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

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \]

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 = 1.44, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^2} \, dx \]

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]

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d e x + a^{2} c e + {\left (b^{2} d e x + b^{2} c e\right )} \operatorname {arsinh}\left (d x + c\right )^{2} + 2 \, {\left (a b d e x + a b c e\right )} \operatorname {arsinh}\left (d x + c\right )}, x\right ) \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

maple [A]  time = 0.17, 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)

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (3 \, c^{2} d + d\right )} x + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac {3}{2}} + c}{a b d^{4} e x^{3} + 3 \, a b c d^{3} e x^{2} + {\left (3 \, c^{2} d^{2} e + d^{2} e\right )} a b x + {\left (c^{3} d e + c d e\right )} a b + {\left (b^{2} d^{4} e x^{3} + 3 \, b^{2} c d^{3} e x^{2} + {\left (3 \, c^{2} d^{2} e + d^{2} e\right )} b^{2} x + {\left (c^{3} d e + c d e\right )} b^{2} + {\left (b^{2} d^{3} e x^{2} + 2 \, b^{2} c d^{2} e x + b^{2} c^{2} d e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (a b d^{3} e x^{2} + 2 \, a b c d^{2} e x + a b c^{2} d e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}} - \int \frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} {\left (d x + c\right )} + {\left (2 \, d^{2} x^{2} + 4 \, c d x + 2 \, c^{2} + 1\right )} \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} + {\left (15 \, c^{2} d^{4} e + 2 \, d^{4} e\right )} a b x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e + 2 \, c d^{3} e\right )} a b x^{3} + {\left (15 \, c^{4} d^{2} e + 12 \, c^{2} d^{2} e + d^{2} e\right )} a b x^{2} + 2 \, {\left (3 \, c^{5} d e + 4 \, c^{3} d e + c d e\right )} a b x + {\left (c^{6} e + 2 \, c^{4} e + c^{2} e\right )} a b + {\left (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\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + {\left (b^{2} d^{6} e x^{6} + 6 \, b^{2} c d^{5} e x^{5} + {\left (15 \, c^{2} d^{4} e + 2 \, d^{4} e\right )} b^{2} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e + 2 \, c d^{3} e\right )} b^{2} x^{3} + {\left (15 \, c^{4} d^{2} e + 12 \, c^{2} d^{2} e + d^{2} e\right )} b^{2} x^{2} + 2 \, {\left (3 \, c^{5} d e + 4 \, c^{3} d e + c d e\right )} b^{2} x + {\left (c^{6} e + 2 \, c^{4} e + c^{2} e\right )} b^{2} + {\left (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\right )} {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} + 2 \, {\left (b^{2} d^{5} e x^{5} + 5 \, b^{2} c d^{4} e x^{4} + {\left (10 \, c^{2} d^{3} e + d^{3} e\right )} b^{2} x^{3} + {\left (10 \, c^{3} d^{2} e + 3 \, c d^{2} e\right )} b^{2} x^{2} + {\left (5 \, c^{4} d e + 3 \, c^{2} d e\right )} b^{2} x + {\left (c^{5} e + c^{3} e\right )} b^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + 2 \, {\left (a b d^{5} e x^{5} + 5 \, a b c d^{4} e x^{4} + {\left (10 \, c^{2} d^{3} e + d^{3} e\right )} a b x^{3} + {\left (10 \, c^{3} d^{2} e + 3 \, c d^{2} e\right )} a b x^{2} + {\left (5 \, c^{4} d e + 3 \, c^{2} d e\right )} a b x + {\left (c^{5} e + c^{3} e\right )} a b\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}\,{d x} \]

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________