3.3.97 \(\int \frac {(a+b \sinh ^{-1}(c x))^2}{x^2 \sqrt {d+c^2 d x^2}} \, dx\) [297]

Optimal. Leaf size=167 \[ \frac {c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}}-\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {2 b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}}-\frac {b^2 c \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {d+c^2 d x^2}} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5800, 5775, 3797, 2221, 2317, 2438} \begin {gather*} -\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{d x}+\frac {c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {c^2 d x^2+d}}+\frac {2 b c \sqrt {c^2 x^2+1} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d}}-\frac {b^2 c \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{\sqrt {c^2 d x^2+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2221

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

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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && 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]

Rubi steps

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

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Mathematica [A]
time = 0.30, size = 168, normalized size = 1.01 \begin {gather*} \frac {b^2 \left (-1-c^2 x^2+c x \sqrt {1+c^2 x^2}\right ) \sinh ^{-1}(c x)^2-2 b \sinh ^{-1}(c x) \left (a+a c^2 x^2-b c x \sqrt {1+c^2 x^2} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )\right )-a \left (a+a c^2 x^2-2 b c x \sqrt {1+c^2 x^2} \log (c x)\right )-b^2 c x \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{x \sqrt {d+c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(525\) vs. \(2(171)=342\).
time = 2.13, size = 526, normalized size = 3.15

method	result	size
default	\(-\frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{d x}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{\left (c^{2} x^{2}+1\right ) x d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \polylog \left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) c}{\sqrt {c^{2} x^{2}+1}\, d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x \,c^{2}}{\left (c^{2} x^{2}+1\right ) d}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{\left (c^{2} x^{2}+1\right ) x d}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right ) c}{\sqrt {c^{2} x^{2}+1}\, d}\)	\(526\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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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^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(c^2*d*x^2 + d)*(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^2*d*x^4 + d*x^2), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asinh(c*x))**2/x**2/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asinh(c*x))**2/(x**2*sqrt(d*(c**2*x**2 + 1))), 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((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/(sqrt(c^2*d*x^2 + d)*x^2), 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^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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