3.1.28 \(\int \frac {1}{(d+e x) (a+b \sinh ^{-1}(c x))^2} \, dx\) [28]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable(1/(e*x+d)/(a+b*arcsinh(c*x))^2,x)

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.02, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e x +d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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(1/(e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(a*b*c^3*x^3*e + a*b*c^3*d*x^2 + a*b*c*x*e + a*b*c*d + (b^2*c^3*x^3*e +
b^2*c^3*d*x^2 + b^2*c*x*e + b^2*c*d + (b^2*c^2*x^2*e + b^2*c^2*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2
+ 1)) + (a*b*c^2*x^2*e + a*b*c^2*d*x)*sqrt(c^2*x^2 + 1)) + integrate((c^5*d*x^4 + 2*c^3*d*x^2 + (c^3*d*x^2 -
2*c*x*e - c*d)*(c^2*x^2 + 1) + c*d + (2*c^4*d*x^3 - 2*c^2*x^2*e + c^2*d*x - e)*sqrt(c^2*x^2 + 1))/(a*b*c^5*x^6
*e^2 + 2*a*b*c^5*d*x^5*e + 4*a*b*c^3*d*x^3*e + 2*a*b*c*d*x*e + a*b*c*d^2 + (a*b*c^5*d^2 + 2*a*b*c^3*e^2)*x^4 +
(2*a*b*c^3*d^2 + a*b*c*e^2)*x^2 + (a*b*c^3*x^4*e^2 + 2*a*b*c^3*d*x^3*e + a*b*c^3*d^2*x^2)*(c^2*x^2 + 1) + (b^
2*c^5*x^6*e^2 + 2*b^2*c^5*d*x^5*e + 4*b^2*c^3*d*x^3*e + 2*b^2*c*d*x*e + b^2*c*d^2 + (b^2*c^5*d^2 + 2*b^2*c^3*e
^2)*x^4 + (2*b^2*c^3*d^2 + b^2*c*e^2)*x^2 + (b^2*c^3*x^4*e^2 + 2*b^2*c^3*d*x^3*e + b^2*c^3*d^2*x^2)*(c^2*x^2 +
1) + 2*(b^2*c^4*x^5*e^2 + 2*b^2*c^4*d*x^4*e + 2*b^2*c^2*d*x^2*e + b^2*c^2*d^2*x + (b^2*c^4*d^2 + b^2*c^2*e^2)
*x^3)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^5*e^2 + 2*a*b*c^4*d*x^4*e + 2*a*b*c^2*d*x
^2*e + a*b*c^2*d^2*x + (a*b*c^4*d^2 + a*b*c^2*e^2)*x^3)*sqrt(c^2*x^2 + 1)), x)

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Fricas [A]
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(1/(e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*x*e + a^2*d + (b^2*x*e + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*x*e + a*b*d)*arcsinh(c*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
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(1/(e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((e*x + d)*(b*arcsinh(c*x) + a)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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