3.3.55 \(\int \frac {(a+b \sinh ^{-1}(c x))^2}{\sqrt {\pi +c^2 \pi x^2}} \, dx\) [255]

Optimal. Leaf size=25 \[ \frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {\pi }} \]

[Out]

1/3*(a+b*arcsinh(c*x))^3/b/c/Pi^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {5783} \begin {gather*} \frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 \sqrt {\pi } b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])

Rule 5783

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {\pi +c^2 \pi x^2}} \, dx &=\frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {\pi }}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {\left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {\pi }} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b*ArcSinh[c*x])^3/(3*b*c*Sqrt[Pi])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(21)=42\).
time = 0.81, size = 72, normalized size = 2.88

method	result	size
default	\(\frac {a^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\sqrt {\pi \,c^{2}}}+\frac {b^{2} \arcsinh \left (c x \right )^{3}}{3 c \sqrt {\pi }}+\frac {a b \arcsinh \left (c x \right )^{2}}{c \sqrt {\pi }}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/3*b^2/c/Pi^(1/2)*arcsinh(c*x)^3+a*b*arc
sinh(c*x)^2/c/Pi^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (21) = 42\).
time = 0.29, size = 47, normalized size = 1.88 \begin {gather*} \frac {b^{2} \operatorname {arsinh}\left (c x\right )^{3}}{3 \, \sqrt {\pi } c} + \frac {a b \operatorname {arsinh}\left (c x\right )^{2}}{\sqrt {\pi } c} + \frac {a^{2} \operatorname {arsinh}\left (c x\right )}{\sqrt {\pi } c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsinh(c*x))^2/(pi*c^2*x^2+pi)^(1/2),x, algorithm="maxima")

[Out]

1/3*b^2*arcsinh(c*x)^3/(sqrt(pi)*c) + a*b*arcsinh(c*x)^2/(sqrt(pi)*c) + a^2*arcsinh(c*x)/(sqrt(pi)*c)

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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/(pi*c^2*x^2+pi)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
time = 1.62, size = 88, normalized size = 3.52 \begin {gather*} \begin {cases} a^{2} \left (\begin {cases} \frac {\sqrt {- \frac {1}{c^{2}}} \operatorname {asin}{\left (x \sqrt {- c^{2}} \right )}}{\sqrt {\pi }} & \text {for}\: \pi c^{2} < 0 \\\frac {\sqrt {\frac {1}{c^{2}}} \operatorname {asinh}{\left (x \sqrt {c^{2}} \right )}}{\sqrt {\pi }} & \text {for}\: \pi c^{2} > 0 \end {cases}\right ) & \text {for}\: b = 0 \\\frac {a^{2} x}{\sqrt {\pi }} & \text {for}\: c = 0 \\\frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{3}}{3 \sqrt {\pi } b c} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*Piecewise((sqrt(-1/c**2)*asin(x*sqrt(-c**2))/sqrt(pi), pi*c**2 < 0), (sqrt(c**(-2))*asinh(x*sq
rt(c**2))/sqrt(pi), pi*c**2 > 0)), Eq(b, 0)), (a**2*x/sqrt(pi), Eq(c, 0)), ((a + b*asinh(c*x))**3/(3*sqrt(pi)*
b*c), True))

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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/(pi*c^2*x^2+pi)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2/sqrt(pi + pi*c^2*x^2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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