Integrand size = 18, antiderivative size = 194 \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b f (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^2}-\frac {(d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}+\frac {4 b (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}+\frac {b^2 f \log (c+d x)}{d^2}+\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}-\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^2} \]
-1/2*(-c*f+d*e)^2*(a+b*arccsch(d*x+c))^2/d^2/f+1/2*(f*x+e)^2*(a+b*arccsch( d*x+c))^2/f+4*b*(-c*f+d*e)*(a+b*arccsch(d*x+c))*arctanh(1/(d*x+c)+(1+1/(d* x+c)^2)^(1/2))/d^2+b^2*f*ln(d*x+c)/d^2+2*b^2*(-c*f+d*e)*polylog(2,-1/(d*x+ c)-(1+1/(d*x+c)^2)^(1/2))/d^2-2*b^2*(-c*f+d*e)*polylog(2,1/(d*x+c)+(1+1/(d *x+c)^2)^(1/2))/d^2+b*f*(d*x+c)*(a+b*arccsch(d*x+c))*(1+1/(d*x+c)^2)^(1/2) /d^2
Leaf count is larger than twice the leaf count of optimal. \(427\) vs. \(2(194)=388\).
Time = 2.96 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.20 \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {2 a^2 (d e-c f) (c+d x)+a^2 f (c+d x)^2+2 a b f (c+d x) \left (\sqrt {1+\frac {1}{(c+d x)^2}}+(c+d x) \text {csch}^{-1}(c+d x)\right )+2 b^2 f \left ((c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \text {csch}^{-1}(c+d x)+\frac {1}{2} (c+d x)^2 \text {csch}^{-1}(c+d x)^2-\log \left (\frac {1}{c+d x}\right )\right )+4 a b d e \left ((c+d x) \text {csch}^{-1}(c+d x)+\log \left (\frac {\text {csch}\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}\right )-\log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )\right )-4 a b c f \left ((c+d x) \text {csch}^{-1}(c+d x)+\log \left (\frac {\text {csch}\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}\right )-\log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )\right )+2 b^2 d e \left (\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 b^2 c f \left (\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )}{2 d^2} \]
(2*a^2*(d*e - c*f)*(c + d*x) + a^2*f*(c + d*x)^2 + 2*a*b*f*(c + d*x)*(Sqrt [1 + (c + d*x)^(-2)] + (c + d*x)*ArcCsch[c + d*x]) + 2*b^2*f*((c + d*x)*Sq rt[1 + (c + d*x)^(-2)]*ArcCsch[c + d*x] + ((c + d*x)^2*ArcCsch[c + d*x]^2) /2 - Log[(c + d*x)^(-1)]) + 4*a*b*d*e*((c + d*x)*ArcCsch[c + d*x] + Log[Cs ch[ArcCsch[c + d*x]/2]/(2*(c + d*x))] - Log[Sinh[ArcCsch[c + d*x]/2]]) - 4 *a*b*c*f*((c + d*x)*ArcCsch[c + d*x] + Log[Csch[ArcCsch[c + d*x]/2]/(2*(c + d*x))] - Log[Sinh[ArcCsch[c + d*x]/2]]) + 2*b^2*d*e*(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*PolyLog[2, -E^(-ArcCsch[c + d*x])] + 2*PolyLog[ 2, E^(-ArcCsch[c + d*x])]) - 2*b^2*c*f*(ArcCsch[c + d*x]*((c + d*x)*ArcCsc h[c + d*x] - 2*Log[1 - E^(-ArcCsch[c + d*x])] + 2*Log[1 + E^(-ArcCsch[c + d*x])]) - 2*PolyLog[2, -E^(-ArcCsch[c + d*x])] + 2*PolyLog[2, E^(-ArcCsch[ c + d*x])]))/(2*d^2)
Time = 0.60 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, 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.
| \(\displaystyle \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 6876 |
| \(\displaystyle -\frac {\int (c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} (d e-c f+f (c+d x)) \left (a+b \text {csch}^{-1}(c+d x)\right )^2d\text {csch}^{-1}(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 5992 |
| \(\displaystyle -\frac {\frac {b \int (d e-c f+f (c+d x))^2 \left (a+b \text {csch}^{-1}(c+d x)\right )d\text {csch}^{-1}(c+d x)}{f}-\frac {(f (c+d x)-c f+d e)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {-\frac {(f (c+d x)-c f+d e)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}+\frac {b \int \left (a+b \text {csch}^{-1}(c+d x)\right ) \left (d e-c f+i f \csc \left (i \text {csch}^{-1}(c+d x)\right )\right )^2d\text {csch}^{-1}(c+d x)}{f}}{d^2}\) |
| \(\Big \downarrow \) 4678 |
| \(\displaystyle -\frac {\frac {b \int \left (d^2 \left (\frac {c f (c f-2 d e)}{d^2 e^2}+1\right ) \left (a+b \text {csch}^{-1}(c+d x)\right ) e^2+2 d f \left (1-\frac {c f}{d e}\right ) (c+d x) \left (a+b \text {csch}^{-1}(c+d x)\right ) e+f^2 (c+d x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )\right )d\text {csch}^{-1}(c+d x)}{f}-\frac {(f (c+d x)-c f+d e)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {b \left (-4 f (d e-c f) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )+\frac {(d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 b}-\left (f^2 (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )\right )-2 b f (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )+2 b f (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )+b f^2 \log \left (\frac {1}{c+d x}\right )\right )}{f}-\frac {(f (c+d x)-c f+d e)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}}{d^2}\) |
-((-1/2*((d*e - c*f + f*(c + d*x))^2*(a + b*ArcCsch[c + d*x])^2)/f + (b*(- (f^2*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x])) + ((d*e - c*f)^2*(a + b*ArcCsch[c + d*x])^2)/(2*b) - 4*f*(d*e - c*f)*(a + b*ArcCsc h[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]] + b*f^2*Log[(c + d*x)^(-1)] - 2*b* f*(d*e - c*f)*PolyLog[2, -E^ArcCsch[c + d*x]] + 2*b*f*(d*e - c*f)*PolyLog[ 2, E^ArcCsch[c + d*x]]))/f)/d^2)
3.1.9.3.1 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]
\[\int \left (f x +e \right ) \left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}d x\]
\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
integral(a^2*f*x + a^2*e + (b^2*f*x + b^2*e)*arccsch(d*x + c)^2 + 2*(a*b*f *x + a*b*e)*arccsch(d*x + c), x)
\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]
\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/2*a^2*f*x^2 + a^2*e*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/d + 1/2*(b^2*f* x^2 + 2*b^2*e*x)*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2 - integrate( -((b^2*d^2*f*x^3 + b^2*c^2*e + b^2*e + (b^2*d^2*e + 2*b^2*c*d*f)*x^2 + (2* b^2*c*d*e + b^2*c^2*f + b^2*f)*x)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^3 + 2*a* b*c*d*f*x^2 + (a*b*c^2*f + a*b*f)*x)*log(d*x + c) + (2*a*b*d^2*f*x^3 + 4*a *b*c*d*f*x^2 + 2*(a*b*c^2*f + a*b*f)*x - 2*(b^2*d^2*f*x^3 + b^2*c^2*e + b^ 2*e + (b^2*d^2*e + 2*b^2*c*d*f)*x^2 + (2*b^2*c*d*e + b^2*c^2*f + b^2*f)*x) *log(d*x + c) + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((2*a*b*d^2*f - b^2*d^2* f)*x^3 - (2*b^2*d^2*e - (4*a*b*d*f - b^2*d*f)*c)*x^2 - 2*(b^2*c*d*e - a*b* c^2*f - a*b*f)*x - 2*(b^2*d^2*f*x^3 + b^2*c^2*e + b^2*e + (b^2*d^2*e + 2*b ^2*c*d*f)*x^2 + (2*b^2*c*d*e + b^2*c^2*f + b^2*f)*x)*log(d*x + c)))*log(sq rt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1) + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*( (b^2*d^2*f*x^3 + b^2*c^2*e + b^2*e + (b^2*d^2*e + 2*b^2*c*d*f)*x^2 + (2*b^ 2*c*d*e + b^2*c^2*f + b^2*f)*x)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^3 + 2*a*b* c*d*f*x^2 + (a*b*c^2*f + a*b*f)*x)*log(d*x + c)))/(d^2*x^2 + 2*c*d*x + c^2 + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + 1), x)
\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2 \,d x \]