3.1.16 \(\int x^2 (a+b \text {csch}^{-1}(c x))^2 \, dx\) [16]

Optimal. Leaf size=122 \[ \frac {b^2 x}{3 c^2}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {csch}^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 \text {PolyLog}\left (2,-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {PolyLog}\left (2,e^{\text {csch}^{-1}(c x)}\right )}{3 c^3} \]

[Out]

1/3*b^2*x/c^2+1/3*x^3*(a+b*arccsch(c*x))^2-2/3*b*(a+b*arccsch(c*x))*arctanh(1/c/x+(1+1/c^2/x^2)^(1/2))/c^3-1/3
*b^2*polylog(2,-1/c/x-(1+1/c^2/x^2)^(1/2))/c^3+1/3*b^2*polylog(2,1/c/x+(1+1/c^2/x^2)^(1/2))/c^3+1/3*b*x^2*(a+b
*arccsch(c*x))*(1+1/c^2/x^2)^(1/2)/c

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Rubi [A]
time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6421, 5560, 4270, 4267, 2317, 2438} \begin {gather*} -\frac {2 b \tanh ^{-1}\left (e^{\text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^3}+\frac {b x^2 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {b^2 \text {Li}_2\left (-e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \text {Li}_2\left (e^{\text {csch}^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 x}{3 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*ArcCsch[c*x])^2,x]

[Out]

(b^2*x)/(3*c^2) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^2*(a + b*ArcCsch[c*x]))/(3*c) + (x^3*(a + b*ArcCsch[c*x])^2)/3 -
(2*b*(a + b*ArcCsch[c*x])*ArcTanh[E^ArcCsch[c*x]])/(3*c^3) - (b^2*PolyLog[2, -E^ArcCsch[c*x]])/(3*c^3) + (b^2*
PolyLog[2, E^ArcCsch[c*x]])/(3*c^3)

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 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 4270

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] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5560

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim
p[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x]
/; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 6421

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rubi steps

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

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Mathematica [A]
time = 0.99, size = 224, normalized size = 1.84 \begin {gather*} \frac {b^2 c x+a b c^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2+a^2 c^3 x^3+b^2 c^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 \text {csch}^{-1}(c x)+2 a b c^3 x^3 \text {csch}^{-1}(c x)+b^2 c^3 x^3 \text {csch}^{-1}(c x)^2-\frac {a b c \sqrt {1+\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )}{\sqrt {1+c^2 x^2}}+b^2 \text {csch}^{-1}(c x) \log \left (1-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \text {csch}^{-1}(c x) \log \left (1+e^{-\text {csch}^{-1}(c x)}\right )+b^2 \text {PolyLog}\left (2,-e^{-\text {csch}^{-1}(c x)}\right )-b^2 \text {PolyLog}\left (2,e^{-\text {csch}^{-1}(c x)}\right )}{3 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*c*x + a*b*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2 + a^2*c^3*x^3 + b^2*c^2*Sqrt[1 + 1/(c^2*x^2)]*x^2*ArcCsch[c*x] +
2*a*b*c^3*x^3*ArcCsch[c*x] + b^2*c^3*x^3*ArcCsch[c*x]^2 - (a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*ArcTanh[(c*x)/Sqrt[1
+ c^2*x^2]])/Sqrt[1 + c^2*x^2] + b^2*ArcCsch[c*x]*Log[1 - E^(-ArcCsch[c*x])] - b^2*ArcCsch[c*x]*Log[1 + E^(-Ar
cCsch[c*x])] + b^2*PolyLog[2, -E^(-ArcCsch[c*x])] - b^2*PolyLog[2, E^(-ArcCsch[c*x])])/(3*c^3)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

1/3*a^2*x^3 + 1/6*(4*x^3*arccsch(c*x) + (2*sqrt(1/(c^2*x^2) + 1)/(c^2*(1/(c^2*x^2) + 1) - c^2) - log(sqrt(1/(c
^2*x^2) + 1) + 1)/c^2 + log(sqrt(1/(c^2*x^2) + 1) - 1)/c^2)/c)*a*b + 1/3*(x^3*log(sqrt(c^2*x^2 + 1) + 1)^2 - 3
*integrate(-1/3*(3*c^2*x^4*log(c)^2 + 3*x^2*log(c)^2 + 3*(c^2*x^4 + x^2)*log(x)^2 + 6*(c^2*x^4*log(c) + x^2*lo
g(c))*log(x) - 2*(3*c^2*x^4*log(c) + 3*x^2*log(c) + 3*(c^2*x^4 + x^2)*log(x) + (c^2*x^4*(3*log(c) + 1) + 3*x^2
*log(c) + 3*(c^2*x^4 + x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + 3*(c^2*x^4*log(c)^2 + x^2*
log(c)^2 + (c^2*x^4 + x^2)*log(x)^2 + 2*(c^2*x^4*log(c) + x^2*log(c))*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^2 + (c
^2*x^2 + 1)^(3/2) + 1), x))*b^2

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

[Out]

integral(b^2*x^2*arccsch(c*x)^2 + 2*a*b*x^2*arccsch(c*x) + a^2*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(a + b*acsch(c*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(x^2*(a+b*arccsch(c*x))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*asinh(1/(c*x)))^2,x)

[Out]

int(x^2*(a + b*asinh(1/(c*x)))^2, x)

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