Integrand size = 12, antiderivative size = 140 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=a x-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{2 c^{2/3}}+\frac {\sqrt {3} b \arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{2 c^{2/3}}-\frac {b \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{c^{2/3}}+b x \text {arctanh}\left (c x^{3/2}\right )-\frac {b \text {arctanh}\left (\frac {\sqrt [3]{c} \sqrt {x}}{1+c^{2/3} x}\right )}{2 c^{2/3}} \] Output:
a*x-1/2*3^(1/2)*b*arctan(1/3*(1-2*c^(1/3)*x^(1/2))*3^(1/2))/c^(2/3)+1/2*3^ (1/2)*b*arctan(1/3*(1+2*c^(1/3)*x^(1/2))*3^(1/2))/c^(2/3)-b*arctanh(c^(1/3 )*x^(1/2))/c^(2/3)+b*x*arctanh(c*x^(3/2))-1/2*b*arctanh(c^(1/3)*x^(1/2)/(1 +c^(2/3)*x))/c^(2/3)
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.81 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=a x+b x \text {arctanh}\left (c x^{3/2}\right )-\frac {b \left (\sqrt {3} \left (\arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )-\arctan \left (\frac {1+2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )\right )+2 \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )+\text {arctanh}\left (\frac {\sqrt [3]{c} \sqrt {x}}{1+c^{2/3} x}\right )\right )}{2 c^{2/3}} \] Input:
Integrate[a + b*ArcTanh[c*x^(3/2)],x]
Output:
a*x + b*x*ArcTanh[c*x^(3/2)] - (b*(Sqrt[3]*(ArcTan[(1 - 2*c^(1/3)*Sqrt[x]) /Sqrt[3]] - ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]]) + 2*ArcTanh[c^(1/3)*S qrt[x]] + ArcTanh[(c^(1/3)*Sqrt[x])/(1 + c^(2/3)*x)]))/(2*c^(2/3))
Time = 0.66 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.21, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2009}
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.
| \(\displaystyle \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a x-\frac {\sqrt {3} b \arctan \left (\frac {1-2 \sqrt [3]{c} \sqrt {x}}{\sqrt {3}}\right )}{2 c^{2/3}}+\frac {\sqrt {3} b \arctan \left (\frac {2 \sqrt [3]{c} \sqrt {x}+1}{\sqrt {3}}\right )}{2 c^{2/3}}-\frac {b \text {arctanh}\left (\sqrt [3]{c} \sqrt {x}\right )}{c^{2/3}}+b x \text {arctanh}\left (c x^{3/2}\right )+\frac {b \log \left (c^{2/3} x-\sqrt [3]{c} \sqrt {x}+1\right )}{4 c^{2/3}}-\frac {b \log \left (c^{2/3} x+\sqrt [3]{c} \sqrt {x}+1\right )}{4 c^{2/3}}\) |
Input:
Int[a + b*ArcTanh[c*x^(3/2)],x]
Output:
a*x - (Sqrt[3]*b*ArcTan[(1 - 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) + (S qrt[3]*b*ArcTan[(1 + 2*c^(1/3)*Sqrt[x])/Sqrt[3]])/(2*c^(2/3)) - (b*ArcTanh [c^(1/3)*Sqrt[x]])/c^(2/3) + b*x*ArcTanh[c*x^(3/2)] + (b*Log[1 - c^(1/3)*S qrt[x] + c^(2/3)*x])/(4*c^(2/3)) - (b*Log[1 + c^(1/3)*Sqrt[x] + c^(2/3)*x] )/(4*c^(2/3))
Time = 0.20 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.28
| method | result | size |
| derivativedivides | \(a x +b x \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(179\) |
| default | \(a x +b x \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(179\) |
| parts | \(a x +b x \,\operatorname {arctanh}\left (c \,x^{\frac {3}{2}}\right )+\frac {b \ln \left (\sqrt {x}-\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b \ln \left (\sqrt {x}+\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}} \sqrt {x}+\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \sqrt {x}}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{2 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\) | \(179\) |
Input:
int(a+b*arctanh(c*x^(3/2)),x,method=_RETURNVERBOSE)
Output:
a*x+b*x*arctanh(c*x^(3/2))+1/2*b/c/(1/c)^(1/3)*ln(x^(1/2)-(1/c)^(1/3))-1/4 *b/c/(1/c)^(1/3)*ln(x+(1/c)^(1/3)*x^(1/2)+(1/c)^(2/3))+1/2*b*3^(1/2)/c/(1/ c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^(1/2)+1))-1/2*b/c/(1/c)^(1/3) *ln(x^(1/2)+(1/c)^(1/3))+1/4*b/c/(1/c)^(1/3)*ln(x-(1/c)^(1/3)*x^(1/2)+(1/c )^(2/3))+1/2*b*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x^( 1/2)-1))
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 1914, normalized size of antiderivative = 13.67 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\text {Too large to display} \] Input:
integrate(a+b*arctanh(c*x^(3/2)),x, algorithm="fricas")
Output:
a*x + 1/8*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3 ) + 1) - 4*b - 2*sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2 )^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3*b^2))*log(1/4*((1/2)^(1/3)* (b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2*c - ((1 /2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b) *b*c + b^2*c + 2*b^2*sqrt(x) + 1/2*sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^2 - 1) *b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3*b^2)*(((1 /2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b) *c - 2*b*c)) + 1/8*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)* (I*sqrt(3) + 1) - 4*b + 2*sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/3)*(b^3 - (c^2 - 1 )*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3*b^2))*log(1/4*((1/ 2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^ 2*c - ((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b*c + b^2*c + 2*b^2*sqrt(x) - 1/2*sqrt(-3/4*((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)^2 + 3*((1/2)^(1/ 3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3) + 1) + 2*b)*b - 3* b^2)*(((1/2)^(1/3)*(b^3 - (c^2 - 1)*b^3/c^2 + b^3/c^2)^(1/3)*(I*sqrt(3)...
Timed out. \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\text {Timed out} \] Input:
integrate(a+b*atanh(c*x**(3/2)),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.13 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} \sqrt {x} - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} - \frac {\log \left (c^{\frac {2}{3}} x + c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {5}{3}}} + \frac {\log \left (c^{\frac {2}{3}} x - c^{\frac {1}{3}} \sqrt {x} + 1\right )}{c^{\frac {5}{3}}} - \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} + 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}} + \frac {2 \, \log \left (\frac {c^{\frac {1}{3}} \sqrt {x} - 1}{c^{\frac {1}{3}}}\right )}{c^{\frac {5}{3}}}\right )} + 4 \, x \operatorname {artanh}\left (c x^{\frac {3}{2}}\right )\right )} b + a x \] Input:
integrate(a+b*arctanh(c*x^(3/2)),x, algorithm="maxima")
Output:
1/4*(c*(2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) + c^(1/3))/c^(1/3) )/c^(5/3) + 2*sqrt(3)*arctan(1/3*sqrt(3)*(2*c^(2/3)*sqrt(x) - c^(1/3))/c^( 1/3))/c^(5/3) - log(c^(2/3)*x + c^(1/3)*sqrt(x) + 1)/c^(5/3) + log(c^(2/3) *x - c^(1/3)*sqrt(x) + 1)/c^(5/3) - 2*log((c^(1/3)*sqrt(x) + 1)/c^(1/3))/c ^(5/3) + 2*log((c^(1/3)*sqrt(x) - 1)/c^(1/3))/c^(5/3)) + 4*x*arctanh(c*x^( 3/2)))*b + a*x
Time = 0.14 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.33 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {1}{4} \, {\left (c {\left (\frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} + \frac {2 \, \sqrt {3} {\left | c \right |}^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )} {\left | c \right |}^{\frac {1}{3}}\right )}{c^{2}} - \frac {{\left | c \right |}^{\frac {1}{3}} \log \left (x + \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} + \frac {{\left | c \right |}^{\frac {1}{3}} \log \left (x - \frac {\sqrt {x}}{{\left | c \right |}^{\frac {1}{3}}} + \frac {1}{{\left | c \right |}^{\frac {2}{3}}}\right )}{c^{2}} - \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \log \left (\sqrt {x} + \frac {1}{{\left | c \right |}^{\frac {1}{3}}}\right )}{c^{2}} + \frac {2 \, {\left | c \right |}^{\frac {1}{3}} \log \left ({\left | \sqrt {x} - \frac {1}{{\left | c \right |}^{\frac {1}{3}}} \right |}\right )}{c^{2}}\right )} + 2 \, x \log \left (-\frac {c x^{\frac {3}{2}} + 1}{c x^{\frac {3}{2}} - 1}\right )\right )} b + a x \] Input:
integrate(a+b*arctanh(c*x^(3/2)),x, algorithm="giac")
Output:
1/4*(c*(2*sqrt(3)*abs(c)^(1/3)*arctan(1/3*sqrt(3)*(2*sqrt(x) + 1/abs(c)^(1 /3))*abs(c)^(1/3))/c^2 + 2*sqrt(3)*abs(c)^(1/3)*arctan(1/3*sqrt(3)*(2*sqrt (x) - 1/abs(c)^(1/3))*abs(c)^(1/3))/c^2 - abs(c)^(1/3)*log(x + sqrt(x)/abs (c)^(1/3) + 1/abs(c)^(2/3))/c^2 + abs(c)^(1/3)*log(x - sqrt(x)/abs(c)^(1/3 ) + 1/abs(c)^(2/3))/c^2 - 2*abs(c)^(1/3)*log(sqrt(x) + 1/abs(c)^(1/3))/c^2 + 2*abs(c)^(1/3)*log(abs(sqrt(x) - 1/abs(c)^(1/3)))/c^2) + 2*x*log(-(c*x^ (3/2) + 1)/(c*x^(3/2) - 1)))*b + a*x
Time = 8.19 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=a\,x+b\,x\,\mathrm {atanh}\left (c\,x^{3/2}\right )-\frac {b\,\mathrm {atanh}\left (c^{1/3}\,\sqrt {x}\right )}{c^{2/3}}+\frac {b\,\mathrm {atanh}\left (\frac {486\,c^8\,\sqrt {x}}{-243\,c^{23/3}+\sqrt {3}\,c^{23/3}\,243{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,c^{2/3}}+\frac {b\,\mathrm {atanh}\left (\frac {486\,c^8\,\sqrt {x}}{243\,c^{23/3}+\sqrt {3}\,c^{23/3}\,243{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,c^{2/3}} \] Input:
int(a + b*atanh(c*x^(3/2)),x)
Output:
a*x + b*x*atanh(c*x^(3/2)) - (b*atanh(c^(1/3)*x^(1/2)))/c^(2/3) + (b*atanh ((486*c^8*x^(1/2))/(3^(1/2)*c^(23/3)*243i - 243*c^(23/3)))*(3^(1/2)*1i + 1 ))/(2*c^(2/3)) + (b*atanh((486*c^8*x^(1/2))/(3^(1/2)*c^(23/3)*243i + 243*c ^(23/3)))*(3^(1/2)*1i - 1))/(2*c^(2/3))
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.75 \[ \int \left (a+b \text {arctanh}\left (c x^{3/2}\right )\right ) \, dx=\frac {2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}-1}{\sqrt {3}}\right ) b +2 \sqrt {3}\, \mathit {atan} \left (\frac {2 \sqrt {x}\, c^{\frac {1}{3}}+1}{\sqrt {3}}\right ) b +4 c^{\frac {2}{3}} \mathit {atanh} \left (\sqrt {x}\, c x \right ) b x +2 \mathit {atanh} \left (\sqrt {x}\, c x \right ) b +4 c^{\frac {2}{3}} a x -3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}+c^{\frac {1}{3}}\right ) b +3 \,\mathrm {log}\left (\sqrt {x}\, c^{\frac {2}{3}}-c^{\frac {1}{3}}\right ) b}{4 c^{\frac {2}{3}}} \] Input:
int(a+b*atanh(c*x^(3/2)),x)
Output:
(2*sqrt(3)*atan((2*sqrt(x)*c**(1/3) - 1)/sqrt(3))*b + 2*sqrt(3)*atan((2*sq rt(x)*c**(1/3) + 1)/sqrt(3))*b + 4*c**(2/3)*atanh(sqrt(x)*c*x)*b*x + 2*ata nh(sqrt(x)*c*x)*b + 4*c**(2/3)*a*x - 3*log(sqrt(x)*c**(2/3) + c**(1/3))*b + 3*log(sqrt(x)*c**(2/3) - c**(1/3))*b)/(4*c**(2/3))