3.3.32 \(\int \frac {(a+b \sinh ^{-1}(c x))^2}{x^3 (d+c^2 d x^2)} \, dx\) [232]

Optimal. Leaf size=194 \[ -\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b^2 c^2 \text {PolyLog}\left (3,-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.28, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5809, 5799, 5569, 4267, 2611, 2320, 6724, 5800, 29} \begin {gather*} \frac {b c^2 \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {b c^2 \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {b c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {2 c^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{d}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \log (x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)),x]

[Out]

-((b*c*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(d*x)) - (a + b*ArcSinh[c*x])^2/(2*d*x^2) + (2*c^2*(a + b*ArcSi
nh[c*x])^2*ArcTanh[E^(2*ArcSinh[c*x])])/d + (b^2*c^2*Log[x])/d + (b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, -E^(2*
ArcSinh[c*x])])/d - (b*c^2*(a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])])/d - (b^2*c^2*PolyLog[3, -E^(2*
ArcSinh[c*x])])/(2*d) + (b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/(2*d)

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 5569

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

Rule 5799

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Dist[1/d, Subst[Int[(
a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n
, 0]

Rule 5800

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(
d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]
/; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]

Rule 5809

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Dist[c^2*((m + 2*p + 3)/(f^2*
(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /;
FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^3 \left (d+c^2 d x^2\right )} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}-c^2 \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x \left (d+c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \sqrt {1+c^2 x^2}} \, dx}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}-\frac {c^2 \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^2\right ) \int \frac {1}{x} \, dx}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {b^2 c^2 \log (x)}{d}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{2 d x^2}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b c^2 \left (a+b \sinh ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{d}-\frac {b^2 c^2 \text {Li}_3\left (-e^{2 \sinh ^{-1}(c x)}\right )}{2 d}+\frac {b^2 c^2 \text {Li}_3\left (e^{2 \sinh ^{-1}(c x)}\right )}{2 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.73, size = 419, normalized size = 2.16 \begin {gather*} -\frac {\frac {1}{12} i b^2 c^2 \pi ^3+\frac {a^2}{x^2}+\frac {2 a b c \sqrt {1+c^2 x^2}}{x}+\frac {2 a b \sinh ^{-1}(c x)}{x^2}+\frac {2 b^2 c \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{x}+\frac {b^2 \sinh ^{-1}(c x)^2}{x^2}-\frac {4}{3} b^2 c^2 \sinh ^{-1}(c x)^3-2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )-4 a b c^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-4 a b c^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )+4 a b c^2 \sinh ^{-1}(c x) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+2 b^2 c^2 \sinh ^{-1}(c x)^2 \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )+2 a^2 c^2 \log (x)-2 b^2 c^2 \log (c x)-a^2 c^2 \log \left (1+c^2 x^2\right )+2 b^2 c^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )-4 a b c^2 \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )-4 a b c^2 \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+2 a b c^2 \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+2 b^2 c^2 \sinh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )+b^2 c^2 \text {PolyLog}\left (3,-e^{-2 \sinh ^{-1}(c x)}\right )-b^2 c^2 \text {PolyLog}\left (3,e^{2 \sinh ^{-1}(c x)}\right )}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)),x]

[Out]

-1/2*((I/12)*b^2*c^2*Pi^3 + a^2/x^2 + (2*a*b*c*Sqrt[1 + c^2*x^2])/x + (2*a*b*ArcSinh[c*x])/x^2 + (2*b^2*c*Sqrt
[1 + c^2*x^2]*ArcSinh[c*x])/x + (b^2*ArcSinh[c*x]^2)/x^2 - (4*b^2*c^2*ArcSinh[c*x]^3)/3 - 2*b^2*c^2*ArcSinh[c*
x]^2*Log[1 + E^(-2*ArcSinh[c*x])] - 4*a*b*c^2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] - 4*a*b*c^2*ArcSinh[c*x]*
Log[1 + I*E^ArcSinh[c*x]] + 4*a*b*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x]^2*Log[
1 - E^(2*ArcSinh[c*x])] + 2*a^2*c^2*Log[x] - 2*b^2*c^2*Log[c*x] - a^2*c^2*Log[1 + c^2*x^2] + 2*b^2*c^2*ArcSinh
[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 4*a*b*c^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] - 4*a*b*c^2*PolyLog[2, I*E^
ArcSinh[c*x]] + 2*a*b*c^2*PolyLog[2, E^(2*ArcSinh[c*x])] + 2*b^2*c^2*ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x]
)] + b^2*c^2*PolyLog[3, -E^(-2*ArcSinh[c*x])] - b^2*c^2*PolyLog[3, E^(2*ArcSinh[c*x])])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(669\) vs. \(2(231)=462\).
time = 3.88, size = 670, normalized size = 3.45

method	result	size
derivativedivides	\(c^{2} \left (\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d \,c^{2} x^{2}}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {a b \arcsinh \left (c x \right )}{d \,c^{2} x^{2}}-\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{d c x}-\frac {a^{2} \ln \left (c x \right )}{d}+\frac {2 b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {a b}{d}-\frac {a b \sqrt {c^{2} x^{2}+1}}{d c x}-\frac {2 a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right )}{d}-\frac {a^{2}}{2 d \,c^{2} x^{2}}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}\right )\)	\(670\)
default	\(c^{2} \left (\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}-\frac {b^{2} \polylog \left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2 d}-\frac {b^{2} \arcsinh \left (c x \right )^{2}}{2 d \,c^{2} x^{2}}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {a b \arcsinh \left (c x \right )}{d \,c^{2} x^{2}}-\frac {b^{2} \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}}{d c x}-\frac {a^{2} \ln \left (c x \right )}{d}+\frac {2 b^{2} \polylog \left (3, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {2 b^{2} \polylog \left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {a b}{d}-\frac {a b \sqrt {c^{2} x^{2}+1}}{d c x}-\frac {2 a b \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 a b \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right )}{d}-\frac {a^{2}}{2 d \,c^{2} x^{2}}+\frac {2 a b \arcsinh \left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{d}-\frac {2 b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {b^{2} \arcsinh \left (c x \right ) \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}+\frac {a b \polylog \left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{d}\right )\)	\(670\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d),x,method=_RETURNVERBOSE)

[Out]

c^2*(1/2*a^2/d*ln(c^2*x^2+1)-b^2/d*arcsinh(c*x)/c/x*(c^2*x^2+1)^(1/2)+b^2/d*arcsinh(c*x)^2*ln(1+(c*x+(c^2*x^2+
1)^(1/2))^2)+b^2/d*arcsinh(c*x)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-2*b^2/d*arcsinh(c*x)*polylog(2,-c*x-(c^2
*x^2+1)^(1/2))-2*a*b/d*polylog(2,c*x+(c^2*x^2+1)^(1/2))+b^2/d*arcsinh(c*x)-a*b/d*arcsinh(c*x)/c^2/x^2+a*b/d*po
lylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)+2*a*b/d*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)-a^2/d*ln(c*x)+a*b/d-a
*b/d/c/x*(c^2*x^2+1)^(1/2)+2*b^2/d*polylog(3,c*x+(c^2*x^2+1)^(1/2))+2*b^2/d*polylog(3,-c*x-(c^2*x^2+1)^(1/2))-
b^2/d*arcsinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))-2*b^2/d*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))-b^2/d*a
rcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))-1/2*b^2/d*arcsinh(c*x)^2/c^2/x^2-2*a*b/d*arcsinh(c*x)*ln(1-c*x-(c^2*
x^2+1)^(1/2))-1/2*a^2/d/c^2/x^2-1/2*b^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/d+b^2/d*ln(1+c*x+(c^2*x^2+1)^(1/
2))-2*b^2/d*ln(c*x+(c^2*x^2+1)^(1/2))+b^2/d*ln(c*x+(c^2*x^2+1)^(1/2)-1)-2*a*b/d*polylog(2,-c*x-(c^2*x^2+1)^(1/
2))-2*a*b/d*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2)))

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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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d),x, algorithm="maxima")

[Out]

1/2*(c^2*log(c^2*x^2 + 1)/d - 2*c^2*log(x)/d - 1/(d*x^2))*a^2 + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(
c^2*d*x^5 + d*x^3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^5 + d*x^3), x)

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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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d),x, algorithm="fricas")

[Out]

integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^2*d*x^5 + d*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a^{2}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2/x**3/(c**2*d*x**2+d),x)

[Out]

(Integral(a**2/(c**2*x**5 + x**3), x) + Integral(b**2*asinh(c*x)**2/(c**2*x**5 + x**3), x) + Integral(2*a*b*as
inh(c*x)/(c**2*x**5 + x**3), x))/d

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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((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)*x^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,\left (d\,c^2\,x^2+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)),x)

[Out]

int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)), x)

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