Integrand size = 20, antiderivative size = 448 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\frac {d \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b^2 d \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \]
d*(a+b*arccsch(d*x+c))^2/f/(-c*f+d*e)-(a+b*arccsch(d*x+c))^2/f/(f*x+e)-2*b *d*(a+b*arccsch(d*x+c))*ln(1+(1/(d*x+c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/ (f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e)/(d^2*e^2-2*c*d*e*f+( c^2+1)*f^2)^(1/2)+2*b*d*(a+b*arccsch(d*x+c))*ln(1+(1/(d*x+c)+(1+1/(d*x+c)^ 2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(-c*f+d*e) /(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)-2*b^2*d*polylog(2,-(1/(d*x+c)+(1+1/ (d*x+c)^2)^(1/2))*(-c*f+d*e)/(f-(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)))/(- c*f+d*e)/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)+2*b^2*d*polylog(2,-(1/(d*x+ c)+(1+1/(d*x+c)^2)^(1/2))*(-c*f+d*e)/(f+(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1 /2)))/(-c*f+d*e)/(d^2*e^2-2*c*d*e*f+(c^2+1)*f^2)^(1/2)
Result contains complex when optimal does not.
Time = 12.80 (sec) , antiderivative size = 2061, normalized size of antiderivative = 4.60 \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\text {Result too large to show} \]
-(a^2/(f*(e + f*x))) - (2*a*b*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(A rcCsch[c + d*x]/(f + (d*e)/(c + d*x) - (c*f)/(c + d*x)) - (2*ArcTan[(d*e - c*f - f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2) *f^2]])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]))/(d*(-(d*e) + c*f)*( e + f*x)^2) - (b^2*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(ArcCsch[c + d*x]^2/((-(d*e) + c*f)*(f + (d*e)/(c + d*x) - (c*f)/(c + d*x))) + (2*(((-I )*Pi*ArcTanh[(-(d*e) + c*f + f*Tanh[ArcCsch[c + d*x]/2])/Sqrt[f^2 + (d*e - c*f)^2]])/Sqrt[f^2 + (d*e - c*f)^2] - (2*(Pi/2 - I*ArcCsch[c + d*x])*ArcT anh[((f - I*(d*e - c*f))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^ 2) + 2*c*d*e*f - f^2 - c^2*f^2]] - 2*ArcCos[((-I)*f)/(d*e - c*f)]*ArcTanh[ ((-f - I*(d*e - c*f))*Tan[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]] + (ArcCos[((-I)*f)/(d*e - c*f)] - (2*I)*(Arc Tanh[((f - I*(d*e - c*f))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e ^2) + 2*c*d*e*f - f^2 - c^2*f^2]] - ArcTanh[((-f - I*(d*e - c*f))*Tan[(Pi/ 2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]]) )*Log[Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]/(Sqrt[2]*E^((I/2)*(Pi/2 - I*ArcCsch[c + d*x]))*Sqrt[(-I)*(d*e - c*f)]*Sqrt[f + (d*e - c*f)/(c + d *x)])] + (ArcCos[((-I)*f)/(d*e - c*f)] + (2*I)*(ArcTanh[((f - I*(d*e - c*f ))*Cot[(Pi/2 - I*ArcCsch[c + d*x])/2])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - f^2 - c^2*f^2]] - ArcTanh[((-f - I*(d*e - c*f))*Tan[(Pi/2 - I*ArcCsch[c + d*...
Time = 1.35 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6876, 5992, 3042, 4679, 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.
| \(\displaystyle \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx\) |
| \(\Big \downarrow \) 6876 |
| \(\displaystyle -d \int \frac {(c+d x)^2 \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(d e-c f+f (c+d x))^2}d\text {csch}^{-1}(c+d x)\) |
| \(\Big \downarrow \) 5992 |
| \(\displaystyle -d \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (f (c+d x)-c f+d e)}-\frac {2 b \int \frac {a+b \text {csch}^{-1}(c+d x)}{d e-c f+f (c+d x)}d\text {csch}^{-1}(c+d x)}{f}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -d \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (f (c+d x)-c f+d e)}-\frac {2 b \int \frac {a+b \text {csch}^{-1}(c+d x)}{d e-c f+i f \csc \left (i \text {csch}^{-1}(c+d x)\right )}d\text {csch}^{-1}(c+d x)}{f}\right )\) |
| \(\Big \downarrow \) 4679 |
| \(\displaystyle -d \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (f (c+d x)-c f+d e)}-\frac {2 b \int \left (\frac {a+b \text {csch}^{-1}(c+d x)}{d e-c f}+\frac {f \left (a+b \text {csch}^{-1}(c+d x)\right )}{(c f-d e) \left (f+\frac {d e \left (1-\frac {c f}{d e}\right )}{c+d x}\right )}\right )d\text {csch}^{-1}(c+d x)}{f}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -d \left (\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (f (c+d x)-c f+d e)}-\frac {2 b \left (-\frac {f \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{f-\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+1\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {f \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (\frac {(d e-c f) e^{\text {csch}^{-1}(c+d x)}}{\sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}+f}+1\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 b (d e-c f)}-\frac {b f \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}+\frac {b f \operatorname {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2}}\right )}{(d e-c f) \sqrt {\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2}}\right )}{f}\right )\) |
-(d*((a + b*ArcCsch[c + d*x])^2/(f*(d*e - c*f + f*(c + d*x))) - (2*b*((a + b*ArcCsch[c + d*x])^2/(2*b*(d*e - c*f)) - (f*(a + b*ArcCsch[c + d*x])*Log [1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (f *(a + b*ArcCsch[c + d*x])*Log[1 + (E^ArcCsch[c + d*x]*(d*e - c*f))/(f + Sq rt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c *d*e*f + (1 + c^2)*f^2]) - (b*f*PolyLog[2, -((E^ArcCsch[c + d*x]*(d*e - c* f))/(f - Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)*Sqrt[d ^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]) + (b*f*PolyLog[2, -((E^ArcCsch[c + d* x]*(d*e - c*f))/(f + Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2]))])/((d*e - c*f)*Sqrt[d^2*e^2 - 2*c*d*e*f + (1 + c^2)*f^2])))/f))
3.1.12.3.1 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, 1/(Sin[e + f*x]^n/(b + a*Si n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt Q[m, 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]
\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}}{\left (f x +e \right )^{2}}d x\]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f^2*x^2 + 2*e*f*x + e^2), x)
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{2}}\, dx \]
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
-b^2*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2/(f^2*x + e*f) - a^2/(f^2 *x + e*f) - integrate(-((b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)* log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b)*log(d *x + c) + 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b - (b^2*d^2*f* x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c) + (b^2*c*d*e + (c^2*f + f)*a*b + (a*b*d^2*f + b^2*d^2*f)*x^2 + (2*a*b*c*d*f + (d^2*e + c*d*f)*b^ 2)*x - (b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^2*f + f)*b^2)*log(d*x + c))*sqr t(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) + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((b^2*d^2*f*x^2 + 2*b^2*c*d*f*x + (c^ 2*f + f)*b^2)*log(d*x + c)^2 - 2*(a*b*d^2*f*x^2 + 2*a*b*c*d*f*x + (c^2*f + f)*a*b)*log(d*x + c)))/(d^2*f^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f^3) *x^3 + e^2*f + (d^2*e^2*f + 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(c*d*e^2* f + c^2*e*f^2 + e*f^2)*x + (d^2*f^3*x^4 + c^2*e^2*f + 2*(d^2*e*f^2 + c*d*f ^3)*x^3 + e^2*f + (d^2*e^2*f + 4*c*d*e*f^2 + c^2*f^3 + f^3)*x^2 + 2*(c*d*e ^2*f + c^2*e*f^2 + e*f^2)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)
\[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \]