tanh[x/2]] - (2*a^2*ArcTan[((-a + b)*Tanh[x/2])/Sqrt[a^2 - b^ 2]])/Sqrt[a^2 - b^2] + b*Tanh[x])/b^2
Time = 0.56 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 4277, 3042, 4276, 3042, 4257, 4318, 3042, 3138, 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.
\(\displaystyle \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\csc \left (\frac {\pi }{2}+i x\right )^3}{a+b \csc \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 4277 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \int \frac {\text {sech}^2(x)}{a+b \text {sech}(x)}dx}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \int \frac {\csc \left (i x+\frac {\pi }{2}\right )^2}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{b}\) |
\(\Big \downarrow \) 4276 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\int \text {sech}(x)dx}{b}-\frac {a \int \frac {\text {sech}(x)}{a+b \text {sech}(x)}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\int \csc \left (i x+\frac {\pi }{2}\right )dx}{b}-\frac {a \int \frac {\csc \left (i x+\frac {\pi }{2}\right )}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\arctan (\sinh (x))}{b}-\frac {a \int \frac {\csc \left (i x+\frac {\pi }{2}\right )}{a+b \csc \left (i x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}\) |
\(\Big \downarrow \) 4318 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\arctan (\sinh (x))}{b}-\frac {a \int \frac {1}{\frac {a \cosh (x)}{b}+1}dx}{b^2}\right )}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\arctan (\sinh (x))}{b}-\frac {a \int \frac {1}{\frac {a \sin \left (i x+\frac {\pi }{2}\right )}{b}+1}dx}{b^2}\right )}{b}\) |
\(\Big \downarrow \) 3138 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\arctan (\sinh (x))}{b}-\frac {2 a \int \frac {1}{\frac {a+b}{b}-\left (1-\frac {a}{b}\right ) \tanh ^2\left (\frac {x}{2}\right )}d\tanh \left (\frac {x}{2}\right )}{b^2}\right )}{b}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\tanh (x)}{b}-\frac {a \left (\frac {\arctan (\sinh (x))}{b}-\frac {2 a \arctan \left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b \sqrt {a-b} \sqrt {a+b}}\right )}{b}\) |
-((a*(ArcTan[Sinh[x]]/b - (2*a*ArcTan[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]] )/(Sqrt[a - b]*b*Sqrt[a + b])))/b) + Tanh[x]/b
3.2.1.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[1/b Int[Csc[e + f*x], x], x] - Simp[a/b Int[Csc[e + f*x]/( a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]^3/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Sym bol] :> Simp[-Cot[e + f*x]/(b*f), x] - Simp[a/b Int[Csc[e + f*x]^2/(a + b *Csc[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x]
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo l] :> Simp[1/b Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {2 a^{2} \arctan \left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{2} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {2 \left (-\frac {b \tanh \left (\frac {x}{2}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}+a \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{b^{2}}\) | \(73\) |
risch | \(-\frac {2}{b \left (1+{\mathrm e}^{2 x}\right )}-\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+\frac {b \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, a}\right )}{\sqrt {-a^{2}+b^{2}}\, b^{2}}+\frac {i a \ln \left ({\mathrm e}^{x}-i\right )}{b^{2}}-\frac {i a \ln \left ({\mathrm e}^{x}+i\right )}{b^{2}}\) | \(160\) |
2*a^2/b^2/((a+b)*(a-b))^(1/2)*arctan((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2) )-2/b^2*(-b*tanh(1/2*x)/(1+tanh(1/2*x)^2)+a*arctan(tanh(1/2*x)))
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (54) = 108\).
Time = 0.28 (sec) , antiderivative size = 504, normalized size of antiderivative = 7.88 \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\left [-\frac {2 \, a^{2} b - 2 \, b^{3} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) - a^{2} + 2 \, b^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + b\right )}}{a \cosh \left (x\right )^{2} + a \sinh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) + 2 \, {\left (a \cosh \left (x\right ) + b\right )} \sinh \left (x\right ) + a}\right ) + 2 \, {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}{a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2}}, -\frac {2 \, {\left (a^{2} b - b^{3} + {\left (a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cosh \left (x\right ) + a \sinh \left (x\right ) + b}{\sqrt {a^{2} - b^{2}}}\right ) + {\left (a^{3} - a b^{2} + {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right )\right )}}{a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2}}\right ] \]
[-(2*a^2*b - 2*b^3 + (a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^ 2 + a^2)*sqrt(-a^2 + b^2)*log((a^2*cosh(x)^2 + a^2*sinh(x)^2 + 2*a*b*cosh( x) - a^2 + 2*b^2 + 2*(a^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(-a^2 + b^2)*(a*c osh(x) + a*sinh(x) + b))/(a*cosh(x)^2 + a*sinh(x)^2 + 2*b*cosh(x) + 2*(a*c osh(x) + b)*sinh(x) + a)) + 2*(a^3 - a*b^2 + (a^3 - a*b^2)*cosh(x)^2 + 2*( a^3 - a*b^2)*cosh(x)*sinh(x) + (a^3 - a*b^2)*sinh(x)^2)*arctan(cosh(x) + s inh(x)))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cosh(x)^2 + 2*(a^2*b^2 - b^4)*co sh(x)*sinh(x) + (a^2*b^2 - b^4)*sinh(x)^2), -2*(a^2*b - b^3 + (a^2*cosh(x) ^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 + a^2)*sqrt(a^2 - b^2)*arctan(- (a*cosh(x) + a*sinh(x) + b)/sqrt(a^2 - b^2)) + (a^3 - a*b^2 + (a^3 - a*b^2 )*cosh(x)^2 + 2*(a^3 - a*b^2)*cosh(x)*sinh(x) + (a^3 - a*b^2)*sinh(x)^2)*a rctan(cosh(x) + sinh(x)))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cosh(x)^2 + 2*( a^2*b^2 - b^4)*cosh(x)*sinh(x) + (a^2*b^2 - b^4)*sinh(x)^2)]
\[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, a^{2} \arctan \left (\frac {a e^{x} + b}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} b^{2}} - \frac {2 \, a \arctan \left (e^{x}\right )}{b^{2}} - \frac {2}{b {\left (e^{\left (2 \, x\right )} + 1\right )}} \]
2*a^2*arctan((a*e^x + b)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*b^2) - 2*a*arct an(e^x)/b^2 - 2/(b*(e^(2*x) + 1))
Time = 4.35 (sec) , antiderivative size = 294, normalized size of antiderivative = 4.59 \[ \int \frac {\text {sech}^3(x)}{a+b \text {sech}(x)} \, dx=\frac {a^2\,\ln \left (64\,a^3\,b-64\,a\,b^3+32\,a^3\,\sqrt {b^2-a^2}-32\,a^4\,{\mathrm {e}}^x-128\,b^4\,{\mathrm {e}}^x-64\,a\,b^2\,\sqrt {b^2-a^2}-128\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}+160\,a^2\,b^2\,{\mathrm {e}}^x+96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )}{b^2\,\sqrt {b^2-a^2}}+\frac {a\,\left (\ln \left (32\,{\mathrm {e}}^x-32{}\mathrm {i}\right )\,1{}\mathrm {i}-\ln \left (32\,{\mathrm {e}}^x+32{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{b^2}-\frac {2}{b+b\,{\mathrm {e}}^{2\,x}}-\frac {a^2\,\ln \left (64\,a\,b^3-64\,a^3\,b+32\,a^3\,\sqrt {b^2-a^2}+32\,a^4\,{\mathrm {e}}^x+128\,b^4\,{\mathrm {e}}^x-64\,a\,b^2\,\sqrt {b^2-a^2}-128\,b^3\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}-160\,a^2\,b^2\,{\mathrm {e}}^x+96\,a^2\,b\,{\mathrm {e}}^x\,\sqrt {b^2-a^2}\right )}{b^2\,\sqrt {b^2-a^2}} \]
(a*(log(32*exp(x) - 32i)*1i - log(32*exp(x) + 32i)*1i))/b^2 - 2/(b + b*exp (2*x)) + (a^2*log(64*a^3*b - 64*a*b^3 + 32*a^3*(b^2 - a^2)^(1/2) - 32*a^4* exp(x) - 128*b^4*exp(x) - 64*a*b^2*(b^2 - a^2)^(1/2) - 128*b^3*exp(x)*(b^2 - a^2)^(1/2) + 160*a^2*b^2*exp(x) + 96*a^2*b*exp(x)*(b^2 - a^2)^(1/2)))/( b^2*(b^2 - a^2)^(1/2)) - (a^2*log(64*a*b^3 - 64*a^3*b + 32*a^3*(b^2 - a^2) ^(1/2) + 32*a^4*exp(x) + 128*b^4*exp(x) - 64*a*b^2*(b^2 - a^2)^(1/2) - 128 *b^3*exp(x)*(b^2 - a^2)^(1/2) - 160*a^2*b^2*exp(x) + 96*a^2*b*exp(x)*(b^2 - a^2)^(1/2)))/(b^2*(b^2 - a^2)^(1/2))