\(\int (a+b \text {csch}(c+d x))^3 \, dx\) [71]

Optimal result

Integrand size = 12, antiderivative size = 75 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=a^3 x-\frac {b \left (6 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {5 a b^2 \coth (c+d x)}{2 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d} \]

[Out]

a^3*x-1/2*b*(6*a^2-b^2)*arctanh(cosh(d*x+c))/d-5/2*a*b^2*coth(d*x+c)/d-1/2*b^2*coth(d*x+c)*(a+b*csch(d*x+c))/d

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3867, 3855, 3852, 8} \[ \int (a+b \text {csch}(c+d x))^3 \, dx=a^3 x-\frac {b \left (6 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {5 a b^2 \coth (c+d x)}{2 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d} \]

[In]

Int[(a + b*Csch[c + d*x])^3,x]

[Out]

a^3*x - (b*(6*a^2 - b^2)*ArcTanh[Cosh[c + d*x]])/(2*d) - (5*a*b^2*Coth[c + d*x])/(2*d) - (b^2*Coth[c + d*x]*(a
+ b*Csch[c + d*x]))/(2*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 3867

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*Cot[c + d*x]*((a + b*Csc[c + d*x])^(
n - 2)/(d*(n - 1))), x] + Dist[1/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2)
+ 3*a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = -\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d}+\frac {1}{2} \int \left (2 a^3+b \left (6 a^2-b^2\right ) \text {csch}(c+d x)+5 a b^2 \text {csch}^2(c+d x)\right ) \, dx \\ & = a^3 x-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d}+\frac {1}{2} \left (5 a b^2\right ) \int \text {csch}^2(c+d x) \, dx+\frac {1}{2} \left (b \left (6 a^2-b^2\right )\right ) \int \text {csch}(c+d x) \, dx \\ & = a^3 x-\frac {b \left (6 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d}-\frac {\left (5 i a b^2\right ) \text {Subst}(\int 1 \, dx,x,-i \coth (c+d x))}{2 d} \\ & = a^3 x-\frac {b \left (6 a^2-b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {5 a b^2 \coth (c+d x)}{2 d}-\frac {b^2 \coth (c+d x) (a+b \text {csch}(c+d x))}{2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(75)=150\).

Time = 5.87 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.01 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=-\frac {-8 a^3 c-8 a^3 d x+12 a b^2 \coth \left (\frac {1}{2} (c+d x)\right )+b^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+24 a^2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-4 b^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )-24 a^2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+4 b^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+b^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+12 a b^2 \tanh \left (\frac {1}{2} (c+d x)\right )}{8 d} \]

[In]

Integrate[(a + b*Csch[c + d*x])^3,x]

[Out]

-1/8*(-8*a^3*c - 8*a^3*d*x + 12*a*b^2*Coth[(c + d*x)/2] + b^3*Csch[(c + d*x)/2]^2 + 24*a^2*b*Log[Cosh[(c + d*x
)/2]] - 4*b^3*Log[Cosh[(c + d*x)/2]] - 24*a^2*b*Log[Sinh[(c + d*x)/2]] + 4*b^3*Log[Sinh[(c + d*x)/2]] + b^3*Se
ch[(c + d*x)/2]^2 + 12*a*b^2*Tanh[(c + d*x)/2])/d

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {a^{3} \left (d x +c \right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-3 \coth \left (d x +c \right ) a \,b^{2}+b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(66\)
default	\(\frac {a^{3} \left (d x +c \right )-6 a^{2} b \,\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )-3 \coth \left (d x +c \right ) a \,b^{2}+b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}\)	\(66\)
parts	\(a^{3} x +\frac {b^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )}{d}-\frac {3 a \,b^{2} \coth \left (d x +c \right )}{d}+\frac {3 a^{2} b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\)	\(70\)
parallelrisch	\(\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3}-\coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3}+8 a^{3} d x +24 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -4 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-12 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}-12 \coth \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{8 d}\)	\(106\)
risch	\(a^{3} x -\frac {b^{2} \left ({\mathrm e}^{3 d x +3 c} b +6 \,{\mathrm e}^{2 d x +2 c} a +{\mathrm e}^{d x +c} b -6 a \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}-\frac {3 b \ln \left ({\mathrm e}^{d x +c}+1\right ) a^{2}}{d}+\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}+\frac {3 b \ln \left ({\mathrm e}^{d x +c}-1\right ) a^{2}}{d}-\frac {b^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}\)	\(133\)

[In]

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

[Out]

1/d*(a^3*(d*x+c)-6*a^2*b*arctanh(exp(d*x+c))-3*coth(d*x+c)*a*b^2+b^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp
(d*x+c))))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 769 vs. \(2 (69) = 138\).

Time = 0.27 (sec) , antiderivative size = 769, normalized size of antiderivative = 10.25 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=\frac {2 \, a^{3} d x \cosh \left (d x + c\right )^{4} + 2 \, a^{3} d x \sinh \left (d x + c\right )^{4} - 2 \, b^{3} \cosh \left (d x + c\right )^{3} + 2 \, a^{3} d x - 2 \, b^{3} \cosh \left (d x + c\right ) + 2 \, {\left (4 \, a^{3} d x \cosh \left (d x + c\right ) - b^{3}\right )} \sinh \left (d x + c\right )^{3} + 12 \, a b^{2} - 4 \, {\left (a^{3} d x + 3 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (6 \, a^{3} d x \cosh \left (d x + c\right )^{2} - 2 \, a^{3} d x - 3 \, b^{3} \cosh \left (d x + c\right ) - 6 \, a b^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left ({\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b - b^{3}\right )} \sinh \left (d x + c\right )^{4} + 6 \, a^{2} b - b^{3} - 2 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} - 2 \, {\left (6 \, a^{2} b - b^{3} - 3 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left ({\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (6 \, a^{2} b - b^{3}\right )} \sinh \left (d x + c\right )^{4} + 6 \, a^{2} b - b^{3} - 2 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2} - 2 \, {\left (6 \, a^{2} b - b^{3} - 3 \, {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )^{3} - {\left (6 \, a^{2} b - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (4 \, a^{3} d x \cosh \left (d x + c\right )^{3} - 3 \, b^{3} \cosh \left (d x + c\right )^{2} - b^{3} - 4 \, {\left (a^{3} d x + 3 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{4} + 4 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + d \sinh \left (d x + c\right )^{4} - 2 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \]

[In]

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

[Out]

1/2*(2*a^3*d*x*cosh(d*x + c)^4 + 2*a^3*d*x*sinh(d*x + c)^4 - 2*b^3*cosh(d*x + c)^3 + 2*a^3*d*x - 2*b^3*cosh(d*
x + c) + 2*(4*a^3*d*x*cosh(d*x + c) - b^3)*sinh(d*x + c)^3 + 12*a*b^2 - 4*(a^3*d*x + 3*a*b^2)*cosh(d*x + c)^2
+ 2*(6*a^3*d*x*cosh(d*x + c)^2 - 2*a^3*d*x - 3*b^3*cosh(d*x + c) - 6*a*b^2)*sinh(d*x + c)^2 - ((6*a^2*b - b^3)
*cosh(d*x + c)^4 + 4*(6*a^2*b - b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (6*a^2*b - b^3)*sinh(d*x + c)^4 + 6*a^2*b
- b^3 - 2*(6*a^2*b - b^3)*cosh(d*x + c)^2 - 2*(6*a^2*b - b^3 - 3*(6*a^2*b - b^3)*cosh(d*x + c)^2)*sinh(d*x +
c)^2 + 4*((6*a^2*b - b^3)*cosh(d*x + c)^3 - (6*a^2*b - b^3)*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) +
sinh(d*x + c) + 1) + ((6*a^2*b - b^3)*cosh(d*x + c)^4 + 4*(6*a^2*b - b^3)*cosh(d*x + c)*sinh(d*x + c)^3 + (6*a
^2*b - b^3)*sinh(d*x + c)^4 + 6*a^2*b - b^3 - 2*(6*a^2*b - b^3)*cosh(d*x + c)^2 - 2*(6*a^2*b - b^3 - 3*(6*a^2*
b - b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((6*a^2*b - b^3)*cosh(d*x + c)^3 - (6*a^2*b - b^3)*cosh(d*x + c)
)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(4*a^3*d*x*cosh(d*x + c)^3 - 3*b^3*cosh(d*x + c)^2
- b^3 - 4*(a^3*d*x + 3*a*b^2)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^4 + 4*d*cosh(d*x + c)*sinh(d*x +
c)^3 + d*sinh(d*x + c)^4 - 2*d*cosh(d*x + c)^2 + 2*(3*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 4*(d*cosh(d*x
+ c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*csch(c + d*x))**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.81 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=a^{3} x + \frac {1}{2} \, b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} + \frac {3 \, a^{2} b \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=\frac {2 \, {\left (d x + c\right )} a^{3} - {\left (6 \, a^{2} b - b^{3}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + {\left (6 \, a^{2} b - b^{3}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{3} e^{\left (d x + c\right )} - 6 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.27 \[ \int (a+b \text {csch}(c+d x))^3 \, dx=a^3\,x-\frac {\frac {6\,a\,b^2}{d}+\frac {b^3\,{\mathrm {e}}^{c+d\,x}}{d}}{{\mathrm {e}}^{2\,c+2\,d\,x}-1}+\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (b^3\,\sqrt {-d^2}-6\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {36\,a^4\,b^2-12\,a^2\,b^4+b^6}}\right )\,\sqrt {36\,a^4\,b^2-12\,a^2\,b^4+b^6}}{\sqrt {-d^2}}-\frac {2\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

[In]

int((a + b/sinh(c + d*x))^3,x)

[Out]

a^3*x - ((6*a*b^2)/d + (b^3*exp(c + d*x))/d)/(exp(2*c + 2*d*x) - 1) + (atan((exp(d*x)*exp(c)*(b^3*(-d^2)^(1/2)
- 6*a^2*b*(-d^2)^(1/2)))/(d*(b^6 - 12*a^2*b^4 + 36*a^4*b^2)^(1/2)))*(b^6 - 12*a^2*b^4 + 36*a^4*b^2)^(1/2))/(-
d^2)^(1/2) - (2*b^3*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))