3.1.12 \(\int \frac {(a+b \text {csch}^{-1}(c+d x))^2}{(e+f x)^2} \, dx\) [12]

Optimal. Leaf size=448 \[ \frac {d \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (d e-c f)}-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}-\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b d \left (a+b \text {csch}^{-1}(c+d x)\right ) \log \left (1+\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {2 b^2 d \text {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f-\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {2 b^2 d \text {PolyLog}\left (2,-\frac {e^{\text {csch}^{-1}(c+d x)} (d e-c f)}{f+\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}} \]

[Out]

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)

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Rubi [A]
time = 0.77, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6457, 5577, 4276, 3403, 2296, 2221, 2317, 2438} \begin {gather*} -\frac {2 b d \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 {2 b d \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 {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^2 d \text {Li}_2\left (-\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 {2 b^2 d \text {Li}_2\left (-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c + d*x])^2/(e + f*x)^2,x]

[Out]

(d*(a + b*ArcCsch[c + d*x])^2)/(f*(d*e - c*f)) - (a + b*ArcCsch[c + d*x])^2/(f*(e + f*x)) - (2*b*d*(a + b*ArcC
sch[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]) + (2*b*d*(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]) - (2*b^2*d*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]) + (2*b^2*d*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])

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3403

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Dist[2,
Int[(c + d*x)^m*(E^((-I)*e + f*fz*x)/((-I)*b + 2*a*E^((-I)*e + f*fz*x) + I*b*E^(2*((-I)*e + f*fz*x)))), x], x]
/; FreeQ[{a, b, c, d, e, f, fz}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4276

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*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 5577

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] + Dist[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]

Rule 6457

Int[((a_.) + ArcCsch[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[-(d^(m + 1)
)^(-1), Subst[Int[(a + b*x)^p*Csch[x]*Coth[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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{(e+f x)^2} \, dx &=-\left (d \text {Subst}\left (\int \frac {(a+b x)^2 \coth (x) \text {csch}(x)}{(d e-c f+f \text {csch}(x))^2} \, dx,x,\text {csch}^{-1}(c+d x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \text {Subst}\left (\int \frac {a+b x}{d e-c f+f \text {csch}(x)} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f}\\ &=-\frac {\left (a+b \text {csch}^{-1}(c+d x)\right )^2}{f (e+f x)}+\frac {(2 b d) \text {Subst}\left (\int \left (\frac {a+b x}{d e-c f}+\frac {f (a+b x)}{(-d e+c f) \left (f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)\right )}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{f}\\ &=\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) \text {Subst}\left (\int \frac {a+b x}{f+d e \left (1-\frac {c f}{d e}\right ) \sinh (x)} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\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 {(4 b d) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 e^x f-d e \left (1-\frac {c f}{d e}\right )+d e e^{2 x} \left (1-\frac {c f}{d e}\right )} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{d e-c f}\\ &=\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 {(4 b d) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac {c f}{d e}\right )-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}+\frac {(4 b d) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 f+2 d e e^x \left (1-\frac {c f}{d e}\right )+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}} \, dx,x,\text {csch}^{-1}(c+d x)\right )}{\sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\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 {\left (2 b^2 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 d e e^x \left (1-\frac {c f}{d e}\right )}{2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right ) \, dx,x,\text {csch}^{-1}(c+d x)\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\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 {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{2 f-2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 d e \left (1-\frac {c f}{d e}\right ) x}{2 f+2 \sqrt {d^2 e^2-2 c d e f+f^2+c^2 f^2}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c+d x)}\right )}{(d e-c f) \sqrt {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2}}\\ &=\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 \text {Li}_2\left (-\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 \text {Li}_2\left (-\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}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 12.41, size = 1874, normalized size = 4.18 \begin {gather*} -\frac {a^2}{f (e+f x)}-\frac {2 a b (c+d x)^2 \left (f+\frac {d e-c f}{c+d x}\right )^2 \left (\frac {\text {csch}^{-1}(c+d x)}{f+\frac {d e}{c+d x}-\frac {c f}{c+d x}}-\frac {2 \text {ArcTan}\left (\frac {d e-c f-f \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )}{d (-d e+c f) (e+f x)^2}-\frac {b^2 (c+d x)^2 \left (f+\frac {d e-c f}{c+d x}\right )^2 \left (\frac {\text {csch}^{-1}(c+d x)^2}{(-d e+c f) \left (f+\frac {d e-c f}{c+d x}\right )}+\frac {2 \left (-\frac {i \pi \tanh ^{-1}\left (\frac {-d e+c f+f \tanh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{\sqrt {f^2+(d e-c f)^2}}\right )}{\sqrt {f^2+(d e-c f)^2}}-\frac {2 i \text {ArcCos}\left (\frac {i f}{-d e+c f}\right ) \text {ArcTan}\left (\frac {(d e-(i+c) f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )+\left (\pi -2 i \text {csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (\frac {(-i d e+f+i c f) \tan \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )+\left (\text {ArcCos}\left (\frac {i f}{-d e+c f}\right )+2 \text {ArcTan}\left (\frac {(d e-(i+c) f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )-2 i \tanh ^{-1}\left (\frac {(-i d e+f+i c f) \tan \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )\right ) \log \left (-\frac {(-1)^{3/4} e^{-\frac {1}{2} \text {csch}^{-1}(c+d x)} \sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}{\sqrt {2} \sqrt {i (-d e+c f)} \sqrt {f+\frac {d e-c f}{c+d x}}}\right )+\left (\text {ArcCos}\left (\frac {i f}{-d e+c f}\right )-2 \text {ArcTan}\left (\frac {(d e-(i+c) f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )+2 i \tanh ^{-1}\left (\frac {(-i d e+f+i c f) \tan \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )\right ) \log \left (\frac {\sqrt [4]{-1} e^{\frac {1}{2} \text {csch}^{-1}(c+d x)} \sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}{\sqrt {2} \sqrt {i (-d e+c f)} \sqrt {f+\frac {d e-c f}{c+d x}}}\right )-\left (\text {ArcCos}\left (\frac {i f}{-d e+c f}\right )-2 \text {ArcTan}\left (\frac {(d e-(i+c) f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )\right ) \log \left (1-\frac {\left (-i f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}\right ) \left (i d e-f-i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}{(d e-c f) \left (-i d e+f+i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}\right )-\left (\text {ArcCos}\left (\frac {i f}{-d e+c f}\right )+2 \text {ArcTan}\left (\frac {(d e-(i+c) f) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )\right ) \log \left (1+\frac {\left (i f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}\right ) \left (i d e-f-i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}{(d e-c f) \left (-i d e+f+i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}\right )+i \left (-\text {PolyLog}\left (2,\frac {\left (-i f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}\right ) \left (i d e-f-i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}{(d e-c f) \left (-i d e+f+i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}\right )+\text {PolyLog}\left (2,-\frac {\left (i f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}\right ) \left (i d e-f-i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}{(d e-c f) \left (-i d e+f+i c f+\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2} \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(c+d x)\right )\right )\right )}\right )\right )}{\sqrt {-d^2 e^2+2 c d e f-\left (1+c^2\right ) f^2}}\right )}{d e-c f}\right )}{d (e+f x)^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCsch[c + d*x])^2/(e + f*x)^2,x]

[Out]

-(a^2/(f*(e + f*x))) - (2*a*b*(c + d*x)^2*(f + (d*e - c*f)/(c + d*x))^2*(ArcCsch[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*f)/(c + d*x))) + (2*(((-I)*Pi*Arc
Tanh[(-(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*I
)*ArcCos[(I*f)/(-(d*e) + c*f)]*ArcTan[((d*e - (I + c)*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2)
+ 2*c*d*e*f - (1 + c^2)*f^2]] + (Pi - (2*I)*ArcCsch[c + d*x])*ArcTanh[(((-I)*d*e + f + I*c*f)*Tan[(Pi + (2*I)
*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]] + (ArcCos[(I*f)/(-(d*e) + c*f)] + 2*ArcTa
n[((d*e - (I + c)*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]] - (2*
I)*ArcTanh[(((-I)*d*e + f + I*c*f)*Tan[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^
2)*f^2]])*Log[-(((-1)^(3/4)*Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2])/(Sqrt[2]*E^(ArcCsch[c + d*x]/2)*Sqrt
[I*(-(d*e) + c*f)]*Sqrt[f + (d*e - c*f)/(c + d*x)]))] + (ArcCos[(I*f)/(-(d*e) + c*f)] - 2*ArcTan[((d*e - (I +
c)*f)*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]] + (2*I)*ArcTanh[(((-
I)*d*e + f + I*c*f)*Tan[(Pi + (2*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]])*Log[(
(-1)^(1/4)*E^(ArcCsch[c + d*x]/2)*Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2])/(Sqrt[2]*Sqrt[I*(-(d*e) + c*f)
]*Sqrt[f + (d*e - c*f)/(c + d*x)])] - (ArcCos[(I*f)/(-(d*e) + c*f)] - 2*ArcTan[((d*e - (I + c)*f)*Cot[(Pi + (2
*I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]])*Log[1 - (((-I)*f + Sqrt[-(d^2*e^2) +
2*c*d*e*f - (1 + c^2)*f^2])*(I*d*e - f - I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*
ArcCsch[c + d*x])/4]))/((d*e - c*f)*((-I)*d*e + f + I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(
Pi + (2*I)*ArcCsch[c + d*x])/4]))] - (ArcCos[(I*f)/(-(d*e) + c*f)] + 2*ArcTan[((d*e - (I + c)*f)*Cot[(Pi + (2*
I)*ArcCsch[c + d*x])/4])/Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]])*Log[1 + ((I*f + Sqrt[-(d^2*e^2) + 2*c*
d*e*f - (1 + c^2)*f^2])*(I*d*e - f - I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*ArcC
sch[c + d*x])/4]))/((d*e - c*f)*((-I)*d*e + f + I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi +
(2*I)*ArcCsch[c + d*x])/4]))] + I*(-PolyLog[2, (((-I)*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2])*(I*d*
e - f - I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4]))/((d*e - c*
f)*((-I)*d*e + f + I*c*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4]))
] + PolyLog[2, -(((I*f + Sqrt[-(d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2])*(I*d*e - f - I*c*f + Sqrt[-(d^2*e^2) +
2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4]))/((d*e - c*f)*((-I)*d*e + f + I*c*f + Sqrt[-(
d^2*e^2) + 2*c*d*e*f - (1 + c^2)*f^2]*Cot[(Pi + (2*I)*ArcCsch[c + d*x])/4])))]))/Sqrt[-(d^2*e^2) + 2*c*d*e*f -
(1 + c^2)*f^2]))/(d*e - c*f)))/(d*(e + f*x)^2)

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (d x +c \right )\right )^{2}}{\left (f x +e \right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(d*x+c))^2/(f*x+e)^2,x)

[Out]

int((a+b*arccsch(d*x+c))^2/(f*x+e)^2,x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e)^2,x, algorithm="maxima")

[Out]

-b^2*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2/(f^2*x + f*e) - a^2/(f^2*x + f*e) - 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 + b^2*c*d*f + b^
2*d^2*e)*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))*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) + 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 + 2*(c*d*f^3 + d^2*f^2*e)*x^3 + (c^2*f^3 + 4*c*d*f^2*e + d^2*f*e^2 + f^3)*x^2 + 2*(c*d*f*e^2 + (c^2*f
^2 + f^2)*e)*x + (c^2*f + f)*e^2 + (d^2*f^3*x^4 + 2*(c*d*f^3 + d^2*f^2*e)*x^3 + (c^2*f^3 + 4*c*d*f^2*e + d^2*f
*e^2 + f^3)*x^2 + 2*(c*d*f*e^2 + (c^2*f^2 + f^2)*e)*x + (c^2*f + f)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x
)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e)^2,x, algorithm="fricas")

[Out]

integral((b^2*arccsch(d*x + c)^2 + 2*a*b*arccsch(d*x + c) + a^2)/(f^2*x^2 + 2*f*x*e + e^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsch(d*x+c))**2/(f*x+e)**2,x)

[Out]

Integral((a + b*acsch(c + d*x))**2/(e + f*x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsch(d*x+c))^2/(f*x+e)^2,x, algorithm="giac")

[Out]

integrate((b*arccsch(d*x + c) + a)^2/(f*x + e)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2}{{\left (e+f\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x)^2,x)

[Out]

int((a + b*asinh(1/(c + d*x)))^2/(e + f*x)^2, x)

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