\(\int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [447]

Optimal result

Integrand size = 27, antiderivative size = 160 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^3 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]

[Out]

-b^3*arctan(sinh(d*x+c))/(a^2+b^2)^2/d-1/2*b*arctan(sinh(d*x+c))/(a^2+b^2)/d-a*(a^2+2*b^2)*ln(cosh(d*x+c))/(a^
2+b^2)^2/d+ln(sinh(d*x+c))/a/d-b^4*ln(a+b*sinh(d*x+c))/a/(a^2+b^2)^2/d+1/2*sech(d*x+c)^2*(a-b*sinh(d*x+c))/(a^
2+b^2)/d

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2916, 12, 908, 653, 209, 649, 266} \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b \arctan (\sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b^3 \arctan (\sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}-\frac {b^4 \log (a+b \sinh (c+d x))}{a d \left (a^2+b^2\right )^2}+\frac {\log (\sinh (c+d x))}{a d} \]

[In]

Int[(Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-((b^3*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d)) - (b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) - (a*(a^2 + 2*b
^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) + Log[Sinh[c + d*x]]/(a*d) - (b^4*Log[a + b*Sinh[c + d*x]])/(a*(a^2
+ b^2)^2*d) + (Sech[c + d*x]^2*(a - b*Sinh[c + d*x]))/(2*(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 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

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^3 \text {Subst}\left (\int \frac {b}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b^4 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b^4 \text {Subst}\left (\int \left (\frac {1}{a b^4 x}-\frac {1}{a \left (a^2+b^2\right )^2 (a+x)}+\frac {-b^2-a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \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^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac {b^2 \text {Subst}\left (\int \frac {-b^2-a x}{\left (b^2+x^2\right )^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^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b^4 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}-\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}-\frac {\left (a \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d} \\ & = -\frac {b^3 \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {b \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {a \left (a^2+2 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\log (\sinh (c+d x))}{a d}-\frac {b^4 \log (a+b \sinh (c+d x))}{a \left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a b \left (a^2+b^2\right ) \arctan (\sinh (c+d x))-2 \left (a^2+b^2\right )^2 \log (\sinh (c+d x))+a \left (a^3+2 a b^2+\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}-b \sinh (c+d x)\right )+2 b^4 \log (a+b \sinh (c+d x))+a \left (a^3+2 a b^2-\left (-b^2\right )^{3/2}\right ) \log \left (\sqrt {-b^2}+b \sinh (c+d x)\right )-a^2 \left (a^2+b^2\right ) \text {sech}^2(c+d x)+a b \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 a \left (a^2+b^2\right )^2 d} \]

[In]

Integrate[(Csch[c + d*x]*Sech[c + d*x]^3)/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*(a*b*(a^2 + b^2)*ArcTan[Sinh[c + d*x]] - 2*(a^2 + b^2)^2*Log[Sinh[c + d*x]] + a*(a^3 + 2*a*b^2 + (-b^2)^(
3/2))*Log[Sqrt[-b^2] - b*Sinh[c + d*x]] + 2*b^4*Log[a + b*Sinh[c + d*x]] + a*(a^3 + 2*a*b^2 - (-b^2)^(3/2))*Lo
g[Sqrt[-b^2] + b*Sinh[c + d*x]] - a^2*(a^2 + b^2)*Sech[c + d*x]^2 + a*b*(a^2 + b^2)*Sech[c + d*x]*Tanh[c + d*x
])/(a*(a^2 + b^2)^2*d)

Maple [A] (verified)

Time = 15.01 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.44

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\)	\(230\)
default	\(\frac {-\frac {2 \left (\frac {\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{4}+\frac {\left (a^{2} b +3 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b^{4} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}}{d}\)	\(230\)
risch	\(\frac {2 d^{2} a^{3} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 d \,a^{3} c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 x}{a}-\frac {2 c}{d a}+\frac {2 b^{4} x}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{4} c}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a \,b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {b^{4} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{a d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(632\)

[In]

int(csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-2/(a^4+2*a^2*b^2+b^4)*(((-1/2*a^2*b-1/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(a^3+a*b^2)*tanh(1/2*d*x+1/2*c)^2+(1/
2*a^2*b+1/2*b^3)*tanh(1/2*d*x+1/2*c))/(1+tanh(1/2*d*x+1/2*c)^2)^2+1/4*(2*a^3+4*a*b^2)*ln(1+tanh(1/2*d*x+1/2*c)
^2)+1/2*(a^2*b+3*b^3)*arctan(tanh(1/2*d*x+1/2*c)))-b^4/a/(a^4+2*a^2*b^2+b^4)*ln(tanh(1/2*d*x+1/2*c)^2*a-2*b*ta
nh(1/2*d*x+1/2*c)-a)+1/a*ln(tanh(1/2*d*x+1/2*c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1279 vs. \(2 (157) = 314\).

Time = 0.44 (sec) , antiderivative size = 1279, normalized size of antiderivative = 7.99 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-((a^3*b + a*b^3)*cosh(d*x + c)^3 + (a^3*b + a*b^3)*sinh(d*x + c)^3 - 2*(a^4 + a^2*b^2)*cosh(d*x + c)^2 - (2*a
^4 + 2*a^2*b^2 - 3*(a^3*b + a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + ((a^3*b + 3*a*b^3)*cosh(d*x + c)^4 + 4*(a^
3*b + 3*a*b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3*b + 3*a*b^3)*sinh(d*x + c)^4 + a^3*b + 3*a*b^3 + 2*(a^3*b
+ 3*a*b^3)*cosh(d*x + c)^2 + 2*(a^3*b + 3*a*b^3 + 3*(a^3*b + 3*a*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a
^3*b + 3*a*b^3)*cosh(d*x + c)^3 + (a^3*b + 3*a*b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(
d*x + c)) - (a^3*b + a*b^3)*cosh(d*x + c) + (b^4*cosh(d*x + c)^4 + 4*b^4*cosh(d*x + c)*sinh(d*x + c)^3 + b^4*s
inh(d*x + c)^4 + 2*b^4*cosh(d*x + c)^2 + b^4 + 2*(3*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x + c)^2 + 4*(b^4*cosh(d
*x + c)^3 + b^4*cosh(d*x + c))*sinh(d*x + c))*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + (
(a^4 + 2*a^2*b^2)*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2)*sinh
(d*x + c)^4 + a^4 + 2*a^2*b^2 + 2*(a^4 + 2*a^2*b^2)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + 3*(a^4 + 2*a^2*b^2)
*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2)*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2)*cosh(d*x + c))*si
nh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - ((a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^4 +
4*(a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*sinh(d*x + c)^4 + a^4 + 2*a
^2*b^2 + b^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + 3*(a^4 + 2*a^2*b^2 + b^4
)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((a^4 + 2*a^2*b^2 + b^4)*cosh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*cosh
(d*x + c))*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) - (a^3*b + a*b^3 - 3*(a^3*b + a
*b^3)*cosh(d*x + c)^2 + 4*(a^4 + a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x
+ c)^4 + 4*(a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(d*x +
c)^4 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c)^2 + (a^5
+ 2*a^3*b^2 + a*b^4)*d)*sinh(d*x + c)^2 + (a^5 + 2*a^3*b^2 + a*b^4)*d + 4*((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d
*x + c)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(d*x + c))*sinh(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)**3/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.66 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{4} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d} + \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-b^4*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^5 + 2*a^3*b^2 + a*b^4)*d) + (a^2*b + 3*b^3)*arctan(e^
(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) - (a^3 + 2*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d
) - (b*e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*d*x - 2*c)
+ (a^2 + b^2)*e^(-4*d*x - 4*c))*d) + log(e^(-d*x - c) + 1)/(a*d) + log(e^(-d*x - c) - 1)/(a*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (157) = 314\).

Time = 0.29 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.14 \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 \, b^{5} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{5} b + 2 \, a^{3} b^{3} + a b^{5}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{2} b + 3 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {4 \, \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a} - \frac {2 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, a^{3} + 12 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \]

[In]

integrate(csch(d*x+c)*sech(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

-1/4*(4*b^5*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^5*b + 2*a^3*b^3 + a*b^5) + (pi + 2*arctan(1/2*(e
^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(a^2*b + 3*b^3)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^3 + 2*a*b^2)*log((e^(d*x + c
) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) - 4*log(abs(e^(d*x + c) - e^(-d*x - c)))/a - 2*(a^3*(e^(d*x +
c) - e^(-d*x - c))^2 + 2*a*b^2*(e^(d*x + c) - e^(-d*x - c))^2 - 2*a^2*b*(e^(d*x + c) - e^(-d*x - c)) - 2*b^3*
(e^(d*x + c) - e^(-d*x - c)) + 8*a^3 + 12*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d*x - c))^2 + 4)
))/d

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (a+b\,\mathrm {sinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

int(1/(cosh(c + d*x)^3*sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

int(1/(cosh(c + d*x)^3*sinh(c + d*x)*(a + b*sinh(c + d*x))), x)