\(\int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) [489]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {csch}(c+d x)}{a^2 d}-\frac {\text {csch}^2(c+d x)}{2 a d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d} \]

[Out]

b*csch(d*x+c)/a^2/d-1/2*csch(d*x+c)^2/a/d+(a^2+b^2)*ln(sinh(d*x+c))/a^3/d-(a^2+b^2)*ln(a+b*sinh(d*x+c))/a^3/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2800, 908} \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {csch}(c+d x)}{a^2 d}+\frac {\left (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d}-\frac {\text {csch}^2(c+d x)}{2 a d} \]

[In]

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

[Out]

(b*Csch[c + d*x])/(a^2*d) - Csch[c + d*x]^2/(2*a*d) + ((a^2 + b^2)*Log[Sinh[c + d*x]])/(a^3*d) - ((a^2 + b^2)*
Log[a + b*Sinh[c + d*x]])/(a^3*d)

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 2800

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {-b^2-x^2}{x^3 (a+x)} \, dx,x,b \sinh (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {b^2}{a x^3}+\frac {b^2}{a^2 x^2}+\frac {-a^2-b^2}{a^3 x}+\frac {a^2+b^2}{a^3 (a+x)}\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 (a^2+b^2\right ) \log (\sinh (c+d x))}{a^3 d}-\frac {\left (a^2+b^2\right ) \log (a+b \sinh (c+d x))}{a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 a b \text {csch}(c+d x)-a^2 \text {csch}^2(c+d x)+2 \left (a^2+b^2\right ) (\log (\sinh (c+d x))-\log (a+b \sinh (c+d x)))}{2 a^3 d} \]

[In]

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

[Out]

(2*a*b*Csch[c + d*x] - a^2*Csch[c + d*x]^2 + 2*(a^2 + b^2)*(Log[Sinh[c + d*x]] - Log[a + b*Sinh[c + d*x]]))/(2
*a^3*d)

Maple [A] (verified)

Time = 1.42 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.79

method	result	size
derivativedivides	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{3}}}{d}\)	\(143\)
default	\(\frac {-\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (4 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {\left (-4 a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{4 a^{3}}}{d}\)	\(143\)
risch	\(-\frac {2 \,{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+b \right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}+\frac {b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right ) b^{2}}{d \,a^{3}}\)	\(159\)

[In]

int(coth(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))-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^3*(4*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)+1/4/a^3*(-4*a^2-4*b^2)*ln(tanh(1/2*d*x+1/2*c)^2*
a-2*b*tanh(1/2*d*x+1/2*c)-a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 617 vs. \(2 (78) = 156\).

Time = 0.27 (sec) , antiderivative size = 617, normalized size of antiderivative = 7.71 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 \, a b \cosh \left (d x + c\right )^{3} + 2 \, a b \sinh \left (d x + c\right )^{3} - 2 \, a^{2} \cosh \left (d x + c\right )^{2} - 2 \, a b \cosh \left (d x + c\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right ) - a^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \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 ) + {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} + b^{2} + 4 \, {\left ({\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} + b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (3 \, a b \cosh \left (d x + c\right )^{2} - 2 \, a^{2} \cosh \left (d x + c\right ) - a b\right )} \sinh \left (d x + c\right )}{a^{3} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} d \sinh \left (d x + c\right )^{4} - 2 \, a^{3} d \cosh \left (d x + c\right )^{2} + a^{3} d + 2 \, {\left (3 \, a^{3} d \cosh \left (d x + c\right )^{2} - a^{3} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} d \cosh \left (d x + c\right )^{3} - a^{3} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

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

[Out]

Integral(coth(c + d*x)**3/(a + b*sinh(c + d*x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (78) = 156\).

Time = 0.20 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.16 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {2 \, {\left (b e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} + b^{2}\right )} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a^{2} + b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \]

[In]

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

[Out]

-2*(b*e^(-d*x - c) - a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x - 4*c)
- a^2)*d) - (a^2 + b^2)*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/(a^3*d) + (a^2 + b^2)*log(e^(-d*x - c)
+ 1)/(a^3*d) + (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. 184 vs. \(2 (78) = 156\).

Time = 0.34 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.30 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2 \, {\left (a^{2} + b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - e^{\left (-d x - c\right )} \right |}\right )}{a^{3}} - \frac {2 \, {\left (a^{2} b + b^{3}\right )} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{3} b} - \frac {3 \, a^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} - 4 \, a b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{2}}{a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}}}{2 \, d} \]

[In]

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

[Out]

1/2*(2*(a^2 + b^2)*log(abs(e^(d*x + c) - e^(-d*x - c)))/a^3 - 2*(a^2*b + b^3)*log(abs(b*(e^(d*x + c) - e^(-d*x
- c)) + 2*a))/(a^3*b) - (3*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 = 1.77 (sec) , antiderivative size = 1329, normalized size of antiderivative = 16.61 \[ \int \frac {\coth ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*atan((a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^
(1/2))/(2*a^3*d*(a^2 + b^2)^2) + ((a^7*d + a^5*b^2*d)*(-a^6*d^2)^(1/2))/(2*a^6*d^2*((a^2 + b^2)^2)^(1/2)*(a^2
+ b^2)) - (a^6*b^2*exp(2*c)*exp(2*d*x)*(-a^6*d^2)^(1/2)*((4*(a^2 + 2*b^2)*(a^4 + b^4 + 2*a^2*b^2))/(a^9*b^2*d*
(a^2 + b^2)^2) + (2*(2*a^4*b^3*d + 2*a^6*b*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*d^2*((a^2 + b^2)^2)^(1/
2)*(a^2 + b^2)) + (4*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 +
2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^9*b^2*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2)) + (4*(a^7*d + a^5
*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^12*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2))))/(8*(a^4 + b^4 + 2*a^
2*b^2)^(1/2)) + (a^6*b^2*exp(3*c)*exp(3*d*x)*((2*(a^7*d + a^5*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^11*b^3*
d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2)
+ 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10*b^3*d*(a^2 + b^2)
^2*(-a^6*d^2)^(1/2)))*(-a^6*d^2)^(1/2))/(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2)) - (a^6*b^2*exp(d*x)*exp(c)*(-a^6*d^2
)^(1/2)*((8*(a^4 + b^4 + 2*a^2*b^2))/(a^8*b*d*(a^2 + b^2)^2) - (4*(2*a^4*b^3*d + 2*a^6*b*d)*(a^4 + b^4 + 2*a^2
*b^2)^(1/2))/(a^12*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) + (2*(a^7*d + a^5*b^2*d)*(a^4 + b^4 + 2*a^2*b^2)
^(1/2))/(a^11*b^3*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)) - (2*(a^2 + 2*b^2)*(a^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 +
2*a^2*b^2)^(1/2) + 2*b^2*(-a^6*d^2)^(1/2)*(a^4 + b^4 + 2*a^2*b^2)^(1/2))*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(a^10
*b^3*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2))))/(8*(a^4 + b^4 + 2*a^2*b^2)^(1/2))) - 2*atan((4*a^6*b*d*(a^2 + b^2)^2*
(-a^6*d^2)^(1/2) + 4*a^4*b^3*d*(a^2 + b^2)^2*(-a^6*d^2)^(1/2))*(1/(8*a^5*b*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^
2)^3) - exp(d*x)*exp(c)*(1/(16*a^4*b^2*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3) - (a^2 + 2*b^2)^2/(16*a^8*b^2*
d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3)) + (a^2 + 2*b^2)/(8*a^7*b*d^2*((a^2 + b^2)^2)^(1/2)*(a^2 + b^2)^3))))
*(a^4 + b^4 + 2*a^2*b^2)^(1/2))/(-a^6*d^2)^(1/2) - (2/(a*d) - (2*b*exp(c + d*x))/(a^2*d))/(exp(2*c + 2*d*x) -
1) - 2/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))