\(\int (a+b \text {csch}(c+d x))^2 \, dx\) [72]

Optimal result

Integrand size = 12, antiderivative size = 34 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=a^2 x-\frac {2 a b \text {arctanh}(\cosh (c+d x))}{d}-\frac {b^2 \coth (c+d x)}{d} \]

[Out]

a^2*x-2*a*b*arctanh(cosh(d*x+c))/d-b^2*coth(d*x+c)/d

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3858, 3855, 3852, 8} \[ \int (a+b \text {csch}(c+d x))^2 \, dx=a^2 x-\frac {2 a b \text {arctanh}(\cosh (c+d x))}{d}-\frac {b^2 \coth (c+d x)}{d} \]

[In]

Int[(a + b*Csch[c + d*x])^2,x]

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3858

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x],
x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(34)=68\).

Time = 0.55 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=-\frac {b^2 \coth \left (\frac {1}{2} (c+d x)\right )-2 a \left (a c+a d x-2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+b^2 \tanh \left (\frac {1}{2} (c+d x)\right )}{2 d} \]

[In]

Integrate[(a + b*Csch[c + d*x])^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09

method	result	size
derivativedivides	\(\frac {a^{2} \left (d x +c \right )-4 a b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-\coth \left (d x +c \right ) b^{2}}{d}\)	\(37\)
default	\(\frac {a^{2} \left (d x +c \right )-4 a b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-\coth \left (d x +c \right ) b^{2}}{d}\)	\(37\)
parts	\(a^{2} x -\frac {b^{2} \coth \left (d x +c \right )}{d}+\frac {2 a b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(38\)
parallelrisch	\(\frac {2 a^{2} d x +4 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}-\coth \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{2 d}\)	\(56\)
risch	\(a^{2} x -\frac {2 b^{2}}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )}-\frac {2 a b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}+\frac {2 a b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}\)	\(60\)

[In]

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

[Out]

1/d*(a^2*(d*x+c)-4*a*b*arctanh(exp(d*x+c))-coth(d*x+c)*b^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (34) = 68\).

Time = 0.28 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.53 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=\frac {a^{2} d x \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d x \sinh \left (d x + c\right )^{2} - a^{2} d x - 2 \, b^{2} - 2 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - a b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + 2 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} - a b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right )}{d \cosh \left (d x + c\right )^{2} + 2 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + d \sinh \left (d x + c\right )^{2} - d} \]

[In]

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

[Out]

(a^2*d*x*cosh(d*x + c)^2 + 2*a^2*d*x*cosh(d*x + c)*sinh(d*x + c) + a^2*d*x*sinh(d*x + c)^2 - a^2*d*x - 2*b^2 -
2*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 - a*b)*log(cosh(d*x + c) + s
inh(d*x + c) + 1) + 2*(a*b*cosh(d*x + c)^2 + 2*a*b*cosh(d*x + c)*sinh(d*x + c) + a*b*sinh(d*x + c)^2 - a*b)*lo
g(cosh(d*x + c) + sinh(d*x + c) - 1))/(d*cosh(d*x + c)^2 + 2*d*cosh(d*x + c)*sinh(d*x + c) + d*sinh(d*x + c)^2
- d)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*csch(c + d*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=a^{2} x + \frac {2 \, a b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {2 \, b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.74 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=\frac {{\left (d x + c\right )} a^{2} - 2 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, b^{2}}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 2.30 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.18 \[ \int (a+b \text {csch}(c+d x))^2 \, dx=a^2\,x-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}} \]

[In]

int((a + b/sinh(c + d*x))^2,x)

[Out]

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