\(\int \frac {\log (h (f+g x)^m)}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx\) [57]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\text {Int}\left (\frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]

[Out]

Unintegrable(ln(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Int[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Defer[Int][Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \]

[In]

Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])),x]

[Out]

Integrate[Log[h*(f + g*x)^m]/(Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 1.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94

\[\int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right ) \sqrt {c^{2} x^{2}+1}}d x\]

[In]

int(ln(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

[Out]

int(ln(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)*log((g*x + f)^m*h)/(a*c^2*x^2 + (b*c^2*x^2 + b)*arcsinh(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 7.69 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\log {\left (h \left (f + g x\right )^{m} \right )}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt {c^{2} x^{2} + 1}}\, dx \]

[In]

integrate(ln(h*(g*x+f)**m)/(a+b*asinh(c*x))/(c**2*x**2+1)**(1/2),x)

[Out]

Integral(log(h*(f + g*x)**m)/((a + b*asinh(c*x))*sqrt(c**2*x**2 + 1)), x)

Maxima [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(log((g*x + f)^m*h)/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)

Giac [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}} \,d x } \]

[In]

integrate(log(h*(g*x+f)^m)/(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(log((g*x + f)^m*h)/(sqrt(c^2*x^2 + 1)*(b*arcsinh(c*x) + a)), x)

Mupad [N/A]

Not integrable

Time = 2.55 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {c^2\,x^2+1}} \,d x \]

[In]

int(log(h*(f + g*x)^m)/((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/2)),x)

[Out]

int(log(h*(f + g*x)^m)/((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/2)), x)