3.3.4 \(\int \frac {(d+c^2 d x^2) (a+b \sinh ^{-1}(c x))^2}{x^2} \, dx\) [204]

Optimal. Leaf size=131 \[ 2 b^2 c^2 d x-2 b c d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )+2 c^2 d x \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {d \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x}-4 b c d \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )-2 b^2 c d \text {PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d \text {PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right ) \]

[Out]

2*b^2*c^2*d*x+2*c^2*d*x*(a+b*arcsinh(c*x))^2-d*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/x-4*b*c*d*(a+b*arcsinh(c*x))*a
rctanh(c*x+(c^2*x^2+1)^(1/2))-2*b^2*c*d*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*b^2*c*d*polylog(2,c*x+(c^2*x^2+1)^
(1/2))-2*b*c*d*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.25, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5807, 5772, 5798, 8, 5806, 5816, 4267, 2317, 2438} \begin {gather*} -2 b c d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{x}+2 c^2 d x \left (a+b \sinh ^{-1}(c x)\right )^2-4 b c d \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )+2 b^2 c^2 d x-2 b^2 c d \text {Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )+2 b^2 c d \text {Li}_2\left (e^{\sinh ^{-1}(c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*b^2*c^2*d*x - 2*b*c*d*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) + 2*c^2*d*x*(a + b*ArcSinh[c*x])^2 - (d*(1 + c^
2*x^2)*(a + b*ArcSinh[c*x])^2)/x - 4*b*c*d*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - 2*b^2*c*d*PolyLog[2,
-E^ArcSinh[c*x]] + 2*b^2*c*d*PolyLog[2, E^ArcSinh[c*x]]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5798

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

Rule 5806

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

Rule 5807

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*((a + b*ArcSinh[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5816

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

Rubi steps

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

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Mathematica [A]
time = 0.28, size = 192, normalized size = 1.47 \begin {gather*} \frac {d \left (-a^2+a^2 c^2 x^2+2 a b c x \left (-\sqrt {1+c^2 x^2}+c x \sinh ^{-1}(c x)\right )+b^2 c x \left (2 c x-2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)+c x \sinh ^{-1}(c x)^2\right )-2 a b \left (\sinh ^{-1}(c x)+c x \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )\right )-b^2 \left (\sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+2 c x \left (-\log \left (1-e^{-\sinh ^{-1}(c x)}\right )+\log \left (1+e^{-\sinh ^{-1}(c x)}\right )\right )\right )-2 c x \text {PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )+2 c x \text {PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )\right )\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 4.11, size = 239, normalized size = 1.82

method	result	size
derivativedivides	\(c \left (a^{2} d \left (c x -\frac {1}{c x}\right )+b^{2} d \arcsinh \left (c x \right )^{2} c x -2 b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} c d x -\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{c x}+2 b^{2} d \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(239\)
default	\(c \left (a^{2} d \left (c x -\frac {1}{c x}\right )+b^{2} d \arcsinh \left (c x \right )^{2} c x -2 b^{2} d \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 b^{2} c d x -\frac {b^{2} d \arcsinh \left (c x \right )^{2}}{c x}+2 b^{2} d \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 b d a \left (\arcsinh \left (c x \right ) c x -\frac {\arcsinh \left (c x \right )}{c x}-\sqrt {c^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )\right )\right )\)	\(239\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(a^2*d*(c*x-1/c/x)+b^2*d*arcsinh(c*x)^2*c*x-2*b^2*d*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*b^2*c*d*x-b^2*d*arcsinh
(c*x)^2/c/x+2*b^2*d*arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))+2*b^2*d*polylog(2,c*x+(c^2*x^2+1)^(1/2))-2*b^2*d*
arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-2*b^2*d*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+2*b*d*a*(arcsinh(c*x)*c*x-a
rcsinh(c*x)/c/x-(c^2*x^2+1)^(1/2)-arctanh(1/(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((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^2,x, algorithm="maxima")

[Out]

b^2*c^2*d*x*arcsinh(c*x)^2 + 2*b^2*c^2*d*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*c^2*d*x + 2*(c*x*arcsinh
(c*x) - sqrt(c^2*x^2 + 1))*a*b*c*d - 2*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*a*b*d - b^2*d*(log(c*x + sqr
t(c^2*x^2 + 1))^2/x - integrate(2*(c^3*x^2 + sqrt(c^2*x^2 + 1)*c^2*x + c)*log(c*x + sqrt(c^2*x^2 + 1))/(c^3*x^
4 + c*x^2 + (c^2*x^3 + x)*sqrt(c^2*x^2 + 1)), x)) - a^2*d/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((c^2*d*x^2+d)*(a+b*arcsinh(c*x))^2/x^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(Integral(a**2*c**2, x) + Integral(a**2/x**2, x) + Integral(b**2*c**2*asinh(c*x)**2, x) + Integral(b**2*asin
h(c*x)**2/x**2, x) + Integral(2*a*b*c**2*asinh(c*x), x) + Integral(2*a*b*asinh(c*x)/x**2, x))

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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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\,\left (d\,c^2\,x^2+d\right )}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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