3.2.67 \(\int \frac {1}{(c e+d e x) (a+b \sinh ^{-1}(c+d x))^2} \, dx\) [167]

Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2},x\right )}{e} \]

[Out]

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

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Rubi [A]
time = 0.04, 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}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-(d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c)/(a*b*d^4*x^3*e + 3*
a*b*c*d^3*x^2*e + (3*c^2*d^2 + d^2)*a*b*x*e + (c^3*d + c*d)*a*b*e + (b^2*d^4*x^3*e + 3*b^2*c*d^3*x^2*e + (3*c^
2*d^2 + d^2)*b^2*x*e + (c^3*d + c*d)*b^2*e + (b^2*d^3*x^2*e + 2*b^2*c*d^2*x*e + b^2*c^2*d*e)*sqrt(d^2*x^2 + 2*
c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + (a*b*d^3*x^2*e + 2*a*b*c*d^2*x*e + a*b*c^
2*d*e)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - integrate((2*(d^2*x^2 + 2*c*d*x + c^2 + 1)*(d*x + c) + (2*d^2*x^2
+ 4*c*d*x + 2*c^2 + 1)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(a*b*d^6*x^6*e + 6*a*b*c*d^5*x^5*e + (15*c^2*d^4 + 2
*d^4)*a*b*x^4*e + 4*(5*c^3*d^3 + 2*c*d^3)*a*b*x^3*e + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*a*b*x^2*e + 2*(3*c^5*d +
4*c^3*d + c*d)*a*b*x*e + (c^6 + 2*c^4 + c^2)*a*b*e + (a*b*d^4*x^4*e + 4*a*b*c*d^3*x^3*e + 6*a*b*c^2*d^2*x^2*e
+ 4*a*b*c^3*d*x*e + a*b*c^4*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^2*d^6*x^6*e + 6*b^2*c*d^5*x^5*e + (15*c^2*d
^4 + 2*d^4)*b^2*x^4*e + 4*(5*c^3*d^3 + 2*c*d^3)*b^2*x^3*e + (15*c^4*d^2 + 12*c^2*d^2 + d^2)*b^2*x^2*e + 2*(3*c
^5*d + 4*c^3*d + c*d)*b^2*x*e + (c^6 + 2*c^4 + c^2)*b^2*e + (b^2*d^4*x^4*e + 4*b^2*c*d^3*x^3*e + 6*b^2*c^2*d^2
*x^2*e + 4*b^2*c^3*d*x*e + b^2*c^4*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + 2*(b^2*d^5*x^5*e + 5*b^2*c*d^4*x^4*e + (
10*c^2*d^3 + d^3)*b^2*x^3*e + (10*c^3*d^2 + 3*c*d^2)*b^2*x^2*e + (5*c^4*d + 3*c^2*d)*b^2*x*e + (c^5 + c^3)*b^2
*e)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) + 2*(a*b*d^5*x^5*e + 5
*a*b*c*d^4*x^4*e + (10*c^2*d^3 + d^3)*a*b*x^3*e + (10*c^3*d^2 + 3*c*d^2)*a*b*x^2*e + (5*c^4*d + 3*c^2*d)*a*b*x
*e + (c^5 + c^3)*a*b*e)*sqrt(d^2*x^2 + 2*c*d*x + c^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/(d*e*x+c*e)/(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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