\(\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx\) [161]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\frac {\text {Int}\left (\frac {1}{(c+d x) (a+b \text {arcsinh}(c+d x))},x\right )}{e} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{e x (a+b \text {arcsinh}(x))} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x (a+b \text {arcsinh}(x))} \, dx,x,c+d x\right )}{d e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {1}{\left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\frac {\int \frac {1}{a c + a d x + b c \operatorname {asinh}{\left (c + d x \right )} + b d x \operatorname {asinh}{\left (c + d x \right )}}\, dx}{e} \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int { \frac {1}{{\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.63 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c e+d e x) (a+b \text {arcsinh}(c+d x))} \, dx=\int \frac {1}{\left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )} \,d x \]

[In]

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

[Out]

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