\(\int (e+f x) (a+b \text {csch}^{-1}(c+d x))^2 \, dx\) [9]

Optimal result

Integrand size = 18, antiderivative size = 194 \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {b f (c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^2}-\frac {(d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}+\frac {4 b (d e-c f) \left (a+b \text {csch}^{-1}(c+d x)\right ) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}+\frac {b^2 f \log (c+d x)}{d^2}+\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}-\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^2} \]

[Out]

-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*arcc
sch(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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6457, 5577, 4275, 4267, 2317, 2438, 4269, 3556} \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {4 b (d e-c f) \text {arctanh}\left (e^{\text {csch}^{-1}(c+d x)}\right ) \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^2}-\frac {(d e-c f)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {b f (c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} \left (a+b \text {csch}^{-1}(c+d x)\right )}{d^2}+\frac {(e+f x)^2 \left (a+b \text {csch}^{-1}(c+d x)\right )^2}{2 f}+\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,-e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}-\frac {2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{\text {csch}^{-1}(c+d x)}\right )}{d^2}+\frac {b^2 f \log (c+d x)}{d^2} \]

[In]

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

[Out]

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

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 3556

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

Rule 4267

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 4269

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 4275

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

Mathematica [B] (verified)

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 \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\frac {2 a^2 (d e-c f) (c+d x)+a^2 f (c+d x)^2+2 a b f (c+d x) \left (\sqrt {1+\frac {1}{(c+d x)^2}}+(c+d x) \text {csch}^{-1}(c+d x)\right )+2 b^2 f \left ((c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \text {csch}^{-1}(c+d x)+\frac {1}{2} (c+d x)^2 \text {csch}^{-1}(c+d x)^2-\log \left (\frac {1}{c+d x}\right )\right )+4 a b d e \left ((c+d x) \text {csch}^{-1}(c+d x)+\log \left (\frac {\text {csch}\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}\right )-\log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )\right )-4 a b c f \left ((c+d x) \text {csch}^{-1}(c+d x)+\log \left (\frac {\text {csch}\left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )}{2 (c+d x)}\right )-\log \left (\sinh \left (\frac {1}{2} \text {csch}^{-1}(c+d x)\right )\right )\right )+2 b^2 d e \left (\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 b^2 c f \left (\text {csch}^{-1}(c+d x) \left ((c+d x) \text {csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \log \left (1+e^{-\text {csch}^{-1}(c+d x)}\right )\right )-2 \operatorname {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c+d x)}\right )+2 \operatorname {PolyLog}\left (2,e^{-\text {csch}^{-1}(c+d x)}\right )\right )}{2 d^2} \]

[In]

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

[Out]

(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)*Arc
Csch[c + d*x]) + 2*b^2*f*((c + d*x)*Sqrt[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[Csch[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*Lo
g[1 - E^(-ArcCsch[c + d*x])] + 2*Log[1 + E^(-ArcCsch[c + d*x])]) - 2*PolyLog[2, -E^(-ArcCsch[c + d*x])] + 2*Po
lyLog[2, E^(-ArcCsch[c + d*x])]) - 2*b^2*c*f*(ArcCsch[c + d*x]*((c + d*x)*ArcCsch[c + d*x] - 2*Log[1 - E^(-Arc
Csch[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)

Maple [F]

\[\int \left (f x +e \right ) \left (a +b \,\operatorname {arccsch}\left (d x +c \right )\right )^{2}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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)

Sympy [F]

\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acsch}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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 - integrat
e(-((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(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^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)

Giac [F]

\[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcsch}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e+f x) \left (a+b \text {csch}^{-1}(c+d x)\right )^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c+d\,x}\right )\right )}^2 \,d x \]

[In]

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

[Out]

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