\(\int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx\) [195]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Int}\left (\frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x},x\right ) \] Output:

Defer(Int)((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x,x)
 

Mathematica [N/A]

Not integrable

Time = 4.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx \] Input:

Integrate[((1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/x,x]
 

Output:

Integrate[((1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/x, x]
 

Rubi [N/A]

Not integrable

Time = 1.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x} \, dx\)

\(\Big \downarrow \) 6223

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \int \frac {1}{\sqrt [6]{c^2 x^2+1}}dx+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 235

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \int \frac {1}{\left (c^2 x^2+1\right )^{7/6}}dx\right )+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 214

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}-\frac {1}{2} \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt [3]{c^2 x^2+1} \int \frac {1}{\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}d\frac {x}{\sqrt {c^2 x^2+1}}\right )+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 233

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \int \frac {\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 833

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\left (1+\sqrt {3}\right ) \int \frac {1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 760

\(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx-\frac {3}{2} b c \left (\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (-\int \frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}d\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}\right )}{4 c^2 x}+\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}\right )+\frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {3}{2} b c \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )+\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx\)

\(\Big \downarrow \) 6239

\(\displaystyle \frac {3}{2} \sqrt [3]{c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {3}{2} b c \left (\frac {3 x}{2 \sqrt [6]{c^2 x^2+1}}+\frac {3 \sqrt [3]{\frac {1}{c^2 x^2+1}} \sqrt {-\frac {c^2 x^2}{c^2 x^2+1}} \left (c^2 x^2+1\right )^{5/6} \left (\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{\sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \left (1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}\right ) \sqrt {\frac {\frac {x^2}{c^2 x^2+1}+\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+1}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}+\sqrt {3}+1}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1} \sqrt {-\frac {1-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}}{\left (-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1\right )^2}}}-\frac {2 \sqrt {\frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}-1}}{-\sqrt [3]{1-\frac {c^2 x^2}{c^2 x^2+1}}-\sqrt {3}+1}\right )}{4 c^2 x}\right )+\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{2/3}}dx\)

Input:

Int[((1 + c^2*x^2)^(1/3)*(a + b*ArcSinh[c*x]))/x,x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.71 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{x}d x\]

Input:

int((c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c))/x,x)
 

Output:

int((c^2*x^2+1)^(1/3)*(a+b*arcsinh(x*c))/x,x)
 

Fricas [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x,x, algorithm="fricas")
 

Output:

integral((c^2*x^2 + 1)^(1/3)*(b*arcsinh(c*x) + a)/x, x)
 

Sympy [N/A]

Not integrable

Time = 11.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \sqrt [3]{c^{2} x^{2} + 1}}{x}\, dx \] Input:

integrate((c**2*x**2+1)**(1/3)*(a+b*asinh(c*x))/x,x)
 

Output:

Integral((a + b*asinh(c*x))*(c**2*x**2 + 1)**(1/3)/x, x)
 

Maxima [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 120, normalized size of antiderivative = 4.80 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {1}{3}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \] Input:

integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x,x, algorithm="maxima")
 

Output:

-1/4*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(c^2*x^2 + 1)^(1/3) + 1)) - 6*(c^2*x 
^2 + 1)^(1/3) + log((c^2*x^2 + 1)^(2/3) + (c^2*x^2 + 1)^(1/3) + 1) - 2*log 
((c^2*x^2 + 1)^(1/3) - 1))*a + b*integrate((c^2*x^2 + 1)^(1/3)*log(c*x + s 
qrt(c^2*x^2 + 1))/x, x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*x^2+1)^(1/3)*(a+b*arcsinh(c*x))/x,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [N/A]

Not integrable

Time = 3.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (c^2\,x^2+1\right )}^{1/3}}{x} \,d x \] Input:

int(((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3))/x,x)
 

Output:

int(((a + b*asinh(c*x))*(c^2*x^2 + 1)^(1/3))/x, x)
 

Reduce [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.88 \[ \int \frac {\sqrt [3]{1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {3 \left (c^{2} x^{2}+1\right )^{\frac {1}{3}} a}{2}+\left (\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}}}{c^{2} x^{3}+x}d x \right ) a +\left (\int \frac {\mathit {asinh} \left (c x \right ) x}{\left (c^{2} x^{2}+1\right )^{\frac {2}{3}}}d x \right ) b \,c^{2}+\left (\int \frac {\left (c^{2} x^{2}+1\right )^{\frac {1}{3}} \mathit {asinh} \left (c x \right )}{c^{2} x^{3}+x}d x \right ) b \] Input:

int((c^2*x^2+1)^(1/3)*(a+b*asinh(c*x))/x,x)
 

Output:

(3*(c**2*x**2 + 1)**(1/3)*a + 2*int((c**2*x**2 + 1)**(1/3)/(c**2*x**3 + x) 
,x)*a + 2*int(((c**2*x**2 + 1)**(1/3)*asinh(c*x)*x)/(c**2*x**2 + 1),x)*b*c 
**2 + 2*int(((c**2*x**2 + 1)**(1/3)*asinh(c*x))/(c**2*x**3 + x),x)*b)/2