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

Optimal. Leaf size=26 \[ \frac{\text{Unintegrable}\left (\frac{1}{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2},x\right )}{e} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0636682, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \]

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

Mathematica [A]  time = 1.40631, size = 0, normalized size = 0. \[ \int \frac{1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.125, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dex+ce \right ) \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{2}}}\, dx \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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*e*x^3 + 3*
a*b*c*d^3*e*x^2 + (3*c^2*d^2*e + d^2*e)*a*b*x + (c^3*d*e + c*d*e)*a*b + (b^2*d^4*e*x^3 + 3*b^2*c*d^3*e*x^2 + (
3*c^2*d^2*e + d^2*e)*b^2*x + (c^3*d*e + c*d*e)*b^2 + (b^2*d^3*e*x^2 + 2*b^2*c*d^2*e*x + 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*e*x^2 + 2*a*b*c*d^2*e*x
+ 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*e*x^6 + 6*a*b*c*d^5*e*x^5 + (15*c^2
*d^4*e + 2*d^4*e)*a*b*x^4 + 4*(5*c^3*d^3*e + 2*c*d^3*e)*a*b*x^3 + (15*c^4*d^2*e + 12*c^2*d^2*e + d^2*e)*a*b*x^
2 + 2*(3*c^5*d*e + 4*c^3*d*e + c*d*e)*a*b*x + (c^6*e + 2*c^4*e + c^2*e)*a*b + (a*b*d^4*e*x^4 + 4*a*b*c*d^3*e*x
^3 + 6*a*b*c^2*d^2*e*x^2 + 4*a*b*c^3*d*e*x + a*b*c^4*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^2*d^6*e*x^6 + 6*b^2
*c*d^5*e*x^5 + (15*c^2*d^4*e + 2*d^4*e)*b^2*x^4 + 4*(5*c^3*d^3*e + 2*c*d^3*e)*b^2*x^3 + (15*c^4*d^2*e + 12*c^2
*d^2*e + d^2*e)*b^2*x^2 + 2*(3*c^5*d*e + 4*c^3*d*e + c*d*e)*b^2*x + (c^6*e + 2*c^4*e + c^2*e)*b^2 + (b^2*d^4*e
*x^4 + 4*b^2*c*d^3*e*x^3 + 6*b^2*c^2*d^2*e*x^2 + 4*b^2*c^3*d*e*x + b^2*c^4*e)*(d^2*x^2 + 2*c*d*x + c^2 + 1) +
2*(b^2*d^5*e*x^5 + 5*b^2*c*d^4*e*x^4 + (10*c^2*d^3*e + d^3*e)*b^2*x^3 + (10*c^3*d^2*e + 3*c*d^2*e)*b^2*x^2 + (
5*c^4*d*e + 3*c^2*d*e)*b^2*x + (c^5*e + c^3*e)*b^2)*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*e*x^5 + 5*a*b*c*d^4*e*x^4 + (10*c^2*d^3*e + d^3*e)*a*b*x^3 + (10*c^3*d^
2*e + 3*c*d^2*e)*a*b*x^2 + (5*c^4*d*e + 3*c^2*d*e)*a*b*x + (c^5*e + c^3*e)*a*b)*sqrt(d^2*x^2 + 2*c*d*x + c^2 +
1)), x)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a^{2} d e x + a^{2} c e +{\left (b^{2} d e x + b^{2} c e\right )} \operatorname{arsinh}\left (d x + c\right )^{2} + 2 \,{\left (a b d e x + a b c e\right )} \operatorname{arsinh}\left (d x + c\right )}, x\right ) \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d e x + c e\right )}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

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)