\(\int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [438]

Optimal result

Integrand size = 25, antiderivative size = 90 \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d} \]

[Out]

-b*arctan(sinh(d*x+c))/(a^2+b^2)/d-a*ln(cosh(d*x+c))/(a^2+b^2)/d+ln(sinh(d*x+c))/a/d-b^2*ln(a+b*sinh(d*x+c))/a
/(a^2+b^2)/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2916, 12, 908, 649, 209, 266} \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b \arctan (\sinh (c+d x))}{d \left (a^2+b^2\right )}-\frac {b^2 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )}-\frac {a \log (\cosh (c+d x))}{d \left (a^2+b^2\right )}+\frac {\log (\sinh (c+d x))}{a d} \]

[In]

Int[(Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-((b*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)*d)) - (a*Log[Cosh[c + d*x]])/((a^2 + b^2)*d) + Log[Sinh[c + d*x]]/(a*
d) - (b^2*Log[a + b*Sinh[c + d*x]])/(a*(a^2 + b^2)*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b \text {Subst}\left (\int \frac {b}{x (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = -\frac {b^2 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = -\frac {b^2 \text {Subst}\left (\int \left (-\frac {1}{a b^2 x}+\frac {1}{a \left (a^2+b^2\right ) (a+x)}+\frac {b^2+a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d}-\frac {\text {Subst}\left (\int \frac {b^2+a x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = \frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d}-\frac {a \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right ) d} \\ & = -\frac {b \arctan (\sinh (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a \log (\cosh (c+d x))}{\left (a^2+b^2\right ) d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {\log (i-\sinh (c+d x))}{a+i b}-\frac {2 \log (\sinh (c+d x))}{a}+\frac {\log (i+\sinh (c+d x))}{a-i b}+\frac {2 b^2 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )}}{2 d} \]

[In]

Integrate[(Csch[c + d*x]*Sech[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*(Log[I - Sinh[c + d*x]]/(a + I*b) - (2*Log[Sinh[c + d*x]])/a + Log[I + Sinh[c + d*x]]/(a - I*b) + (2*b^2*
Log[a + b*Sinh[c + d*x]])/(a*(a^2 + b^2)))/d

Maple [A] (verified)

Time = 1.71 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {-a \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) a}}{d}\)	\(108\)
default	\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}+\frac {-a \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2 b \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}+b^{2}}-\frac {b^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right ) a}}{d}\)	\(108\)
risch	\(\frac {2 a \,d^{2} x}{a^{2} d^{2}+b^{2} d^{2}}+\frac {2 a d c}{a^{2} d^{2}+b^{2} d^{2}}-\frac {2 x}{a}-\frac {2 c}{d a}+\frac {2 b^{2} x}{a \left (a^{2}+b^{2}\right )}+\frac {2 b^{2} c}{a d \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) b}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a}{\left (a^{2}+b^{2}\right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b}{\left (a^{2}+b^{2}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a}{\left (a^{2}+b^{2}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d \left (a^{2}+b^{2}\right )}\)	\(267\)

[In]

int(csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/a*ln(tanh(1/2*d*x+1/2*c))+1/(a^2+b^2)*(-a*ln(1+tanh(1/2*d*x+1/2*c)^2)-2*b*arctan(tanh(1/2*d*x+1/2*c)))-
b^2/(a^2+b^2)/a*ln(tanh(1/2*d*x+1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.49 \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 \, a b \arctan \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + b^{2} \log \left (\frac {2 \, {\left (b \sinh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + a^{2} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{{\left (a^{3} + a b^{2}\right )} d} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-(2*a*b*arctan(cosh(d*x + c) + sinh(d*x + c)) + b^2*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c)
)) + a^2*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - (a^2 + b^2)*log(2*sinh(d*x + c)/(cosh(d*x + c)
- sinh(d*x + c))))/((a^3 + a*b^2)*d)

Sympy [F]

\[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\operatorname {csch}{\left (c + d x \right )} \operatorname {sech}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(csch(c + d*x)*sech(c + d*x)/(a + b*sinh(c + d*x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.53 \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{2} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{3} + a b^{2}\right )} d} + \frac {2 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} - \frac {a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-b^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^3 + a*b^2)*d) + 2*b*arctan(e^(-d*x - c))/((a^2 + b^2)
*d) - a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.63 \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {2 \, b^{3} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3} b + a b^{3}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} b}{a^{2} + b^{2}} + \frac {a \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{2} + b^{2}} - \frac {2 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a}}{2 \, d} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

-1/2*(2*b^3*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^3*b + a*b^3) + (pi + 2*arctan(1/2*(e^(2*d*x + 2*
c) - 1)*e^(-d*x - c)))*b/(a^2 + b^2) + a*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^2 + b^2) - 2*log(abs(e^(d*
x + c) - e^(-d*x - c)))/a)/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}(c+d x) \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {1}{\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/(cosh(c + d*x)*sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)*sinh(c + d*x)*(a + b*sinh(c + d*x))), x)