### 3.7 $$\int (e+f x)^3 (a+b \text{csch}^{-1}(c+d x))^2 \, dx$$

Optimal. Leaf size=501 $-\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{2 b f^2 (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{3 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{4 b (d e-c f)^3 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{b f^3 (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{b^2 f^2 x (d e-c f)}{d^3}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}$

[Out]

(b^2*f^2*(d*e - c*f)*x)/d^3 + (b^2*f^3*(c + d*x)^2)/(12*d^4) - (b*f^3*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a +
b*ArcCsch[c + d*x]))/(3*d^4) + (3*b*f*(d*e - c*f)^2*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]
))/d^4 + (b*f^2*(d*e - c*f)*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/d^4 + (b*f^3*(c + d
*x)^3*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/(6*d^4) - ((d*e - c*f)^4*(a + b*ArcCsch[c + d*x])^2)/
(4*d^4*f) + ((e + f*x)^4*(a + b*ArcCsch[c + d*x])^2)/(4*f) - (2*b*f^2*(d*e - c*f)*(a + b*ArcCsch[c + d*x])*Arc
Tanh[E^ArcCsch[c + d*x]])/d^4 + (4*b*(d*e - c*f)^3*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^4 -
(b^2*f^3*Log[c + d*x])/(3*d^4) + (3*b^2*f*(d*e - c*f)^2*Log[c + d*x])/d^4 - (b^2*f^2*(d*e - c*f)*PolyLog[2, -
E^ArcCsch[c + d*x]])/d^4 + (2*b^2*(d*e - c*f)^3*PolyLog[2, -E^ArcCsch[c + d*x]])/d^4 + (b^2*f^2*(d*e - c*f)*Po
lyLog[2, E^ArcCsch[c + d*x]])/d^4 - (2*b^2*(d*e - c*f)^3*PolyLog[2, E^ArcCsch[c + d*x]])/d^4

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Rubi [A]  time = 0.886769, antiderivative size = 501, normalized size of antiderivative = 1., number of steps used = 20, 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} $-\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b f^2 (c+d x)^2 \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{2 b f^2 (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{3 b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} (d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{4 b (d e-c f)^3 \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{b f^3 (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{b^2 f^2 x (d e-c f)}{d^3}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(e + f*x)^3*(a + b*ArcCsch[c + d*x])^2,x]

[Out]

(b^2*f^2*(d*e - c*f)*x)/d^3 + (b^2*f^3*(c + d*x)^2)/(12*d^4) - (b*f^3*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a +
b*ArcCsch[c + d*x]))/(3*d^4) + (3*b*f*(d*e - c*f)^2*(c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]
))/d^4 + (b*f^2*(d*e - c*f)*(c + d*x)^2*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/d^4 + (b*f^3*(c + d
*x)^3*Sqrt[1 + (c + d*x)^(-2)]*(a + b*ArcCsch[c + d*x]))/(6*d^4) - ((d*e - c*f)^4*(a + b*ArcCsch[c + d*x])^2)/
(4*d^4*f) + ((e + f*x)^4*(a + b*ArcCsch[c + d*x])^2)/(4*f) - (2*b*f^2*(d*e - c*f)*(a + b*ArcCsch[c + d*x])*Arc
Tanh[E^ArcCsch[c + d*x]])/d^4 + (4*b*(d*e - c*f)^3*(a + b*ArcCsch[c + d*x])*ArcTanh[E^ArcCsch[c + d*x]])/d^4 -
(b^2*f^3*Log[c + d*x])/(3*d^4) + (3*b^2*f*(d*e - c*f)^2*Log[c + d*x])/d^4 - (b^2*f^2*(d*e - c*f)*PolyLog[2, -
E^ArcCsch[c + d*x]])/d^4 + (2*b^2*(d*e - c*f)^3*PolyLog[2, -E^ArcCsch[c + d*x]])/d^4 + (b^2*f^2*(d*e - c*f)*Po
lyLog[2, E^ArcCsch[c + d*x]])/d^4 - (2*b^2*(d*e - c*f)^3*PolyLog[2, E^ArcCsch[c + d*x]])/d^4

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)^3 \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))^3 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{b \operatorname{Subst}\left (\int (a+b x) (d e-c f+f \text{csch}(x))^4 \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4 f}\\ &=\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{b \operatorname{Subst}\left (\int \left (d^4 e^4 \left (1+\frac{c f \left (-4 d^3 e^3+6 c d^2 e^2 f-4 c^2 d e f^2+c^3 f^3\right )}{d^4 e^4}\right ) (a+b x)+4 d^3 e^3 f \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) \text{csch}(x)+6 d^2 e^2 f^2 \left (1+\frac{c f (-2 d e+c f)}{d^2 e^2}\right ) (a+b x) \text{csch}^2(x)+4 d e f^3 \left (1-\frac{c f}{d e}\right ) (a+b x) \text{csch}^3(x)+f^4 (a+b x) \text{csch}^4(x)\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4 f}\\ &=-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{\left (b f^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^4(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{2 d^4}-\frac{\left (2 b f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^3(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (3 b f (d e-c f)^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (2 b (d e-c f)^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{\left (b f^3\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{\left (b f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (3 b^2 f (d e-c f)^2\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}-\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{\left (b^2 f^3\right ) \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{3 d^4}-\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{\left (2 b^2 (d e-c f)^3\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{\left (b^2 f^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ &=\frac{b^2 f^2 (d e-c f) x}{d^3}+\frac{b^2 f^3 (c+d x)^2}{12 d^4}-\frac{b f^3 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{3 d^4}+\frac{3 b f (d e-c f)^2 (c+d x) \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^2 (d e-c f) (c+d x)^2 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^4}+\frac{b f^3 (c+d x)^3 \sqrt{1+\frac{1}{(c+d x)^2}} \left (a+b \text{csch}^{-1}(c+d x)\right )}{6 d^4}-\frac{(d e-c f)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 d^4 f}+\frac{(e+f x)^4 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{4 f}-\frac{2 b f^2 (d e-c f) \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{4 b (d e-c f)^3 \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{b^2 f^3 \log (c+d x)}{3 d^4}+\frac{3 b^2 f (d e-c f)^2 \log (c+d x)}{d^4}-\frac{b^2 f^2 (d e-c f) \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}+\frac{b^2 f^2 (d e-c f) \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}-\frac{2 b^2 (d e-c f)^3 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^4}\\ \end{align*}

Mathematica [C]  time = 12.6609, size = 1429, normalized size = 2.85 $\text{result too large to display}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)^3*(a + b*ArcCsch[c + d*x])^2,x]

[Out]

a^2*e^3*x + (3*a^2*e^2*f*x^2)/2 + a^2*e*f^2*x^3 + (a^2*f^3*x^4)/4 + (a*b*(3*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2
+ f^3*x^3)*ArcCsch[c + d*x] + (f*(c + d*x)*Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2]*((-2 + 13*c^2)*f^2
- 2*c*d*f*(15*e + 2*f*x) + d^2*(18*e^2 + 6*e*f*x + f^2*x^2)) - 3*c*(-4*d^3*e^3 + 6*c*d^2*e^2*f - 4*c^2*d*e*f^
2 + c^3*f^3)*ArcSinh[(c + d*x)^(-1)] + 6*(2*d^3*e^3 - 6*c*d^2*e^2*f + (-1 + 6*c^2)*d*e*f^2 + c*(1 - 2*c^2)*f^3
)*Log[(c + d*x)*(1 + Sqrt[(1 + c^2 + 2*c*d*x + d^2*x^2)/(c + d*x)^2])])/d^4))/6 - (b^2*e^3*(-(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*PolyL
og[2, -E^(-ArcCsch[c + d*x])] - 2*PolyLog[2, E^(-ArcCsch[c + d*x])]))/d - (3*b^2*d*e^2*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*Cot
h[ArcCsch[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*e*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[A
rcCsch[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^(-ArcCsch[c + d*x])]) + PolyLog[2, -E^(-ArcCsch[c + d*x])] - PolyLog[2, E^(-Ar
cCsch[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*A
rcCsch[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*ArcCs
ch[c + d*x]^2)*Tanh[ArcCsch[c + d*x]/2]))/(8*d^3) - (b^2*f^3*x^3*(-16*(2*ArcCsch[c + d*x] - 18*c^2*ArcCsch[c +
d*x] + 6*c^3*ArcCsch[c + d*x]^2 - 3*c*(-2 + ArcCsch[c + d*x]^2))*Coth[ArcCsch[c + d*x]/2] + 2*(2 - 24*c*ArcCs
ch[c + d*x] - 3*ArcCsch[c + d*x]^2 + 36*c^2*ArcCsch[c + d*x]^2)*Csch[ArcCsch[c + d*x]/2]^2 + 3*ArcCsch[c + d*x
]^2*Csch[ArcCsch[c + d*x]/2]^4 - (2*ArcCsch[c + d*x]*(-1 + 6*c*ArcCsch[c + d*x])*Csch[ArcCsch[c + d*x]/2]^4)/(
c + d*x) - 64*(-1 + 9*c^2)*Log[(c + d*x)^(-1)] + 192*c*(-1 + 2*c^2)*(ArcCsch[c + d*x]*(Log[1 - E^(-ArcCsch[c +
d*x])] - Log[1 + E^(-ArcCsch[c + d*x])]) + PolyLog[2, -E^(-ArcCsch[c + d*x])] - PolyLog[2, E^(-ArcCsch[c + d*
x])]) - 2*(2 + 24*c*ArcCsch[c + d*x] - 3*ArcCsch[c + d*x]^2 + 36*c^2*ArcCsch[c + d*x]^2)*Sech[ArcCsch[c + d*x]
/2]^2 + 3*ArcCsch[c + d*x]^2*Sech[ArcCsch[c + d*x]/2]^4 - 32*(c + d*x)^3*ArcCsch[c + d*x]*(1 + 6*c*ArcCsch[c +
d*x])*Sinh[ArcCsch[c + d*x]/2]^4 + 16*(-2*ArcCsch[c + d*x] + 18*c^2*ArcCsch[c + d*x] + 6*c^3*ArcCsch[c + d*x]
^2 - 3*c*(-2 + ArcCsch[c + d*x]^2))*Tanh[ArcCsch[c + d*x]/2]))/(192*d*(c + d*x)^3*(-1 + c/(c + d*x))^3)

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Maple [F]  time = 0.425, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{3} \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)^3*(a+b*arccsch(d*x+c))^2,x)

[Out]

int((f*x+e)^3*(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)^3*(a+b*arccsch(d*x+c))^2,x, algorithm="maxima")

[Out]

1/4*a^2*f^3*x^4 + a^2*e*f^2*x^3 + 3/2*a^2*e^2*f*x^2 + a^2*e^3*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^3/d + 1/4*(b^2*f^3*x^4 + 4*b^2*e*f^2*x^3 + 6*b^
2*e^2*f*x^2 + 4*b^2*e^3*x)*log(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1) + 1)^2 - integrate(-1/2*(2*(b^2*d^2*f^3*x^5 +
b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f^2 + 2*b^2*c*d*f^3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (3*d^2*e^2*f
+ f^3)*b^2)*x^3 + (6*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2 + (d^2*e^3 + 3*e*f^2)*b^2)*x^2 + (2*b^2*c*d*e^3 + 3*b^2*
c^2*e^2*f + 3*b^2*e^2*f)*x)*log(d*x + c)^2 - 4*(a*b*d^2*f^3*x^5 + (3*a*b*d^2*e*f^2 + 2*a*b*c*d*f^3)*x^4 + (6*a
*b*c*d*e*f^2 + a*b*c^2*f^3 + (3*d^2*e^2*f + f^3)*a*b)*x^3 + 3*(2*a*b*c*d*e^2*f + a*b*c^2*e*f^2 + a*b*e*f^2)*x^
2 + 3*(a*b*c^2*e^2*f + a*b*e^2*f)*x)*log(d*x + c) + (4*a*b*d^2*f^3*x^5 + 4*(3*a*b*d^2*e*f^2 + 2*a*b*c*d*f^3)*x
^4 + 4*(6*a*b*c*d*e*f^2 + a*b*c^2*f^3 + (3*d^2*e^2*f + f^3)*a*b)*x^3 + 12*(2*a*b*c*d*e^2*f + a*b*c^2*e*f^2 + a
*b*e*f^2)*x^2 + 12*(a*b*c^2*e^2*f + a*b*e^2*f)*x - 4*(b^2*d^2*f^3*x^5 + b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f
^2 + 2*b^2*c*d*f^3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (3*d^2*e^2*f + f^3)*b^2)*x^3 + (6*b^2*c*d*e^2*f + 3
*b^2*c^2*e*f^2 + (d^2*e^3 + 3*e*f^2)*b^2)*x^2 + (2*b^2*c*d*e^3 + 3*b^2*c^2*e^2*f + 3*b^2*e^2*f)*x)*log(d*x + c
) + ((4*a*b*d^2*f^3 - b^2*d^2*f^3)*x^5 + (12*a*b*d^2*e*f^2 - 4*b^2*d^2*e*f^2 + (8*a*b*d*f^3 - b^2*d*f^3)*c)*x^
4 - 2*(3*b^2*d^2*e^2*f - 2*a*b*c^2*f^3 - 2*(3*d^2*e^2*f + f^3)*a*b - 2*(6*a*b*d*e*f^2 - b^2*d*e*f^2)*c)*x^3 -
2*(2*b^2*d^2*e^3 - 6*a*b*c^2*e*f^2 - 6*a*b*e*f^2 - 3*(4*a*b*d*e^2*f - b^2*d*e^2*f)*c)*x^2 - 4*(b^2*c*d*e^3 - 3
*a*b*c^2*e^2*f - 3*a*b*e^2*f)*x - 4*(b^2*d^2*f^3*x^5 + b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f^2 + 2*b^2*c*d*f^
3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (3*d^2*e^2*f + f^3)*b^2)*x^3 + (6*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2 +
(d^2*e^3 + 3*e*f^2)*b^2)*x^2 + (2*b^2*c*d*e^3 + 3*b^2*c^2*e^2*f + 3*b^2*e^2*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) + 2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((b^2*d^
2*f^3*x^5 + b^2*c^2*e^3 + b^2*e^3 + (3*b^2*d^2*e*f^2 + 2*b^2*c*d*f^3)*x^4 + (6*b^2*c*d*e*f^2 + b^2*c^2*f^3 + (
3*d^2*e^2*f + f^3)*b^2)*x^3 + (6*b^2*c*d*e^2*f + 3*b^2*c^2*e*f^2 + (d^2*e^3 + 3*e*f^2)*b^2)*x^2 + (2*b^2*c*d*e
^3 + 3*b^2*c^2*e^2*f + 3*b^2*e^2*f)*x)*log(d*x + c)^2 - 2*(a*b*d^2*f^3*x^5 + (3*a*b*d^2*e*f^2 + 2*a*b*c*d*f^3)
*x^4 + (6*a*b*c*d*e*f^2 + a*b*c^2*f^3 + (3*d^2*e^2*f + f^3)*a*b)*x^3 + 3*(2*a*b*c*d*e^2*f + a*b*c^2*e*f^2 + a*
b*e*f^2)*x^2 + 3*(a*b*c^2*e^2*f + a*b*e^2*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^{3} x^{3} + 3 \, a^{2} e f^{2} x^{2} + 3 \, a^{2} e^{2} f x + a^{2} e^{3} +{\left (b^{2} f^{3} x^{3} + 3 \, b^{2} e f^{2} x^{2} + 3 \, b^{2} e^{2} f x + b^{2} e^{3}\right )} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{3} x^{3} + 3 \, a b e f^{2} x^{2} + 3 \, a b e^{2} f x + a b e^{3}\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)^3*(a+b*arccsch(d*x+c))^2,x, algorithm="fricas")

[Out]

integral(a^2*f^3*x^3 + 3*a^2*e*f^2*x^2 + 3*a^2*e^2*f*x + a^2*e^3 + (b^2*f^3*x^3 + 3*b^2*e*f^2*x^2 + 3*b^2*e^2*
f*x + b^2*e^3)*arccsch(d*x + c)^2 + 2*(a*b*f^3*x^3 + 3*a*b*e*f^2*x^2 + 3*a*b*e^2*f*x + a*b*e^3)*arccsch(d*x +
c), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*(a+b*acsch(d*x+c))**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{3}{\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)^3*(a+b*arccsch(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((f*x + e)^3*(b*arccsch(d*x + c) + a)^2, x)