Integrand size = 10, antiderivative size = 147 \[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=-\frac {\left (2-17 a^2\right ) (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^4}+\frac {x^2 (a+b x) \sqrt {1+\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \text {csch}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {csch}^{-1}(a+b x)+\frac {a \left (1-2 a^2\right ) \text {arctanh}\left (\sqrt {1+\frac {1}{(a+b x)^2}}\right )}{2 b^4} \]
-1/4*a^4*arccsch(b*x+a)/b^4+1/4*x^4*arccsch(b*x+a)+1/2*a*(-2*a^2+1)*arctan h((1+1/(b*x+a)^2)^(1/2))/b^4-1/12*(-17*a^2+2)*(b*x+a)*(1+1/(b*x+a)^2)^(1/2 )/b^4+1/12*x^2*(b*x+a)*(1+1/(b*x+a)^2)^(1/2)/b^2-1/3*a*(b*x+a)^2*(1+1/(b*x +a)^2)^(1/2)/b^4
Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.01 \[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\frac {\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (-2 a+13 a^3-2 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3\right )+3 b^4 x^4 \text {csch}^{-1}(a+b x)-3 a^4 \text {arcsinh}\left (\frac {1}{a+b x}\right )+6 a \left (1-2 a^2\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{12 b^4} \]
(Sqrt[(1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(-2*a + 13*a^3 - 2*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3) + 3*b^4*x^4*ArcCsch[a + b*x] - 3*a^4*Ar cSinh[(a + b*x)^(-1)] + 6*a*(1 - 2*a^2)*Log[(a + b*x)*(1 + Sqrt[(1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(12*b^4)
Time = 0.54 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6876, 25, 5992, 3042, 4269, 3042, 4536, 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.
| \(\displaystyle \int x^3 \text {csch}^{-1}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6876 |
| \(\displaystyle -\frac {\int b^3 x^3 (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)d\text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -b^3 x^3 (a+b x)^2 \sqrt {1+\frac {1}{(a+b x)^2}} \text {csch}^{-1}(a+b x)d\text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 5992 |
| \(\displaystyle -\frac {\frac {1}{4} \int b^4 x^4d\text {csch}^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)+\frac {1}{4} \int \left (a-i \csc \left (i \text {csch}^{-1}(a+b x)\right )\right )^4d\text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4269 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \int -b x \left (3 a^3+8 (a+b x)^2 a+\left (2-9 a^2\right ) (a+b x)\right )d\text {csch}^{-1}(a+b x)-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}\right )-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)+\frac {1}{4} \left (-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}+\frac {1}{3} \int \left (a-i \csc \left (i \text {csch}^{-1}(a+b x)\right )\right ) \left (3 a^3-8 \csc \left (i \text {csch}^{-1}(a+b x)\right )^2 a+i \left (2-9 a^2\right ) \csc \left (i \text {csch}^{-1}(a+b x)\right )\right )d\text {csch}^{-1}(a+b x)\right )}{b^4}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (6 a^4+12 \left (1-2 a^2\right ) (a+b x) a-2 \left (2-17 a^2\right ) (a+b x)^2\right )d\text {csch}^{-1}(a+b x)+4 a \sqrt {\frac {1}{(a+b x)^2}+1} (a+b x)^2\right )-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}\right )-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (6 a^4 \text {csch}^{-1}(a+b x)-12 \left (1-2 a^2\right ) a \text {arctanh}\left (\sqrt {\frac {1}{(a+b x)^2}+1}\right )+2 \left (2-17 a^2\right ) (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}\right )+4 a \sqrt {\frac {1}{(a+b x)^2}+1} (a+b x)^2\right )-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {\frac {1}{(a+b x)^2}+1}\right )-\frac {1}{4} b^4 x^4 \text {csch}^{-1}(a+b x)}{b^4}\) |
-((-1/4*(b^4*x^4*ArcCsch[a + b*x]) + (-1/3*(b^2*x^2*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)]) + (4*a*(a + b*x)^2*Sqrt[1 + (a + b*x)^(-2)] + (2*(2 - 17*a^2 )*(a + b*x)*Sqrt[1 + (a + b*x)^(-2)] + 6*a^4*ArcCsch[a + b*x] - 12*a*(1 - 2*a^2)*ArcTanh[Sqrt[1 + (a + b*x)^(-2)]])/2)/3)/4)/b^4)
3.1.1.3.1 Defintions of rubi rules used
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[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]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2 Int[Simp[2*A*a + (2*B*a + b*(2* A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, e, f, A, B, C}, x]
Int[Coth[(c_.) + (d_.)*(x_)]*Csch[(c_.) + (d_.)*(x_)]*(Csch[(c_.) + (d_.)*( x_)]*(b_.) + (a_))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(e + f*x)^m)*((a + b*Csch[c + d*x])^(n + 1)/(b*d*(n + 1))), x] + Simp[f*(m/(b *d*(n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Csch[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1) Subst[Int[(a + b*x)^p*Csch[x]*C oth[x]*(d*e - c*f + f*Csch[x])^m, x], x, ArcCsch[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
Time = 0.41 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.54
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arccsch}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccsch}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+12 a^{3} \operatorname {arcsinh}\left (b x +a \right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}+1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}+1}-6 a \,\operatorname {arcsinh}\left (b x +a \right )+2 \sqrt {\left (b x +a \right )^{2}+1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(227\) |
| default | \(\frac {\frac {\operatorname {arccsch}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccsch}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccsch}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsch}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccsch}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}+1}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {\left (b x +a \right )^{2}+1}}\right )+12 a^{3} \operatorname {arcsinh}\left (b x +a \right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}+1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}+1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}+1}-6 a \,\operatorname {arcsinh}\left (b x +a \right )+2 \sqrt {\left (b x +a \right )^{2}+1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(227\) |
| parts | \(\frac {x^{4} \operatorname {arccsch}\left (b x +a \right )}{4}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right ) \sqrt {b^{2}}-x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{2} \sqrt {b^{2}}+4 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a b x +12 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{3} b -13 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2}-6 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \sqrt {b^{2}}\right )}{12 b^{4} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}+1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(315\) |
1/b^4*(1/4*arccsch(b*x+a)*a^4-arccsch(b*x+a)*a^3*(b*x+a)+3/2*arccsch(b*x+a )*a^2*(b*x+a)^2-arccsch(b*x+a)*a*(b*x+a)^3+1/4*arccsch(b*x+a)*(b*x+a)^4-1/ 12*((b*x+a)^2+1)^(1/2)*(3*a^4*arctanh(1/((b*x+a)^2+1)^(1/2))+12*a^3*arcsin h(b*x+a)-18*a^2*((b*x+a)^2+1)^(1/2)+6*a*(b*x+a)*((b*x+a)^2+1)^(1/2)-(b*x+a )^2*((b*x+a)^2+1)^(1/2)-6*a*arcsinh(b*x+a)+2*((b*x+a)^2+1)^(1/2))/(((b*x+a )^2+1)/(b*x+a)^2)^(1/2)/(b*x+a))
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (127) = 254\).
Time = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.21 \[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\frac {3 \, b^{4} x^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 3 \, a^{4} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + 1\right ) + 3 \, a^{4} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - 1\right ) + 6 \, {\left (2 \, a^{3} - a\right )} \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right ) + {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} + {\left (9 \, a^{2} - 2\right )} b x - 2 \, a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{12 \, b^{4}} \]
1/12*(3*b^4*x^4*log(((b*x + a)*sqrt((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - 3*a^4*log(-b*x + (b*x + a)*sqrt((b^2* x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)) - a + 1) + 3*a^4*log(- b*x + (b*x + a)*sqrt((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^ 2)) - a - 1) + 6*(2*a^3 - a)*log(-b*x + (b*x + a)*sqrt((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)) - a) + (b^3*x^3 - 3*a*b^2*x^2 + 13*a ^3 + (9*a^2 - 2)*b*x - 2*a)*sqrt((b^2*x^2 + 2*a*b*x + a^2 + 1)/(b^2*x^2 + 2*a*b*x + a^2)))/b^4
\[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\int x^{3} \operatorname {acsch}{\left (a + b x \right )}\, dx \]
\[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arcsch}\left (b x + a\right ) \,d x } \]
-1/2*(-I*a^3 + I*a)*(log(I*(b^2*x + a*b)/b + 1) - log(-I*(b^2*x + a*b)/b + 1))/b^4 + 1/8*(2*b^4*x^4*log(sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1) + b^2 *x^2 - 6*a*b*x - (a^4 - 6*a^2 + 1)*log(b^2*x^2 + 2*a*b*x + a^2 + 1) - 2*(b ^4*x^4 - a^4)*log(b*x + a))/b^4 + integrate(1/4*(b^2*x^5 + a*b*x^4)/(b^2*x ^2 + 2*a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 1), x)
\[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arcsch}\left (b x + a\right ) \,d x } \]
Timed out. \[ \int x^3 \text {csch}^{-1}(a+b x) \, dx=\int x^3\,\mathrm {asinh}\left (\frac {1}{a+b\,x}\right ) \,d x \]