\(\int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [501]

Optimal result

Integrand size = 29, antiderivative size = 211 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \]

[Out]

1/2*b*arctan(sinh(d*x+c))/(a^2+b^2)/d+b*(a^2+2*b^2)*arctan(sinh(d*x+c))/(a^2+b^2)^2/d+b*csch(d*x+c)/a^2/d-1/2*
csch(d*x+c)^2/a/d+a*(2*a^2+3*b^2)*ln(cosh(d*x+c))/(a^2+b^2)^2/d-(2*a^2-b^2)*ln(sinh(d*x+c))/a^3/d-b^6*ln(a+b*s
inh(d*x+c))/a^3/(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 = 0.25 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2916, 12, 908, 653, 209, 649, 266} \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {b \arctan (\sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 d \left (a^2+b^2\right )}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 d \left (a^2+b^2\right )^2}-\frac {\text {csch}^2(c+d x)}{2 a d} \]

[In]

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

[Out]

(b*ArcTan[Sinh[c + d*x]])/(2*(a^2 + b^2)*d) + (b*(a^2 + 2*b^2)*ArcTan[Sinh[c + d*x]])/((a^2 + b^2)^2*d) + (b*C
sch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + (a*(2*a^2 + 3*b^2)*Log[Cosh[c + d*x]])/((a^2 + b^2)^2*d) - (
(2*a^2 - b^2)*Log[Sinh[c + d*x]])/(a^3*d) - (b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(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^3}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b^6 \text {Subst}\left (\int \frac {1}{x^3 (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b^6 \text {Subst}\left (\int \left (\frac {1}{a b^4 x^3}-\frac {1}{a^2 b^4 x^2}+\frac {-2 a^2+b^2}{a^3 b^6 x}-\frac {1}{a^3 \left (a^2+b^2\right )^2 (a+x)}+\frac {b^2+a x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac {b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 b^2\right ) x}{b^6 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = \frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {b^2 \left (a^2+2 b^2\right )+a \left (2 a^2+3 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 {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \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^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 (b^2 \left (a^2+2 b^2\right )\right ) \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 {\left (a \left (2 a^2+3 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 \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right ) d}+\frac {b \left (a^2+2 b^2\right ) \arctan (\sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {a \left (2 a^2+3 b^2\right ) \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {b^6 \log (a+b \sinh (c+d x))}{a^3 \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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b \arctan (\sinh (c+d x))}{a^2+b^2}+\frac {2 b \text {csch}(c+d x)}{a^2}-\frac {\text {csch}^2(c+d x)}{a}+\frac {(a-i b) \left (2 a^2+i a b+2 b^2\right ) \log (i-\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 \left (2 a^2-b^2\right ) \log (\sinh (c+d x))}{a^3}+\frac {(a+i b) \left (2 a^2-i a b+2 b^2\right ) \log (i+\sinh (c+d x))}{\left (a^2+b^2\right )^2}-\frac {2 b^6 \log (a+b \sinh (c+d x))}{a^3 \left (a^2+b^2\right )^2}-\frac {a \text {sech}^2(c+d x)}{a^2+b^2}+\frac {b \text {sech}(c+d x) \tanh (c+d x)}{a^2+b^2}}{2 d} \]

[In]

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

[Out]

((b*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) + (2*b*Csch[c + d*x])/a^2 - Csch[c + d*x]^2/a + ((a - I*b)*(2*a^2 + I*a
*b + 2*b^2)*Log[I - Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*(2*a^2 - b^2)*Log[Sinh[c + d*x]])/a^3 + ((a + I*b)*(2*a
^2 - I*a*b + 2*b^2)*Log[I + Sinh[c + d*x]])/(a^2 + b^2)^2 - (2*b^6*Log[a + b*Sinh[c + d*x]])/(a^3*(a^2 + b^2)^
2) - (a*Sech[c + d*x]^2)/(a^2 + b^2) + (b*Sech[c + d*x]*Tanh[c + d*x])/(a^2 + b^2))/(2*d)

Maple [A] (verified)

Time = 54.09 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.38

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{6} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(292\)
default	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {b^{6} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{\left (a^{2}+b^{2}\right )^{2} a^{3}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (a^{3}+a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a^{3}+6 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{2}+\left (3 a^{2} b +5 b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(292\)
risch	\(-\frac {4 d^{2} a^{3} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {4 d \,a^{3} c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {6 a \,b^{2} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {4 x}{a}+\frac {4 c}{d a}-\frac {2 b^{2} x}{a^{3}}-\frac {2 b^{2} c}{d \,a^{3}}+\frac {2 b^{6} x}{a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 b^{6} c}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {{\mathrm e}^{d x +c} \left (-3 a^{2} b \,{\mathrm e}^{6 d x +6 c}-2 b^{3} {\mathrm e}^{6 d x +6 c}+4 a^{3} {\mathrm e}^{5 d x +5 c}+2 \,{\mathrm e}^{5 d x +5 c} a \,b^{2}+a^{2} b \,{\mathrm e}^{4 d x +4 c}-2 \,{\mathrm e}^{4 d x +4 c} b^{3}+4 \,{\mathrm e}^{3 d x +3 c} a \,b^{2}-{\mathrm e}^{2 d x +2 c} a^{2} b +2 \,{\mathrm e}^{2 d x +2 c} b^{3}+4 \,{\mathrm e}^{d x +c} a^{3}+2 b^{2} a \,{\mathrm e}^{d x +c}+3 a^{2} b +2 b^{3}\right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2} a^{2} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {3 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 ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {2 \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \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 {5 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 {5 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 {2 \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {3 \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 {2 \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}-1\right )}{d \,a^{3}}-\frac {b^{6} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \,a^{3} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(835\)

[In]

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

[Out]

1/d*(-1/4/a^2*(1/2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c))-b^6/(a^2+b^2)^2/a^3*ln(tanh(1/2*d*x+1/2*c)
^2*a-2*b*tanh(1/2*d*x+1/2*c)-a)-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^3*(-8*a^2+4*b^2)*ln(tanh(1/2*d*x+1/2*c))+1/2
*b/a^2/tanh(1/2*d*x+1/2*c)+2/(a^2+b^2)^2*(((-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*(4*a^3+6*a*b^2)*ln(1+tanh(1
/2*d*x+1/2*c)^2)+1/2*(3*a^2*b+5*b^3)*arctan(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. 3148 vs. \(2 (206) = 412\).

Time = 0.65 (sec) , antiderivative size = 3148, normalized size of antiderivative = 14.92 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^7 + (3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*sinh(d*x + c)^7 - 2*(2*a^6
+ 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^6 - (4*a^6 + 6*a^4*b^2 + 2*a^2*b^4 - 7*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*co
sh(d*x + c))*sinh(d*x + c)^6 - (a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^5 - (a^5*b - a^3*b^3 - 2*a*b^5 - 21*(
3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^2 + 12*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)
^5 - 4*(a^4*b^2 + a^2*b^4)*cosh(d*x + c)^4 - (4*a^4*b^2 + 4*a^2*b^4 - 35*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(
d*x + c)^3 + 30*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^2 + 5*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c))*s
inh(d*x + c)^4 + (a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^3 + (a^5*b - a^3*b^3 - 2*a*b^5 + 35*(3*a^5*b + 5*a^
3*b^3 + 2*a*b^5)*cosh(d*x + c)^4 - 40*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^3 - 10*(a^5*b - a^3*b^3 - 2*
a*b^5)*cosh(d*x + c)^2 - 16*(a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 - 2*(2*a^6 + 3*a^4*b^2 + a^2*b^
4)*cosh(d*x + c)^2 - (4*a^6 + 6*a^4*b^2 + 2*a^2*b^4 - 21*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c)^5 + 30*
(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^4 + 10*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^3 + 24*(a^4*b^2 +
a^2*b^4)*cosh(d*x + c)^2 - 3*(a^5*b - a^3*b^3 - 2*a*b^5)*cosh(d*x + c))*sinh(d*x + c)^2 + ((3*a^5*b + 5*a^3*b
^3)*cosh(d*x + c)^8 + 56*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(3*a^5*b + 5*a^3*b^3)*cosh
(d*x + c)^2*sinh(d*x + c)^6 + 8*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a^5*b + 5*a^3*b^3)*si
nh(d*x + c)^8 + 3*a^5*b + 5*a^3*b^3 - 2*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^4 - 2*(3*a^5*b + 5*a^3*b^3 - 35*(3
*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^4)*sinh(d*x + c)^4 + 8*(7*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^5 - (3*a^5*b +
5*a^3*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^6 - 3*(3*a^5*b + 5*a^3*b
^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((3*a^5*b + 5*a^3*b^3)*cosh(d*x + c)^7 - (3*a^5*b + 5*a^3*b^3)*cosh(d
*x + c)^3)*sinh(d*x + c))*arctan(cosh(d*x + c) + sinh(d*x + c)) - (3*a^5*b + 5*a^3*b^3 + 2*a*b^5)*cosh(d*x + c
) - (b^6*cosh(d*x + c)^8 + 56*b^6*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*b^6*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8
*b^6*cosh(d*x + c)*sinh(d*x + c)^7 + b^6*sinh(d*x + c)^8 - 2*b^6*cosh(d*x + c)^4 + b^6 + 2*(35*b^6*cosh(d*x +
c)^4 - b^6)*sinh(d*x + c)^4 + 8*(7*b^6*cosh(d*x + c)^5 - b^6*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^6*cosh(d*
x + c)^6 - 3*b^6*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(b^6*cosh(d*x + c)^7 - b^6*cosh(d*x + c)^3)*sinh(d*x + c
))*log(2*(b*sinh(d*x + c) + a)/(cosh(d*x + c) - sinh(d*x + c))) + ((2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^8 + 56*(2
*a^6 + 3*a^4*b^2)*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8
*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)*sinh(d*x + c)^7 + (2*a^6 + 3*a^4*b^2)*sinh(d*x + c)^8 + 2*a^6 + 3*a^4*b^2 -
2*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^4 - 2*(2*a^6 + 3*a^4*b^2 - 35*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^4)*sinh(d
*x + c)^4 + 8*(7*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^5 - (2*a^6 + 3*a^4*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*
(7*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^6 - 3*(2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*((2*a^6 +
3*a^4*b^2)*cosh(d*x + c)^7 - (2*a^6 + 3*a^4*b^2)*cosh(d*x + c)^3)*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x
+ c) - sinh(d*x + c))) - ((2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^8 + 56*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x +
c)^3*sinh(d*x + c)^5 + 28*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(2*a^6 + 3*a^4*b^2 -
b^6)*cosh(d*x + c)*sinh(d*x + c)^7 + (2*a^6 + 3*a^4*b^2 - b^6)*sinh(d*x + c)^8 + 2*a^6 + 3*a^4*b^2 - b^6 - 2*(
2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^4 - 2*(2*a^6 + 3*a^4*b^2 - b^6 - 35*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c)^4)*sinh(d*x + c)^4 + 8*(7*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^5 - (2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c))*sinh(d*x + c)^3 + 4*(7*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^6 - 3*(2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x
+ c)^2)*sinh(d*x + c)^2 + 8*((2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x + c)^7 - (2*a^6 + 3*a^4*b^2 - b^6)*cosh(d*x +
c)^3)*sinh(d*x + c))*log(2*sinh(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + (7*(3*a^5*b + 5*a^3*b^3 + 2*a*b^5)
*cosh(d*x + c)^6 - 3*a^5*b - 5*a^3*b^3 - 2*a*b^5 - 12*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c)^5 - 5*(a^5*b
- a^3*b^3 - 2*a*b^5)*cosh(d*x + c)^4 - 16*(a^4*b^2 + a^2*b^4)*cosh(d*x + c)^3 + 3*(a^5*b - a^3*b^3 - 2*a*b^5)
*cosh(d*x + c)^2 - 4*(2*a^6 + 3*a^4*b^2 + a^2*b^4)*cosh(d*x + c))*sinh(d*x + c))/((a^7 + 2*a^5*b^2 + a^3*b^4)*
d*cosh(d*x + c)^8 + 56*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*(a^7 + 2*a^5*b^2 + a
^3*b^4)*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)*sinh(d*x + c)^7 + (a
^7 + 2*a^5*b^2 + a^3*b^4)*d*sinh(d*x + c)^8 - 2*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^4 + 2*(35*(a^7 + 2
*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^4 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d)*sinh(d*x + c)^4 + 8*(7*(a^7 + 2*a^5*b^2
+ a^3*b^4)*d*cosh(d*x + c)^5 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*(a^7 + 2*a
^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^6 - 3*(a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + (a^7
+ 2*a^5*b^2 + a^3*b^4)*d + 8*((a^7 + 2*a^5*b^2 + a^3*b^4)*d*cosh(d*x + c)^7 - (a^7 + 2*a^5*b^2 + a^3*b^4)*d*co
sh(d*x + c)^3)*sinh(d*x + c))

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (206) = 412\).

Time = 0.32 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.98 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b^{6} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{7} + 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d} - \frac {{\left (3 \, a^{2} b + 5 \, b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {4 \, a b^{2} e^{\left (-4 \, d x - 4 \, c\right )} - {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-d x - c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - {\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )} + 2 \, {\left (2 \, a^{3} + a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{{\left (a^{4} + a^{2} b^{2} - 2 \, {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{4} + a^{2} b^{2}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} - \frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \]

[In]

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

[Out]

-b^6*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^7 + 2*a^5*b^2 + a^3*b^4)*d) - (3*a^2*b + 5*b^3)*arcta
n(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) + (2*a^3 + 3*a*b^2)*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 +
b^4)*d) - (4*a*b^2*e^(-4*d*x - 4*c) - (3*a^2*b + 2*b^3)*e^(-d*x - c) + 2*(2*a^3 + a*b^2)*e^(-2*d*x - 2*c) + (a
^2*b - 2*b^3)*e^(-3*d*x - 3*c) - (a^2*b - 2*b^3)*e^(-5*d*x - 5*c) + 2*(2*a^3 + a*b^2)*e^(-6*d*x - 6*c) + (3*a^
2*b + 2*b^3)*e^(-7*d*x - 7*c))/((a^4 + a^2*b^2 - 2*(a^4 + a^2*b^2)*e^(-4*d*x - 4*c) + (a^4 + a^2*b^2)*e^(-8*d*
x - 8*c))*d) - (2*a^2 - b^2)*log(e^(-d*x - c) + 1)/(a^3*d) - (2*a^2 - b^2)*log(e^(-d*x - c) - 1)/(a^3*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 464 vs. \(2 (206) = 412\).

Time = 0.30 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.20 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 \, b^{7} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{7} b + 2 \, a^{5} b^{3} + a^{3} b^{5}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{2} b + 5 \, b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (2 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 12 \, a^{3} + 16 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}} + \frac {4 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {2 \, {\left (6 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a^{2}\right )}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{4 \, d} \]

[In]

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

[Out]

-1/4*(4*b^7*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^7*b + 2*a^5*b^3 + a^3*b^5) - (pi + 2*arctan(1/2*
(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a^2*b + 5*b^3)/(a^4 + 2*a^2*b^2 + b^4) - 2*(2*a^3 + 3*a*b^2)*log((e^(d
*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2*b^2 + b^4) + 2*(2*a^3*(e^(d*x + c) - e^(-d*x - c))^2 + 3*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)) + 12*a
^3 + 16*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d*x - c))^2 + 4)) + 4*(2*a^2 - b^2)*log(abs(e^(d*x
+ c) - e^(-d*x - c)))/a^3 - 2*(6*a^2*(e^(d*x + c) - e^(-d*x - c))^2 - 3*b^2*(e^(d*x + c) - e^(-d*x - c))^2 +
4*a*b*(e^(d*x + c) - e^(-d*x - c)) - 4*a^2)/(a^3*(e^(d*x + c) - e^(-d*x - c))^2))/d

Mupad [B] (verification not implemented)

Time = 8.27 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.63 \[ \int \frac {\text {csch}^3(c+d x) \text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4\,b^5}{a\,d\,\left (a^2\,b^3+b^5\right )}-\frac {4\,b^4\,{\mathrm {e}}^{3\,c+3\,d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^4\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2\,b^3+b^5\right )}+\frac {4\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^2+b^2\right )}{a\,d\,\left (a^2\,b^3+b^5\right )}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {4\,\left (a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (2\,a^4\,b^3+3\,a^2\,b^5+b^7\right )}{a\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,\left (3\,a^4\,b^4+5\,a^2\,b^6+2\,b^8\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}-\frac {b^4\,{\mathrm {e}}^{c+d\,x}\,\left (-a^4+a^2\,b^2+2\,b^4\right )}{a^2\,d\,\left (a^2\,b^3+b^5\right )\,\left (a^2+b^2\right )}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (4\,a+b\,5{}\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}-\frac {b^6\,\ln \left (2\,a\,{\mathrm {e}}^{c+d\,x}-b+b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{d\,a^7+2\,d\,a^5\,b^2+d\,a^3\,b^4}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (5\,b+a\,4{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )}-\frac {\ln \left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (2\,a^2-b^2\right )}{a^3\,d} \]

[In]

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

[Out]

(log(exp(c + d*x)*1i + 1)*(4*a + b*5i))/(2*(a^2*d - b^2*d + a*b*d*2i)) - ((4*(b^7 + a^2*b^5))/(a*d*(b^5 + a^2*
b^3)*(a^2 + b^2)) + (2*exp(2*c + 2*d*x)*(b^7 + 3*a^2*b^5 + 2*a^4*b^3))/(a*d*(b^5 + a^2*b^3)*(a^2 + b^2)) - (ex
p(3*c + 3*d*x)*(2*b^8 + 5*a^2*b^6 + 3*a^4*b^4))/(a^2*d*(b^5 + a^2*b^3)*(a^2 + b^2)) - (b^4*exp(c + d*x)*(2*b^4
- a^4 + a^2*b^2))/(a^2*d*(b^5 + a^2*b^3)*(a^2 + b^2)))/(exp(4*c + 4*d*x) - 1) - ((4*b^5)/(a*d*(b^5 + a^2*b^3)
) - (4*b^4*exp(3*c + 3*d*x))/(d*(b^5 + a^2*b^3)) + (4*b^4*exp(c + d*x))/(d*(b^5 + a^2*b^3)) + (4*b^3*exp(2*c +
2*d*x)*(2*a^2 + b^2))/(a*d*(b^5 + a^2*b^3)))/(exp(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - (b^6*log(2*a*exp(c
+ d*x) - b + b*exp(2*c + 2*d*x)))/(a^7*d + a^3*b^4*d + 2*a^5*b^2*d) + (log(exp(c + d*x) + 1i)*(a*4i + 5*b))/(
2*(a^2*d*1i - b^2*d*1i + 2*a*b*d)) - (log(exp(2*c + 2*d*x) - 1)*(2*a^2 - b^2))/(a^3*d)