Integrand size = 21, antiderivative size = 55 \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b) d}-\frac {\text {arctanh}(\cosh (c+d x))}{(a+b) d} \] Output:
b^(1/2)*arctan(a^(1/2)*cosh(d*x+c)/b^(1/2))/a^(1/2)/(a+b)/d-arctanh(cosh(d *x+c))/(a+b)/d
Result contains complex when optimal does not.
Time = 0.94 (sec) , antiderivative size = 232, normalized size of antiderivative = 4.22 \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}-i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}+\frac {\sqrt {b} \arctan \left (\frac {\left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2}\right ) \sinh (c) \tanh \left (\frac {d x}{2}\right )+\cosh (c) \left (\sqrt {a}+i \sqrt {a+b} \sqrt {(\cosh (c)-\sinh (c))^2} \tanh \left (\frac {d x}{2}\right )\right )}{\sqrt {b}}\right )}{\sqrt {a}}-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b) d} \] Input:
Integrate[Csch[c + d*x]/(a + b*Sech[c + d*x]^2),x]
Output:
((Sqrt[b]*ArcTan[((Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Si nh[c]*Tanh[(d*x)/2] + Cosh[c]*(Sqrt[a] - I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sin h[c])^2]*Tanh[(d*x)/2]))/Sqrt[b]])/Sqrt[a] + (Sqrt[b]*ArcTan[((Sqrt[a] + I *Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2])*Sinh[c]*Tanh[(d*x)/2] + Cosh[c]* (Sqrt[a] + I*Sqrt[a + b]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[(d*x)/2]))/Sqrt[ b]])/Sqrt[a] - Log[Cosh[(c + d*x)/2]] + Log[Sinh[(c + d*x)/2]])/((a + b)*d )
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4621, 383, 218, 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.
| \(\displaystyle \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\sin (i c+i d x) \left (a+b \sec (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\left (b \sec (i c+i d x)^2+a\right ) \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 4621 |
| \(\displaystyle -\frac {\int \frac {\cosh ^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right ) \left (a \cosh ^2(c+d x)+b\right )}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 383 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}-\frac {b \int \frac {1}{a \cosh ^2(c+d x)+b}d\cosh (c+d x)}{a+b}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {\frac {\int \frac {1}{1-\cosh ^2(c+d x)}d\cosh (c+d x)}{a+b}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {\text {arctanh}(\cosh (c+d x))}{a+b}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} \cosh (c+d x)}{\sqrt {b}}\right )}{\sqrt {a} (a+b)}}{d}\) |
Input:
Int[Csch[c + d*x]/(a + b*Sech[c + d*x]^2),x]
Output:
-((-((Sqrt[b]*ArcTan[(Sqrt[a]*Cosh[c + d*x])/Sqrt[b]])/(Sqrt[a]*(a + b))) + ArcTanh[Cosh[c + d*x]]/(a + b))/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_.)*(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[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_Sym bol] :> Simp[(-a)*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(a + b*x^2), x], x] + Simp[c*(e^2/(b*c - a*d)) Int[(e*x)^(m - 2)/(c + d*x^2), x], x] /; Free Q[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LeQ[2, m, 3]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/ 2] && IntegerQ[n] && IntegerQ[p]
Time = 1.18 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18
| method | result | size |
| derivativedivides | \(\frac {\frac {b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right ) \sqrt {a b}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}}{d}\) | \(65\) |
| default | \(\frac {\frac {b \arctan \left (\frac {2 \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 a -2 b}{4 \sqrt {a b}}\right )}{\left (a +b \right ) \sqrt {a b}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a +b}}{d}\) | \(65\) |
| risch | \(\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d \left (a +b \right )}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d \left (a +b \right )}+\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a \left (a +b \right ) d}-\frac {\sqrt {-a b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}\, {\mathrm e}^{d x +c}}{a}+1\right )}{2 a \left (a +b \right ) d}\) | \(135\) |
Input:
int(csch(d*x+c)/(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/d*(b/(a+b)/(a*b)^(1/2)*arctan(1/4*(2*(a+b)*tanh(1/2*d*x+1/2*c)^2+2*a-2*b )/(a*b)^(1/2))+1/(a+b)*ln(tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (47) = 94\).
Time = 0.15 (sec) , antiderivative size = 533, normalized size of antiderivative = 9.69 \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(csch(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="fricas")
Output:
[1/2*(sqrt(-b/a)*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*(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) + 4*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d*x + c)^3 + a*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a)*sin h(d*x + c))*sqrt(-b/a) + 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)*co sh(d*x + c))*sinh(d*x + c) + a)) - 2*log(cosh(d*x + c) + sinh(d*x + c) + 1 ) + 2*log(cosh(d*x + c) + sinh(d*x + c) - 1))/((a + b)*d), -(sqrt(b/a)*arc tan(1/2*(a*cosh(d*x + c)^3 + 3*a*cosh(d*x + c)*sinh(d*x + c)^2 + a*sinh(d* x + c)^3 + (a + 4*b)*cosh(d*x + c) + (3*a*cosh(d*x + c)^2 + a + 4*b)*sinh( d*x + c))*sqrt(b/a)/b) - sqrt(b/a)*arctan(1/2*(a*cosh(d*x + c) + a*sinh(d* x + c))*sqrt(b/a)/b) + log(cosh(d*x + c) + sinh(d*x + c) + 1) - log(cosh(d *x + c) + sinh(d*x + c) - 1))/((a + b)*d)]
\[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \] Input:
integrate(csch(d*x+c)/(a+b*sech(d*x+c)**2),x)
Output:
Integral(csch(c + d*x)/(a + b*sech(c + d*x)**2), x)
\[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )}{b \operatorname {sech}\left (d x + c\right )^{2} + a} \,d x } \] Input:
integrate(csch(d*x+c)/(a+b*sech(d*x+c)^2),x, algorithm="maxima")
Output:
-log((e^(d*x + c) + 1)*e^(-c))/(a*d + b*d) + log((e^(d*x + c) - 1)*e^(-c)) /(a*d + b*d) + 2*integrate((b*e^(3*d*x + 3*c) - b*e^(d*x + c))/(a^2 + a*b + (a^2*e^(4*c) + a*b*e^(4*c))*e^(4*d*x) + 2*(a^2*e^(2*c) + 3*a*b*e^(2*c) + 2*b^2*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csch(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 = 3.15 (sec) , antiderivative size = 616, normalized size of antiderivative = 11.20 \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^4\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}+16\,a^2\,b^2\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}+8\,a\,b^3\,\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}\right )}{16\,d\,a^3\,b^2+24\,d\,a^2\,b^3+9\,d\,a\,b^4+d\,b^5}\right )}{\sqrt {-a^2\,d^2-2\,a\,b\,d^2-b^2\,d^2}}-\frac {\sqrt {b}\,\left (2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (\frac {64\,\left (2\,b^{7/2}\,d+8\,a^2\,b^{3/2}\,d+10\,a\,b^{5/2}\,d\right )}{a^5\,\left (a+b\right )\,\sqrt {a\,d^2\,{\left (a+b\right )}^2}\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}+\frac {32\,\left (b^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+4\,a\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^5\,\sqrt {b}\,d\,{\left (a+b\right )}^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )+\frac {32\,{\mathrm {e}}^{3\,c}\,{\mathrm {e}}^{3\,d\,x}\,\left (b^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+4\,a\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{a^5\,\sqrt {b}\,d\,{\left (a+b\right )}^2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}}\right )\,\left (a^6\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}+a^5\,b\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}\right )}{64\,b^2+256\,a\,b}\right )-2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a\,d^2\,{\left (a+b\right )}^2}}{2\,\sqrt {b}\,d\,\left (a+b\right )}\right )\right )}{2\,\sqrt {a^3\,d^2+2\,a^2\,b\,d^2+a\,b^2\,d^2}} \] Input:
int(1/(sinh(c + d*x)*(a + b/cosh(c + d*x)^2)),x)
Output:
- (2*atan((exp(d*x)*exp(c)*(b^4*(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2) + 16*a^2*b^2*(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2) + 8*a*b^3*(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2)))/(b^5*d + 24*a^2*b^3*d + 16*a^3*b^2*d + 9*a*b^ 4*d)))/(- a^2*d^2 - b^2*d^2 - 2*a*b*d^2)^(1/2) - (b^(1/2)*(2*atan(((exp(d* x)*exp(c)*((64*(2*b^(7/2)*d + 8*a^2*b^(3/2)*d + 10*a*b^(5/2)*d))/(a^5*(a + b)*(a*d^2*(a + b)^2)^(1/2)*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)) + ( 32*(b^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2) + 4*a*b*(a^3*d^2 + a*b^2 *d^2 + 2*a^2*b*d^2)^(1/2)))/(a^5*b^(1/2)*d*(a + b)^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2))) + (32*exp(3*c)*exp(3*d*x)*(b^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2) + 4*a*b*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)))/ (a^5*b^(1/2)*d*(a + b)^2*(a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2)))*(a^6* (a^3*d^2 + a*b^2*d^2 + 2*a^2*b*d^2)^(1/2) + a^5*b*(a^3*d^2 + a*b^2*d^2 + 2 *a^2*b*d^2)^(1/2)))/(256*a*b + 64*b^2)) - 2*atan((exp(d*x)*exp(c)*(a*d^2*( a + b)^2)^(1/2))/(2*b^(1/2)*d*(a + b)))))/(2*(a^3*d^2 + a*b^2*d^2 + 2*a^2* b*d^2)^(1/2))
Time = 0.27 (sec) , antiderivative size = 355, normalized size of antiderivative = 6.45 \[ \int \frac {\text {csch}(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx=\frac {2 \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {e^{d x +c} a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right )-2 \sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}\, \mathit {atan} \left (\frac {e^{d x +c} a}{\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}+a +2 b}}\right ) b +\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )-\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right )+\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b -\sqrt {a}\, \sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}\, \mathrm {log}\left (\sqrt {2 \sqrt {b}\, \sqrt {a +b}-a -2 b}+e^{d x +c} \sqrt {a}\right ) b +2 \,\mathrm {log}\left (e^{d x +c}-1\right ) a^{2}-2 \,\mathrm {log}\left (e^{d x +c}+1\right ) a^{2}}{2 a^{2} d \left (a +b \right )} \] Input:
int(csch(d*x+c)/(a+b*sech(d*x+c)^2),x)
Output:
(2*sqrt(b)*sqrt(a)*sqrt(a + b)*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))) - 2*sqrt (a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*s qrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*b + sqrt(b)*sqrt(a)*sqrt(a + b)*sqr t(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(b)*sqrt(a)*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(a)*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))*b - sqrt(a)*sqr t(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*log(e**(c + d*x) - 1)*a**2 - 2*log(e**(c + d*x) + 1)*a**2)/(2*a**2*d*(a + b))