3.440 \(\int \frac{1}{\sqrt{1+c^2 x^2} (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=18 \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

-(1/(b*c*(a + b*ArcSinh[c*x])))

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Rubi [A]  time = 0.0447206, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {5675} \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(1/(b*c*(a + b*ArcSinh[c*x])))

Rule 5675

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

Rubi steps

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

Mathematica [A]  time = 0.012315, size = 18, normalized size = 1. \[ -\frac{1}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(1/(b*c*(a + b*ArcSinh[c*x])))

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Maple [A]  time = 0.007, size = 19, normalized size = 1.1 \begin{align*} -{\frac{1}{bc \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b/c/(a+b*arcsinh(c*x))

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Maxima [A]  time = 1.16412, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*arcsinh(c*x) + a)*b*c)

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Fricas [A]  time = 2.24254, size = 66, normalized size = 3.67 \begin{align*} -\frac{1}{b^{2} c \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(b^2*c*log(c*x + sqrt(c^2*x^2 + 1)) + a*b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [A]  time = 1.35751, size = 41, normalized size = 2.28 \begin{align*} -\frac{1}{{\left (b \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) + a\right )} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/((b*log(c*x + sqrt(c^2*x^2 + 1)) + a)*b*c)