Integrand size = 20, antiderivative size = 351 \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 b f (d e-c f) (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^3}+\frac {b f^2 (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}-\frac {2 b f^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {4 b (d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}+\frac {2 b^2 f (d e-c f) \log (c+d x)}{d^3}-\frac {b^2 f^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^3}+\frac {b^2 f^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac {2 b^2 (d e-c f)^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^3} \]
1/3*b^2*f^2*x/d^2-1/3*(-c*f+d*e)^3*(a+b*arccsch(d*x+c))^2/d^3/f+1/3*(f*x+e )^3*(a+b*arccsch(d*x+c))^2/f-2/3*b*f^2*(a+b*arccsch(d*x+c))*arctanh(1/(d*x +c)+(1+1/(d*x+c)^2)^(1/2))/d^3+4*b*(-c*f+d*e)^2*(a+b*arccsch(d*x+c))*arcta nh(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))/d^3+2*b^2*f*(-c*f+d*e)*ln(d*x+c)/d^3-1 /3*b^2*f^2*polylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^3+2*b^2*(-c*f+d*e )^2*polylog(2,-1/(d*x+c)-(1+1/(d*x+c)^2)^(1/2))/d^3+1/3*b^2*f^2*polylog(2, 1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))/d^3-2*b^2*(-c*f+d*e)^2*polylog(2,1/(d*x+c )+(1+1/(d*x+c)^2)^(1/2))/d^3+2*b*f*(-c*f+d*e)*(d*x+c)*(a+b*arccsch(d*x+c)) *(1+1/(d*x+c)^2)^(1/2)/d^3+1/3*b*f^2*(d*x+c)^2*(a+b*arccsch(d*x+c))*(1+1/( d*x+c)^2)^(1/2)/d^3
Result contains complex when optimal does not.
Time = 7.97 (sec) , antiderivative size = 893, normalized size of antiderivative = 2.54 \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {1}{3} a b \left (2 x \left (3 e^2+3 e f x+f^2 x^2\right ) \text {csch}^{-1}(c+d x)+\frac {-f (c+d x) \sqrt {\frac {1+c^2+2 c d x+d^2 x^2}{(c+d x)^2}} (5 c f-d (6 e+f x))+2 c \left (3 d^2 e^2-3 c d e f+c^2 f^2\right ) \text {arcsinh}\left (\frac {1}{c+d x}\right )+\left (6 d^2 e^2-12 c d e f+\left (-1+6 c^2\right ) f^2\right ) \log \left ((c+d x) \left (1+\sqrt {\frac {1+c^2+2 c d x+d^2 x^2}{(c+d x)^2}}\right )\right )}{d^3}\right )-\frac {b^2 e^2 \left (-\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )}{d}-\frac {2 b^2 d e f x \left (\frac {(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \text {csch}^{-1}(c+d x)}{d^2}+\frac {(c+d x)^2 \text {csch}^{-1}(c+d x)^2}{2 d^2}-\frac {c \text {csch}^{-1}(c+d x)^2 \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}-\frac {\log \left (\frac {1}{c+d x}\right )}{d^2}-\frac {2 i c \left (i \text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )\right )}{d^2}+\frac {c \text {csch}^{-1}(c+d x)^2 \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 d^2}\right )}{(c+d x) \left (-1+\frac {c}{c+d x}\right )}-\frac {b^2 f^2 \left (2 \left (-2+12 c \text {csch}^{-1}(c+d x)+\text {csch}^{-1}(c+d x)^2-6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \coth \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \text {csch}^{-1}(c+d x) \left (-1+3 c \text {csch}^{-1}(c+d x)\right ) \text {csch}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-\frac {\text {csch}^{-1}(c+d x)^2 \text {csch}^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}-48 c \left (\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )+\log \left (\sqrt {1+\frac {1}{(c+d x)^2}}\right )\right )+8 \left (-1+6 c^2\right ) \left (\text {csch}^{-1}(c+d x) \left (\log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )-\log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \text {csch}^{-1}(c+d x) \left (1+3 c \text {csch}^{-1}(c+d x)\right ) \text {sech}^2\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )-8 (c+d x)^3 \text {csch}^{-1}(c+d x)^2 \sinh ^4\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )+2 \left (2+12 c \text {csch}^{-1}(c+d x)-\text {csch}^{-1}(c+d x)^2+6 c^2 \text {csch}^{-1}(c+d x)^2\right ) \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )}{24 d^3} \]
a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f ^2*x^2)*ArcCsch[c + d*x] + (-(f*(c + d*x)*Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^ 2)/(c + d*x)^2]*(5*c*f - d*(6*e + f*x))) + 2*c*(3*d^2*e^2 - 3*c*d*e*f + c^ 2*f^2)*ArcSinh[(c + d*x)^(-1)] + (6*d^2*e^2 - 12*c*d*e*f + (-1 + 6*c^2)*f^ 2)*Log[(c + d*x)*(1 + Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2])])/d ^3))/3 - (b^2*e^2*(-(ArcCsch[c + d*x]*((c + d*x)*ArcCsch[c + d*x] - 2*Log[ 1 - E^(-ArcCsch[c + d*x])] + 2*Log[1 + E^(-ArcCsch[c + d*x])])) + 2*PolyLo g[2, -E^(-ArcCsch[c + d*x])] - 2*PolyLog[2, E^(-ArcCsch[c + d*x])]))/d - ( 2*b^2*d*e*f*x*(((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*ArcCsch[c + d*x])/d^2 + ((c + d*x)^2*ArcCsch[c + d*x]^2)/(2*d^2) - (c*ArcCsch[c + d*x]^2*Coth[Arc Csch[c + d*x]/2])/(2*d^2) - Log[(c + d*x)^(-1)]/d^2 - ((2*I)*c*(I*ArcCsch[ c + d*x]*(Log[1 - E^(-ArcCsch[c + d*x])] - Log[1 + E^(-ArcCsch[c + d*x])]) + I*(PolyLog[2, -E^(-ArcCsch[c + d*x])] - PolyLog[2, E^(-ArcCsch[c + d*x] )])))/d^2 + (c*ArcCsch[c + d*x]^2*Tanh[ArcCsch[c + d*x]/2])/(2*d^2)))/((c + d*x)*(-1 + c/(c + d*x))) - (b^2*f^2*(2*(-2 + 12*c*ArcCsch[c + d*x] + Arc Csch[c + d*x]^2 - 6*c^2*ArcCsch[c + d*x]^2)*Coth[ArcCsch[c + d*x]/2] + 2*A rcCsch[c + d*x]*(-1 + 3*c*ArcCsch[c + d*x])*Csch[ArcCsch[c + d*x]/2]^2 - ( ArcCsch[c + d*x]^2*Csch[ArcCsch[c + d*x]/2]^4)/(2*(c + d*x)) - 48*c*(Log[1 /((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + Log[Sqrt[1 + (c + d*x)^(-2)]]) + 8*(-1 + 6*c^2)*(ArcCsch[c + d*x]*(Log[1 - E^(-ArcCsch[c + d*x])] - Log[...
Time = 0.80 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6876, 5992, 3042, 4678, 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.
| \(\displaystyle \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 6876 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} (d e-c f+f (c+d x))^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2d\text {csch}^{-1}(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 5992 |
| \(\displaystyle -\frac {\frac {2 b \int (d e-c f+f (c+d x))^3 \left (a+b \text {csch}^{-1}(c+d x)\right )d\text {csch}^{-1}(c+d x)}{3 f}-\frac {(f (c+d x)-c f+d e)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {(f (c+d x)-c f+d e)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \int \left (a+b \text {csch}^{-1}(c+d x)\right ) \left (d e-c f+i f \csc \left (i \text {csch}^{-1}(c+d x)\right )\right )^3d\text {csch}^{-1}(c+d x)}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 4678 |
| \(\displaystyle -\frac {\frac {2 b \int \left (d^3 \left (1-\frac {c f \left (3 d^2 e^2-3 c d f e+c^2 f^2\right )}{d^3 e^3}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right ) e^3+3 d^2 f \left (\frac {c f (c f-2 d e)}{d^2 e^2}+1\right ) (c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right ) e^2+3 d f^2 \left (1-\frac {c f}{d e}\right ) (c+d x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right ) e+f^3 (c+d x)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )\right )d\text {csch}^{-1}(c+d x)}{3 f}-\frac {(f (c+d x)-c f+d e)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {2 b \left (-6 f (d e-c f)^2 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )+f^3 \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )-3 f^2 (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right )+\frac {(d e-c f)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 b}-\frac {1}{2} f^3 (c+d x)^2 \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )+3 b f^2 (d e-c f) \log \left (\frac {1}{c+d x}\right )-3 b f (d e-c f)^2 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )+3 b f (d e-c f)^2 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )+\frac {1}{2} b f^3 \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )-\frac {1}{2} b f^3 \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )-\frac {1}{2} b f^3 (c+d x)\right )}{3 f}-\frac {(f (c+d x)-c f+d e)^3 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{3 f}}{d^3}\) |
-((-1/3*((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCsch[c + d*x])^2)/f + (2*b* (-1/2*(b*f^3*(c + d*x)) - 3*f^2*(d*e - c*f)*(c + d*x)*Sqrt[1 + (c + d*x)^( -2)]*(a + b*ArcCsch[c + d*x]) - (f^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]* (a + b*ArcCsch[c + d*x]))/2 + ((d*e - c*f)^3*(a + b*ArcCsch[c + d*x])^2)/( 2*b) + f^3*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] - 6*f*(d*e - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] + 3*b*f^2*( d*e - c*f)*Log[(c + d*x)^(-1)] + (b*f^3*PolyLog[2, -E^ArcCsch[c + d*x]])/2 - 3*b*f*(d*e - c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]] - (b*f^3*PolyLog[2, E^ArcCsch[c + d*x]])/2 + 3*b*f*(d*e - c*f)^2*PolyLog[2, E^ArcCsch[c + d*x ]]))/(3*f))/d^3)
3.1.8.3.1 Defintions of rubi rules used
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
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]
\[\int \left (f x +e \right )^{2} \left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}d x\]
\[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
integral(a^2*f^2*x^2 + 2*a^2*e*f*x + a^2*e^2 + (b^2*f^2*x^2 + 2*b^2*e*f*x + b^2*e^2)*arccsch(d*x + c)^2 + 2*(a*b*f^2*x^2 + 2*a*b*e*f*x + a*b*e^2)*ar ccsch(d*x + c), x)
\[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \]
\[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/3*a^2*f^2*x^3 + a^2*e*f*x^2 + a^2*e^2*x + (2*(d*x + c)*arccsch(d*x + c) + log(sqrt(1/(d*x + c)^2 + 1) + 1) - log(sqrt(1/(d*x + c)^2 + 1) - 1))*a*b *e^2/d + 1/3*(b^2*f^2*x^3 + 3*b^2*e*f*x^2 + 3*b^2*e^2*x)*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2 - integrate(-1/3*(3*(b^2*d^2*f^2*x^4 + b^2*c^2 *e^2 + b^2*e^2 + 2*(b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + (4*b^2*c*d*e*f + b^2* c^2*f^2 + (d^2*e^2 + f^2)*b^2)*x^2 + 2*(b^2*c*d*e^2 + b^2*c^2*e*f + b^2*e* f)*x)*log(d*x + c)^2 - 6*(a*b*d^2*f^2*x^4 + 2*(a*b*d^2*e*f + a*b*c*d*f^2)* x^3 + (4*a*b*c*d*e*f + a*b*c^2*f^2 + a*b*f^2)*x^2 + 2*(a*b*c^2*e*f + a*b*e *f)*x)*log(d*x + c) + 2*(3*a*b*d^2*f^2*x^4 + 6*(a*b*d^2*e*f + a*b*c*d*f^2) *x^3 + 3*(4*a*b*c*d*e*f + a*b*c^2*f^2 + a*b*f^2)*x^2 + 6*(a*b*c^2*e*f + a* b*e*f)*x - 3*(b^2*d^2*f^2*x^4 + b^2*c^2*e^2 + b^2*e^2 + 2*(b^2*d^2*e*f + b ^2*c*d*f^2)*x^3 + (4*b^2*c*d*e*f + b^2*c^2*f^2 + (d^2*e^2 + f^2)*b^2)*x^2 + 2*(b^2*c*d*e^2 + b^2*c^2*e*f + b^2*e*f)*x)*log(d*x + c) + ((3*a*b*d^2*f^ 2 - b^2*d^2*f^2)*x^4 + (6*a*b*d^2*e*f - 3*b^2*d^2*e*f + (6*a*b*d*f^2 - b^2 *d*f^2)*c)*x^3 - 3*(b^2*d^2*e^2 - a*b*c^2*f^2 - a*b*f^2 - (4*a*b*d*e*f - b ^2*d*e*f)*c)*x^2 - 3*(b^2*c*d*e^2 - 2*a*b*c^2*e*f - 2*a*b*e*f)*x - 3*(b^2* d^2*f^2*x^4 + b^2*c^2*e^2 + b^2*e^2 + 2*(b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + (4*b^2*c*d*e*f + b^2*c^2*f^2 + (d^2*e^2 + f^2)*b^2)*x^2 + 2*(b^2*c*d*e^2 + b^2*c^2*e*f + b^2*e*f)*x)*log(d*x + c))*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) )*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1) + 3*sqrt(d^2*x^2 + 2*c*d*x...
\[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2 \,d x \]