\(\int (a+b \text {csch}^2(c+d x))^2 \, dx\) [3]

Optimal result

Integrand size = 14, antiderivative size = 43 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=a^2 x-\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \]

[Out]

a^2*x-(2*a-b)*b*coth(d*x+c)/d-1/3*b^2*coth(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4213, 398, 212} \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=a^2 x-\frac {b (2 a-b) \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \]

[In]

Int[(a + b*Csch[c + d*x]^2)^2,x]

[Out]

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

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 398

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)
^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILt
Q[q, 0] && GeQ[p, -q]

Rule 4213

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist
[ff/f, Subst[Int[(a + b + b*ff^2*x^2)^p/(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p},
x] && NeQ[a + b, 0] && NeQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a-b+b x^2\right )^2}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-((2 a-b) b)-b^2 x^2+\frac {a^2}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d} \\ & = -\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d} \\ & = a^2 x-\frac {(2 a-b) b \coth (c+d x)}{d}-\frac {b^2 \coth ^3(c+d x)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.95 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=\frac {4 \left (a+b \text {csch}^2(c+d x)\right )^2 \left (3 a^2 (c+d x)-b \coth (c+d x) \left (6 a-2 b+b \text {csch}^2(c+d x)\right )\right ) \sinh ^4(c+d x)}{3 d (a-2 b-a \cosh (2 (c+d x)))^2} \]

[In]

Integrate[(a + b*Csch[c + d*x]^2)^2,x]

[Out]

(4*(a + b*Csch[c + d*x]^2)^2*(3*a^2*(c + d*x) - b*Coth[c + d*x]*(6*a - 2*b + b*Csch[c + d*x]^2))*Sinh[c + d*x]
^4)/(3*d*(a - 2*b - a*Cosh[2*(c + d*x)])^2)

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.05

method	result	size
parts	\(a^{2} x +\frac {b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}-\frac {2 a b \coth \left (d x +c \right )}{d}\)	\(45\)
derivativedivides	\(\frac {a^{2} \left (d x +c \right )-2 \coth \left (d x +c \right ) a b +b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\)	\(47\)
default	\(\frac {a^{2} \left (d x +c \right )-2 \coth \left (d x +c \right ) a b +b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}\)	\(47\)
risch	\(a^{2} x -\frac {4 b \left (3 a \,{\mathrm e}^{4 d x +4 c}-6 \,{\mathrm e}^{2 d x +2 c} a +3 b \,{\mathrm e}^{2 d x +2 c}+3 a -b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\)	\(69\)
parallelrisch	\(\frac {-\coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}+\left (-24 a b +9 b^{2}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b^{2}+\left (-24 a b +9 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+24 a^{2} d x}{24 d}\)	\(86\)

[In]

int((a+b*csch(d*x+c)^2)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (41) = 82\).

Time = 0.28 (sec) , antiderivative size = 180, normalized size of antiderivative = 4.19 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=-\frac {2 \, {\left (3 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{3} + 6 \, {\left (3 \, a b - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} - {\left (3 \, a^{2} d x + 6 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (a b - b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (3 \, a^{2} d x - {\left (3 \, a^{2} d x + 6 \, a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 6 \, a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \sinh \left (d x + c\right )^{3} + 3 \, {\left (d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )\right )}} \]

[In]

integrate((a+b*csch(d*x+c)^2)^2,x, algorithm="fricas")

[Out]

-1/3*(2*(3*a*b - b^2)*cosh(d*x + c)^3 + 6*(3*a*b - b^2)*cosh(d*x + c)*sinh(d*x + c)^2 - (3*a^2*d*x + 6*a*b - 2
*b^2)*sinh(d*x + c)^3 - 6*(a*b - b^2)*cosh(d*x + c) + 3*(3*a^2*d*x - (3*a^2*d*x + 6*a*b - 2*b^2)*cosh(d*x + c)
^2 + 6*a*b - 2*b^2)*sinh(d*x + c))/(d*sinh(d*x + c)^3 + 3*(d*cosh(d*x + c)^2 - d)*sinh(d*x + c))

Sympy [F]

\[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{2}\, dx \]

[In]

integrate((a+b*csch(d*x+c)**2)**2,x)

[Out]

Integral((a + b*csch(c + d*x)**2)**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (41) = 82\).

Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.81 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=a^{2} x + \frac {4}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {4 \, a b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]

[In]

integrate((a+b*csch(d*x+c)^2)^2,x, algorithm="maxima")

[Out]

a^2*x + 4/3*b^2*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(
d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) + 4*a*b/(d*(e^(-2*d*x - 2*c) - 1))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.88 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (d x + c\right )} a^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 3 \, a b - b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]

[In]

integrate((a+b*csch(d*x+c)^2)^2,x, algorithm="giac")

[Out]

1/3*(3*(d*x + c)*a^2 - 4*(3*a*b*e^(4*d*x + 4*c) - 6*a*b*e^(2*d*x + 2*c) + 3*b^2*e^(2*d*x + 2*c) + 3*a*b - b^2)
/(e^(2*d*x + 2*c) - 1)^3)/d

Mupad [B] (verification not implemented)

Time = 2.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 3.86 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^2 \, dx=a^2\,x-\frac {\frac {4\,a\,b}{3\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-b^2\right )}{3\,d}+\frac {4\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\frac {4\,\left (a\,b-b^2\right )}{3\,d}-\frac {4\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {4\,a\,b}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]

[In]

int((a + b/sinh(c + d*x)^2)^2,x)

[Out]

a^2*x - ((4*a*b)/(3*d) - (8*exp(2*c + 2*d*x)*(a*b - b^2))/(3*d) + (4*a*b*exp(4*c + 4*d*x))/(3*d))/(3*exp(2*c +
2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) + ((4*(a*b - b^2))/(3*d) - (4*a*b*exp(2*c + 2*d*x))/(3*d)
)/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - (4*a*b)/(3*d*(exp(2*c + 2*d*x) - 1))