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

Optimal. Leaf size=194 \[ \frac{2 b^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{2 b^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{(d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{4 b (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^2}+\frac{b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f}+\frac{b^2 f \log (c+d x)}{d^2} \]

[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

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Rubi [A]  time = 0.273336, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {6322, 5469, 4190, 4182, 2279, 2391, 4184, 3475} \[ \frac{2 b^2 (d e-c f) \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{2 b^2 (d e-c f) \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{(d e-c f)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac{4 b (d e-c f) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^2}+\frac{b f (c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left (a+b \text{csch}^{-1}(c+d x)\right )}{d^2}+\frac{(e+f x)^2 \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{2 f}+\frac{b^2 f \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

[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 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]

Rubi steps

\begin{align*} \int (e+f x) \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)) \, 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 \operatorname{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 \operatorname{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) \operatorname{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)) \operatorname{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 ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{\left (b^2 f\right ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \tanh ^{-1}\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 ) \operatorname{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 ) \operatorname{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 ) \tanh ^{-1}\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) \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}-\frac{2 b^2 (d e-c f) \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d^2}\\ \end{align*}

Mathematica [B]  time = 1.38779, size = 401, normalized size = 2.07 \[ \frac{2 b^2 d e \left (-2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )+\text{csch}^{-1}(c+d x) \left ((c+d x) \text{csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )+2 \log \left (e^{-\text{csch}^{-1}(c+d x)}+1\right )\right )\right )-2 b^2 c f \left (-2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )+2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )+\text{csch}^{-1}(c+d x) \left ((c+d x) \text{csch}^{-1}(c+d x)-2 \log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )+2 \log \left (e^{-\text{csch}^{-1}(c+d x)}+1\right )\right )\right )+2 a^2 (c+d x) (d e-c f)+a^2 f (c+d x)^2+2 a b d e \left (2 (c+d x) \text{csch}^{-1}(c+d x)-2 \log \left (2 (c+d x) \sinh ^2\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )\right )\right )+2 a b f (c+d x) \left (\sqrt{\frac{1}{(c+d x)^2}+1}+(c+d x) \text{csch}^{-1}(c+d x)\right )+2 a b c f \left (2 \log \left (2 (c+d x) \sinh ^2\left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )\right )-2 (c+d x) \text{csch}^{-1}(c+d x)\right )+2 b^2 f \left (-\log \left (\frac{1}{c+d x}\right )+\frac{1}{2} (c+d x)^2 \text{csch}^{-1}(c+d x)^2+(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \text{csch}^{-1}(c+d x)\right )}{2 d^2} \]

Warning: Unable to verify antiderivative.

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

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Maple [F]  time = 0.222, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) \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)*(a+b*arccsch(d*x+c))^2,x)

[Out]

int((f*x+e)*(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)*(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)

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

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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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

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