Integrand size = 14, antiderivative size = 74 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^3 x-\frac {b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)}{d}-\frac {(3 a-2 b) b^2 \coth ^3(c+d x)}{3 d}-\frac {b^3 \coth ^5(c+d x)}{5 d} \]
a^3*x-b*(3*a^2-3*a*b+b^2)*coth(d*x+c)/d-1/3*(3*a-2*b)*b^2*coth(d*x+c)^3/d- 1/5*b^3*coth(d*x+c)^5/d
Time = 2.72 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.53 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\frac {8 \left (a+b \text {csch}^2(c+d x)\right )^3 \left (15 a^3 (c+d x)-b \coth (c+d x) \left (45 a^2-30 a b+8 b^2+(15 a-4 b) b \text {csch}^2(c+d x)+3 b^2 \text {csch}^4(c+d x)\right )\right ) \sinh ^6(c+d x)}{15 d (-a+2 b+a \cosh (2 (c+d x)))^3} \]
(8*(a + b*Csch[c + d*x]^2)^3*(15*a^3*(c + d*x) - b*Coth[c + d*x]*(45*a^2 - 30*a*b + 8*b^2 + (15*a - 4*b)*b*Csch[c + d*x]^2 + 3*b^2*Csch[c + d*x]^4)) *Sinh[c + d*x]^6)/(15*d*(-a + 2*b + a*Cosh[2*(c + d*x)])^3)
Time = 0.27 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 4616, 300, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-b \sec \left (i c+i d x+\frac {\pi }{2}\right )^2\right )^3dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle \frac {\int \frac {\left (b \coth ^2(c+d x)+a-b\right )^3}{1-\coth ^2(c+d x)}d\coth (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (-b^3 \coth ^4(c+d x)-(3 a-2 b) b^2 \coth ^2(c+d x)-b \left (3 a^2-3 b a+b^2\right )+\frac {a^3}{1-\coth ^2(c+d x)}\right )d\coth (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \text {arctanh}(\coth (c+d x))-b \left (3 a^2-3 a b+b^2\right ) \coth (c+d x)-\frac {1}{3} b^2 (3 a-2 b) \coth ^3(c+d x)-\frac {1}{5} b^3 \coth ^5(c+d x)}{d}\) |
(a^3*ArcTanh[Coth[c + d*x]] - b*(3*a^2 - 3*a*b + b^2)*Coth[c + d*x] - ((3* a - 2*b)*b^2*Coth[c + d*x]^3)/3 - (b^3*Coth[c + d*x]^5)/5)/d
3.1.2.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[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]
Time = 1.55 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (d x +c \right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(83\) |
| default | \(\frac {a^{3} \left (d x +c \right )-3 a^{2} b \coth \left (d x +c \right )+3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}\) | \(83\) |
| parts | \(a^{3} x +\frac {b^{3} \left (-\frac {8}{15}-\frac {\operatorname {csch}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {csch}\left (d x +c \right )^{2}}{15}\right ) \coth \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )}{d}-\frac {3 a^{2} b \coth \left (d x +c \right )}{d}\) | \(84\) |
| parallelrisch | \(\frac {-3 \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+\left (-60 a \,b^{2}+25 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-720 a^{2} b +540 a \,b^{2}-150 b^{3}\right ) \coth \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b^{3}+\left (-60 a \,b^{2}+25 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-720 a^{2} b +540 a \,b^{2}-150 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+480 a^{3} d x}{480 d}\) | \(150\) |
| risch | \(a^{3} x -\frac {2 b \left (45 a^{2} {\mathrm e}^{8 d x +8 c}-180 a^{2} {\mathrm e}^{6 d x +6 c}+90 a b \,{\mathrm e}^{6 d x +6 c}+270 a^{2} {\mathrm e}^{4 d x +4 c}-210 a b \,{\mathrm e}^{4 d x +4 c}+80 \,{\mathrm e}^{4 d x +4 c} b^{2}-180 a^{2} {\mathrm e}^{2 d x +2 c}+150 a b \,{\mathrm e}^{2 d x +2 c}-40 \,{\mathrm e}^{2 d x +2 c} b^{2}+45 a^{2}-30 a b +8 b^{2}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}\) | \(164\) |
1/d*(a^3*(d*x+c)-3*a^2*b*coth(d*x+c)+3*a*b^2*(2/3-1/3*csch(d*x+c)^2)*coth( d*x+c)+b^3*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)^2)*coth(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (70) = 140\).
Time = 0.25 (sec) , antiderivative size = 461, normalized size of antiderivative = 6.23 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=-\frac {{\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (27 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3} - 2 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (27 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (9 \, a^{2} b - 12 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 5 \, {\left (30 \, a^{3} d x + {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 90 \, a^{2} b - 60 \, a b^{2} + 16 \, b^{3} - 3 \, {\left (15 \, a^{3} d x + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
-1/15*((45*a^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^5 + 5*(45*a^2*b - 30*a* b^2 + 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^4 - (15*a^3*d*x + 45*a^2*b - 30*a *b^2 + 8*b^3)*sinh(d*x + c)^5 - 5*(27*a^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^3 + 5*(15*a^3*d*x + 45*a^2*b - 30*a*b^2 + 8*b^3 - 2*(15*a^3*d*x + 45*a ^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 5*(2*(45*a^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^3 - 3*(27*a^2*b - 30*a*b^2 + 8*b^3)*cos h(d*x + c))*sinh(d*x + c)^2 + 10*(9*a^2*b - 12*a*b^2 + 8*b^3)*cosh(d*x + c ) - 5*(30*a^3*d*x + (15*a^3*d*x + 45*a^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^4 + 90*a^2*b - 60*a*b^2 + 16*b^3 - 3*(15*a^3*d*x + 45*a^2*b - 30*a*b^2 + 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^5 + 5*(2*d*cosh( d*x + c)^2 - d)*sinh(d*x + c)^3 + 5*(d*cosh(d*x + c)^4 - 3*d*cosh(d*x + c) ^2 + 2*d)*sinh(d*x + c))
\[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{3}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 334 vs. \(2 (70) = 140\).
Time = 0.19 (sec) , antiderivative size = 334, normalized size of antiderivative = 4.51 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^{3} x - \frac {16}{15} \, b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + 4 \, a b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a^{2} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \]
a^3*x - 16/15*b^3*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d* x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d *(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8 *d*x - 8*c) + e^(-10*d*x - 10*c) - 1))) + 4*a*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))) + 6*a^2*b /(d*(e^(-2*d*x - 2*c) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (70) = 140\).
Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.46 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=\frac {15 \, {\left (d x + c\right )} a^{3} - \frac {2 \, {\left (45 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 180 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 90 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 270 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 210 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 80 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 180 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 150 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 40 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, a^{2} b - 30 \, a b^{2} + 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
1/15*(15*(d*x + c)*a^3 - 2*(45*a^2*b*e^(8*d*x + 8*c) - 180*a^2*b*e^(6*d*x + 6*c) + 90*a*b^2*e^(6*d*x + 6*c) + 270*a^2*b*e^(4*d*x + 4*c) - 210*a*b^2* e^(4*d*x + 4*c) + 80*b^3*e^(4*d*x + 4*c) - 180*a^2*b*e^(2*d*x + 2*c) + 150 *a*b^2*e^(2*d*x + 2*c) - 40*b^3*e^(2*d*x + 2*c) + 45*a^2*b - 30*a*b^2 + 8* b^3)/(e^(2*d*x + 2*c) - 1)^5)/d
Time = 2.26 (sec) , antiderivative size = 508, normalized size of antiderivative = 6.86 \[ \int \left (a+b \text {csch}^2(c+d x)\right )^3 \, dx=a^3\,x-\frac {\frac {2\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{15\,d}+\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {6\,a^2\,b}{5\,d}+\frac {24\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {24\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {6\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {18\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (9\,a^2\,b-12\,a\,b^2+8\,b^3\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {6\,\left (a\,b^2-a^2\,b\right )}{5\,d}+\frac {6\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{5\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {6\,a^2\,b}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
a^3*x - ((2*(9*a^2*b - 12*a*b^2 + 8*b^3))/(15*d) + (12*exp(2*c + 2*d*x)*(a *b^2 - a^2*b))/(5*d) + (6*a^2*b*exp(4*c + 4*d*x))/(5*d))/(3*exp(2*c + 2*d* x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((6*a^2*b)/(5*d) + (24*e xp(2*c + 2*d*x)*(a*b^2 - a^2*b))/(5*d) + (24*exp(6*c + 6*d*x)*(a*b^2 - a^2 *b))/(5*d) + (4*exp(4*c + 4*d*x)*(9*a^2*b - 12*a*b^2 + 8*b^3))/(5*d) + (6* a^2*b*exp(8*c + 8*d*x))/(5*d))/(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1) - ((6* (a*b^2 - a^2*b))/(5*d) + (18*exp(4*c + 4*d*x)*(a*b^2 - a^2*b))/(5*d) + (2* exp(2*c + 2*d*x)*(9*a^2*b - 12*a*b^2 + 8*b^3))/(5*d) + (6*a^2*b*exp(6*c + 6*d*x))/(5*d))/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d* x) + exp(8*c + 8*d*x) + 1) - ((6*(a*b^2 - a^2*b))/(5*d) + (6*a^2*b*exp(2*c + 2*d*x))/(5*d))/(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1) - (6*a^2*b)/ (5*d*(exp(2*c + 2*d*x) - 1))