### 3.8 $$\int (e+f x)^2 (a+b \text{csch}^{-1}(c+d x))^2 \, dx$$

Optimal. Leaf size=351 $\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{b^2 f^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{b^2 f^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{2 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{4 b (d e-c f)^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{2 b f^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}+\frac{b^2 f^2 x}{3 d^2}$

[Out]

(b^2*f^2*x)/(3*d^2) + (2*b*f*(d*e - c*f)*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/d^3 + (b
*f^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/(3*d^3) - ((d*e - c*f)^3*(a + b*ArcCsch[c
+ d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCsch[c + d*x])^2)/(3*f) - (2*b*f^2*(a + b*ArcCsch[c + d*x])*ArcT
anh[E^ArcCsch[c + d*x]])/(3*d^3) + (4*b*(d*e - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^
3 + (2*b^2*f*(d*e - c*f)*Log[c + d*x])/d^3 - (b^2*f^2*PolyLog[2, -E^ArcCsch[c + d*x]])/(3*d^3) + (2*b^2*(d*e -
c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d^3 + (b^2*f^2*PolyLog[2, E^ArcCsch[c + d*x]])/(3*d^3) - (2*b^2*(d*e
- c*f)^2*PolyLog[2, E^ArcCsch[c + d*x]])/d^3

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Rubi [A]  time = 0.50625, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.45, Rules used = {6322, 5469, 4190, 4182, 2279, 2391, 4184, 3475, 4185} $\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{b^2 f^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{b^2 f^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{(d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{2 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{4 b (d e-c f)^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{2 b f^2 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}+\frac{2 b^2 f (d e-c f) \log (c+d x)}{d^3}+\frac{b^2 f^2 x}{3 d^2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^2*f^2*x)/(3*d^2) + (2*b*f*(d*e - c*f)*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/d^3 + (b
*f^2*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/(3*d^3) - ((d*e - c*f)^3*(a + b*ArcCsch[c
+ d*x])^2)/(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCsch[c + d*x])^2)/(3*f) - (2*b*f^2*(a + b*ArcCsch[c + d*x])*ArcT
anh[E^ArcCsch[c + d*x]])/(3*d^3) + (4*b*(d*e - c*f)^2*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^
3 + (2*b^2*f*(d*e - c*f)*Log[c + d*x])/d^3 - (b^2*f^2*PolyLog[2, -E^ArcCsch[c + d*x]])/(3*d^3) + (2*b^2*(d*e -
c*f)^2*PolyLog[2, -E^ArcCsch[c + d*x]])/d^3 + (b^2*f^2*PolyLog[2, E^ArcCsch[c + d*x]])/(3*d^3) - (2*b^2*(d*e
- c*f)^2*PolyLog[2, E^ArcCsch[c + d*x]])/d^3

Rule 6322

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]

Rule 5469

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 4190

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]

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2279

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 2391

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 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rubi steps

\begin{align*} \int (e+f x)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (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 )}{d^3}\\ &=\frac{(e+f x)^3 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) (d e-c f+f \text{csch}(x))^3 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{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) \operatorname{Subst}\left (\int \left (d^3 e^3 \left (1-\frac{c f \left (3 d^2 e^2-3 c d e f+c^2 f^2\right )}{d^3 e^3}\right ) (a+b x)+3 d^2 e^2 f \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text{csch}(x)+3 d e f^2 \left (1-\frac{c f}{d e}\right ) (a+b x) \text{csch}^2(x)+f^3 (a+b x) \text{csch}^3(x)\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3 f}\\ &=-\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{\left (2 b f^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{(2 b f (d e-c f)) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}-\frac{\left (2 b (d e-c f)^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}\\ &=\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{4 b (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{\left (b f^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}-\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}+\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}-\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^3}\\ &=\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^3}+\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{\left (2 b^2 (d e-c f)^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}\\ &=\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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{2 b^2 (d e-c f)^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}\\ &=\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 ) \tanh ^{-1}\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 ) \tanh ^{-1}\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 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}+\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}+\frac{b^2 f^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{3 d^3}-\frac{2 b^2 (d e-c f)^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^3}\\ \end{align*}

Mathematica [C]  time = 9.14219, size = 864, normalized size = 2.46 $\frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x-\frac{2 b^2 d e f \left (\frac{(c+d x)^2 \text{csch}^{-1}(c+d x)^2}{2 d^2}-\frac{c \coth \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right ) \text{csch}^{-1}(c+d x)^2}{2 d^2}+\frac{c \tanh \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right ) \text{csch}^{-1}(c+d x)^2}{2 d^2}+\frac{(c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \text{csch}^{-1}(c+d x)}{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 (\text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )\right )}{d^2}\right ) x}{(c+d x) \left (\frac{c}{c+d x}-1\right )}+\frac{1}{3} a b \left (2 x \left (3 e^2+3 f x e+f^2 x^2\right ) \text{csch}^{-1}(c+d x)+\frac{-f (c+d x) \sqrt{\frac{c^2+2 d x c+d^2 x^2+1}{(c+d x)^2}} (5 c f-d (6 e+f x))+2 c \left (3 d^2 e^2-3 c d f e+c^2 f^2\right ) \sinh ^{-1}\left (\frac{1}{c+d x}\right )+\left (6 d^2 e^2-12 c d f e+\left (6 c^2-1\right ) f^2\right ) \log \left ((c+d x) \left (\sqrt{\frac{c^2+2 d x c+d^2 x^2+1}{(c+d x)^2}}+1\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 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )}{d}-\frac{b^2 f^2 \left (-\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)}+2 \text{csch}^{-1}(c+d x) \left (3 c \text{csch}^{-1}(c+d x)-1\right ) \text{csch}^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 \text{csch}^{-1}(c+d x) \left (3 c \text{csch}^{-1}(c+d x)+1\right ) \text{sech}^2\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )+2 \left (-6 c^2 \text{csch}^{-1}(c+d x)^2+\text{csch}^{-1}(c+d x)^2+12 c \text{csch}^{-1}(c+d x)-2\right ) \coth \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )-48 c \log \left (\frac{1}{c+d x}\right )+8 \left (6 c^2-1\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 )+\text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )-\text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )\right )+2 \left (6 c^2 \text{csch}^{-1}(c+d x)^2-\text{csch}^{-1}(c+d x)^2+12 c \text{csch}^{-1}(c+d x)+2\right ) \tanh \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )\right )}{24 d^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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*PolyLog[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[ArcCsch[c + d*x]/2])/(2*d^2) - Lo
g[(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] + ArcCsch[c + d*x]^2 - 6*c^2*ArcCsch[c + d*x]^2)*Coth[ArcCsch[c + d*x]/2] + 2*ArcCsch[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[(c + d*x)^(-1)] + 8*(-1 + 6*c^2)*(ArcCsch[c + d*x]*(Log[1 - E^(-ArcCsch[c + d*x])] - Log[1 + E^(-Arc
Csch[c + d*x])]) + PolyLog[2, -E^(-ArcCsch[c + d*x])] - PolyLog[2, E^(-ArcCsch[c + d*x])]) - 2*ArcCsch[c + d*x
]*(1 + 3*c*ArcCsch[c + d*x])*Sech[ArcCsch[c + d*x]/2]^2 - 8*(c + d*x)^3*ArcCsch[c + d*x]^2*Sinh[ArcCsch[c + d*
x]/2]^4 + 2*(2 + 12*c*ArcCsch[c + d*x] - ArcCsch[c + d*x]^2 + 6*c^2*ArcCsch[c + d*x]^2)*Tanh[ArcCsch[c + d*x]/
2]))/(24*d^3)

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Maple [F]  time = 0.403, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{2} \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 + c^2 + 1)*((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 - 2*(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)))/(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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} +{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\right )} \operatorname{arcsch}\left (d x + c\right ), x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*arccsch(d*x + c), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2}{\left (b \operatorname{arcsch}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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