Integrand size = 16, antiderivative size = 167 \[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {b e \left (9 c^2 d^2-e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}{2 c}+\frac {b e^3 \sqrt {1+\frac {1}{c^2 x^2}} x^3}{12 c}-\frac {b d^4 \text {csch}^{-1}(c x)}{4 e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}+\frac {b d \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\sqrt {1+\frac {1}{c^2 x^2}}\right )}{2 c^3} \]
-1/4*b*d^4*arccsch(c*x)/e+1/4*(e*x+d)^4*(a+b*arccsch(c*x))/e+1/2*b*d*(2*c^ 2*d^2-e^2)*arctanh((1+1/c^2/x^2)^(1/2))/c^3+1/6*b*e*(9*c^2*d^2-e^2)*x*(1+1 /c^2/x^2)^(1/2)/c^3+1/2*b*d*e^2*x^2*(1+1/c^2/x^2)^(1/2)/c+1/12*b*e^3*x^3*( 1+1/c^2/x^2)^(1/2)/c
Time = 0.26 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.99 \[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {3 a c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+b e \sqrt {1+\frac {1}{c^2 x^2}} x \left (-2 e^2+c^2 \left (18 d^2+6 d e x+e^2 x^2\right )\right )+3 b c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right ) \text {csch}^{-1}(c x)+6 b d \left (2 c^2 d^2-e^2\right ) \log \left (\left (1+\sqrt {1+\frac {1}{c^2 x^2}}\right ) x\right )}{12 c^3} \]
(3*a*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + b*e*Sqrt[1 + 1/(c ^2*x^2)]*x*(-2*e^2 + c^2*(18*d^2 + 6*d*e*x + e^2*x^2)) + 3*b*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)*ArcCsch[c*x] + 6*b*d*(2*c^2*d^2 - e^2 )*Log[(1 + Sqrt[1 + 1/(c^2*x^2)])*x])/(12*c^3)
Time = 0.75 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.99, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6844, 1892, 1803, 540, 25, 2338, 27, 2338, 27, 538, 222, 243, 73, 221}
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.
| \(\displaystyle \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6844 |
| \(\displaystyle \frac {b \int \frac {(d+e x)^4}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{4 c e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}\) |
| \(\Big \downarrow \) 1892 |
| \(\displaystyle \frac {b \int \frac {\left (\frac {d}{x}+e\right )^4 x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}dx}{4 c e}+\frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}\) |
| \(\Big \downarrow \) 1803 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \int \frac {\left (\frac {d}{x}+e\right )^4 x^4}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}}{4 c e}\) |
| \(\Big \downarrow \) 540 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (-\frac {1}{3} \int -\frac {\left (\frac {3 d^4}{x^3}+\frac {12 e d^3}{x^2}+12 e^3 d+\frac {2 e^2 \left (9 d^2-\frac {e^2}{c^2}\right )}{x}\right ) x^3}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \int \frac {\left (\frac {3 d^4}{x^3}+\frac {12 e d^3}{x^2}+12 e^3 d+\frac {2 e^2 \left (9 d^2-\frac {e^2}{c^2}\right )}{x}\right ) x^3}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (-\frac {1}{2} \int -\frac {2 \left (\frac {3 d^4}{x^2}+\frac {6 e \left (2 d^2-\frac {e^2}{c^2}\right ) d}{x}+2 e^2 \left (9 d^2-\frac {e^2}{c^2}\right )\right ) x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (\int \frac {\left (\frac {3 d^4}{x^2}+\frac {6 e \left (2 d^2-\frac {e^2}{c^2}\right ) d}{x}+2 e^2 \left (9 d^2-\frac {e^2}{c^2}\right )\right ) x^2}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (-\int -\frac {3 d \left (\frac {d^3}{x}+2 e \left (2 d^2-\frac {e^2}{c^2}\right )\right ) x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \int \frac {\left (\frac {d^3}{x}+2 e \left (2 d^2-\frac {e^2}{c^2}\right )\right ) x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 538 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \left (d^3 \int \frac {1}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}+2 e \left (2 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}\right )-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \left (2 e \left (2 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x}+c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )\right )-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \left (e \left (2 d^2-\frac {e^2}{c^2}\right ) \int \frac {x}{\sqrt {1+\frac {1}{c^2 x^2}}}d\frac {1}{x^2}+c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )\right )-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \left (2 c^2 e \left (2 d^2-\frac {e^2}{c^2}\right ) \int \frac {1}{c^2 \sqrt {1+\frac {1}{c^2 x^2}}-c^2}d\sqrt {1+\frac {1}{c^2 x^2}}+c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )\right )-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {(d+e x)^4 \left (a+b \text {csch}^{-1}(c x)\right )}{4 e}-\frac {b \left (\frac {1}{3} \left (3 d \left (c d^3 \text {arcsinh}\left (\frac {1}{c x}\right )-2 e \text {arctanh}\left (\sqrt {\frac {1}{c^2 x^2}+1}\right ) \left (2 d^2-\frac {e^2}{c^2}\right )\right )-2 e^2 x \sqrt {\frac {1}{c^2 x^2}+1} \left (9 d^2-\frac {e^2}{c^2}\right )-6 d e^3 x^2 \sqrt {\frac {1}{c^2 x^2}+1}\right )-\frac {1}{3} e^4 x^3 \sqrt {\frac {1}{c^2 x^2}+1}\right )}{4 c e}\) |
((d + e*x)^4*(a + b*ArcCsch[c*x]))/(4*e) - (b*(-1/3*(e^4*Sqrt[1 + 1/(c^2*x ^2)]*x^3) + (-2*e^2*(9*d^2 - e^2/c^2)*Sqrt[1 + 1/(c^2*x^2)]*x - 6*d*e^3*Sq rt[1 + 1/(c^2*x^2)]*x^2 + 3*d*(c*d^3*ArcSinh[1/(c*x)] - 2*e*(2*d^2 - e^2/c ^2)*ArcTanh[Sqrt[1 + 1/(c^2*x^2)]]))/3))/(4*c*e)
3.1.44.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) , x] + Simp[1/(a*(m + 1)) Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x )^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ (p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 2] || !IntegerQ[p])
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsch[c*x])/(e*(m + 1))), x] + Simp[ b/(c*e*(m + 1)) Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
Time = 0.69 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.50
| method | result | size |
| parts | \(\frac {a \left (e x +d \right )^{4}}{4 e}+\frac {b \left (\frac {c \,e^{3} \operatorname {arccsch}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arccsch}\left (c x \right ) x^{3} d +\frac {3 c e \,\operatorname {arccsch}\left (c x \right ) x^{2} d^{2}}{2}+\operatorname {arccsch}\left (c x \right ) x c \,d^{3}+\frac {c \,\operatorname {arccsch}\left (c x \right ) d^{4}}{4 e}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (3 c^{4} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )-12 c^{3} d^{3} e \,\operatorname {arcsinh}\left (c x \right )-e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-6 c^{2} d \,e^{3} x \sqrt {c^{2} x^{2}+1}-18 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}+6 c d \,e^{3} \operatorname {arcsinh}\left (c x \right )+2 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{12 c^{4} e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, x}\right )}{c}\) | \(250\) |
| derivativedivides | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccsch}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccsch}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccsch}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arccsch}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-3 c^{4} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+12 c^{3} d^{3} e \,\operatorname {arcsinh}\left (c x \right )+18 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}+6 c^{2} d \,e^{3} x \sqrt {c^{2} x^{2}+1}+e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-6 c d \,e^{3} \operatorname {arcsinh}\left (c x \right )-2 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{12 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}}{c}\) | \(269\) |
| default | \(\frac {\frac {a \left (c e x +c d \right )^{4}}{4 c^{3} e}+\frac {b \left (\frac {\operatorname {arccsch}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arccsch}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arccsch}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arccsch}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arccsch}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-3 c^{4} d^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )+12 c^{3} d^{3} e \,\operatorname {arcsinh}\left (c x \right )+18 c^{2} d^{2} e^{2} \sqrt {c^{2} x^{2}+1}+6 c^{2} d \,e^{3} x \sqrt {c^{2} x^{2}+1}+e^{4} c^{2} x^{2} \sqrt {c^{2} x^{2}+1}-6 c d \,e^{3} \operatorname {arcsinh}\left (c x \right )-2 e^{4} \sqrt {c^{2} x^{2}+1}\right )}{12 e \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{3}}}{c}\) | \(269\) |
1/4*a*(e*x+d)^4/e+b/c*(1/4*c*e^3*arccsch(c*x)*x^4+c*e^2*arccsch(c*x)*x^3*d +3/2*c*e*arccsch(c*x)*x^2*d^2+arccsch(c*x)*x*c*d^3+1/4*c/e*arccsch(c*x)*d^ 4-1/12/c^4/e*(c^2*x^2+1)^(1/2)*(3*c^4*d^4*arctanh(1/(c^2*x^2+1)^(1/2))-12* c^3*d^3*e*arcsinh(c*x)-e^4*c^2*x^2*(c^2*x^2+1)^(1/2)-6*c^2*d*e^3*x*(c^2*x^ 2+1)^(1/2)-18*c^2*d^2*e^2*(c^2*x^2+1)^(1/2)+6*c*d*e^3*arcsinh(c*x)+2*e^4*( c^2*x^2+1)^(1/2))/((c^2*x^2+1)/c^2/x^2)^(1/2)/x)
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (147) = 294\).
Time = 0.33 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.51 \[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {3 \, a c^{3} e^{3} x^{4} + 12 \, a c^{3} d e^{2} x^{3} + 18 \, a c^{3} d^{2} e x^{2} + 12 \, a c^{3} d^{3} x + 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} e + 4 \, b c^{3} d e^{2} + b c^{3} e^{3}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 6 \, {\left (2 \, b c^{2} d^{3} - b d e^{2}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 3 \, {\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} e + 4 \, b c^{3} d e^{2} + b c^{3} e^{3}\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 3 \, {\left (b c^{3} e^{3} x^{4} + 4 \, b c^{3} d e^{2} x^{3} + 6 \, b c^{3} d^{2} e x^{2} + 4 \, b c^{3} d^{3} x - 4 \, b c^{3} d^{3} - 6 \, b c^{3} d^{2} e - 4 \, b c^{3} d e^{2} - b c^{3} e^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} e^{3} x^{3} + 6 \, b c^{2} d e^{2} x^{2} + 2 \, {\left (9 \, b c^{2} d^{2} e - b e^{3}\right )} x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{3}} \]
1/12*(3*a*c^3*e^3*x^4 + 12*a*c^3*d*e^2*x^3 + 18*a*c^3*d^2*e*x^2 + 12*a*c^3 *d^3*x + 3*(4*b*c^3*d^3 + 6*b*c^3*d^2*e + 4*b*c^3*d*e^2 + b*c^3*e^3)*log(c *x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) - 6*(2*b*c^2*d^3 - b*d*e^2)*lo g(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x) - 3*(4*b*c^3*d^3 + 6*b*c^3*d^2* e + 4*b*c^3*d*e^2 + b*c^3*e^3)*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + 3*(b*c^3*e^3*x^4 + 4*b*c^3*d*e^2*x^3 + 6*b*c^3*d^2*e*x^2 + 4*b*c^3 *d^3*x - 4*b*c^3*d^3 - 6*b*c^3*d^2*e - 4*b*c^3*d*e^2 - b*c^3*e^3)*log((c*x *sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x)) + (b*c^2*e^3*x^3 + 6*b*c^2*d*e^ 2*x^2 + 2*(9*b*c^2*d^2*e - b*e^3)*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2)))/c^3
\[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x\right )^{3}\, dx \]
Time = 0.21 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.56 \[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac {3}{2} \, a d^{2} e x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arcsch}\left (c x\right ) + \frac {x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b d^{2} e + \frac {1}{4} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arcsch}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b e^{3} + a d^{3} x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d^{3}}{2 \, c} \]
1/4*a*e^3*x^4 + a*d*e^2*x^3 + 3/2*a*d^2*e*x^2 + 3/2*(x^2*arccsch(c*x) + x* sqrt(1/(c^2*x^2) + 1)/c)*b*d^2*e + 1/4*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^ 2*x^2) + 1)/(c^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c^2*x^2) + 1) + 1) /c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*b*d*e^2 + 1/12*(3*x^4*arccsc h(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) + 1))/c^3 )*b*e^3 + a*d^3*x + 1/2*(2*c*x*arccsch(c*x) + log(sqrt(1/(c^2*x^2) + 1) + 1) - log(sqrt(1/(c^2*x^2) + 1) - 1))*b*d^3/c
\[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { {\left (e x + d\right )}^{3} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \,d x } \]
Timed out. \[ \int (d+e x)^3 \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^3 \,d x \]