Integrand size = 22, antiderivative size = 652 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\frac {2 a \sqrt {x}}{f}+\frac {2 b \sqrt {1+c} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {1+c}}\right )}{\sqrt {d} f}-\frac {2 b \sqrt {1-c} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {1-c}}\right )}{\sqrt {d} f}+\frac {b e \log \left (\frac {f \left (\sqrt {-1-c}-\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e+\sqrt {-1-c} f}\right ) \log \left (e+f \sqrt {x}\right )}{f^2}-\frac {b e \log \left (\frac {f \left (\sqrt {1-c}-\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e+\sqrt {1-c} f}\right ) \log \left (e+f \sqrt {x}\right )}{f^2}+\frac {b e \log \left (-\frac {f \left (\sqrt {-1-c}+\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e-\sqrt {-1-c} f}\right ) \log \left (e+f \sqrt {x}\right )}{f^2}-\frac {b e \log \left (-\frac {f \left (\sqrt {1-c}+\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e-\sqrt {1-c} f}\right ) \log \left (e+f \sqrt {x}\right )}{f^2}-\frac {b \sqrt {x} \log \left (-\frac {1-c-d x}{c+d x}\right )}{f}-\frac {e \log \left (e+f \sqrt {x}\right ) \left (a-b \log \left (-\frac {1-c-d x}{c+d x}\right )\right )}{f^2}+\frac {b \sqrt {x} \log \left (\frac {1+c+d x}{c+d x}\right )}{f}-\frac {e \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (\frac {1+c+d x}{c+d x}\right )\right )}{f^2}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e-\sqrt {-1-c} f}\right )}{f^2}+\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e+\sqrt {-1-c} f}\right )}{f^2}-\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e-\sqrt {1-c} f}\right )}{f^2}-\frac {b e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e+\sqrt {1-c} f}\right )}{f^2} \] Output:
2*a*x^(1/2)/f+2*b*(1+c)^(1/2)*arctan(d^(1/2)*x^(1/2)/(1+c)^(1/2))/d^(1/2)/ f-2*b*(1-c)^(1/2)*arctanh(d^(1/2)*x^(1/2)/(1-c)^(1/2))/d^(1/2)/f+b*e*ln(f* ((-1-c)^(1/2)-d^(1/2)*x^(1/2))/(d^(1/2)*e+(-1-c)^(1/2)*f))*ln(e+f*x^(1/2)) /f^2-b*e*ln(f*((1-c)^(1/2)-d^(1/2)*x^(1/2))/(d^(1/2)*e+(1-c)^(1/2)*f))*ln( e+f*x^(1/2))/f^2+b*e*ln(-f*((-1-c)^(1/2)+d^(1/2)*x^(1/2))/(d^(1/2)*e-(-1-c )^(1/2)*f))*ln(e+f*x^(1/2))/f^2-b*e*ln(-f*((1-c)^(1/2)+d^(1/2)*x^(1/2))/(d ^(1/2)*e-(1-c)^(1/2)*f))*ln(e+f*x^(1/2))/f^2-b*x^(1/2)*ln(-(-d*x-c+1)/(d*x +c))/f-e*ln(e+f*x^(1/2))*(a-b*ln(-(-d*x-c+1)/(d*x+c)))/f^2+b*x^(1/2)*ln((d *x+c+1)/(d*x+c))/f-e*ln(e+f*x^(1/2))*(a+b*ln((d*x+c+1)/(d*x+c)))/f^2+b*e*p olylog(2,d^(1/2)*(e+f*x^(1/2))/(d^(1/2)*e-(-1-c)^(1/2)*f))/f^2+b*e*polylog (2,d^(1/2)*(e+f*x^(1/2))/(d^(1/2)*e+(-1-c)^(1/2)*f))/f^2-b*e*polylog(2,d^( 1/2)*(e+f*x^(1/2))/(d^(1/2)*e-(1-c)^(1/2)*f))/f^2-b*e*polylog(2,d^(1/2)*(e +f*x^(1/2))/(d^(1/2)*e+(1-c)^(1/2)*f))/f^2
Time = 10.34 (sec) , antiderivative size = 602, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\frac {2 a \left (f \sqrt {x}-e \log \left (e+f \sqrt {x}\right )\right )+b \left (\frac {2 \sqrt {1+c} f \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {1+c}}\right )}{\sqrt {d}}-\frac {2 \sqrt {1-c} f \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {1-c}}\right )}{\sqrt {d}}+e \log \left (\frac {f \left (\sqrt {-1-c}-\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e+\sqrt {-1-c} f}\right ) \log \left (e+f \sqrt {x}\right )-e \log \left (\frac {f \left (\sqrt {1-c}-\sqrt {d} \sqrt {x}\right )}{\sqrt {d} e+\sqrt {1-c} f}\right ) \log \left (e+f \sqrt {x}\right )+e \log \left (\frac {f \left (\sqrt {-1-c}+\sqrt {d} \sqrt {x}\right )}{-\sqrt {d} e+\sqrt {-1-c} f}\right ) \log \left (e+f \sqrt {x}\right )-e \log \left (\frac {f \left (\sqrt {1-c}+\sqrt {d} \sqrt {x}\right )}{-\sqrt {d} e+\sqrt {1-c} f}\right ) \log \left (e+f \sqrt {x}\right )-f \sqrt {x} \log \left (\frac {-1+c+d x}{c+d x}\right )+e \log \left (e+f \sqrt {x}\right ) \log \left (\frac {-1+c+d x}{c+d x}\right )+f \sqrt {x} \log \left (\frac {1+c+d x}{c+d x}\right )-e \log \left (e+f \sqrt {x}\right ) \log \left (\frac {1+c+d x}{c+d x}\right )+e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e-\sqrt {-1-c} f}\right )+e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e+\sqrt {-1-c} f}\right )-e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e-\sqrt {1-c} f}\right )-e \operatorname {PolyLog}\left (2,\frac {\sqrt {d} \left (e+f \sqrt {x}\right )}{\sqrt {d} e+\sqrt {1-c} f}\right )\right )}{f^2} \] Input:
Integrate[(a + b*ArcCoth[c + d*x])/(e + f*Sqrt[x]),x]
Output:
(2*a*(f*Sqrt[x] - e*Log[e + f*Sqrt[x]]) + b*((2*Sqrt[1 + c]*f*ArcTan[(Sqrt [d]*Sqrt[x])/Sqrt[1 + c]])/Sqrt[d] - (2*Sqrt[1 - c]*f*ArcTanh[(Sqrt[d]*Sqr t[x])/Sqrt[1 - c]])/Sqrt[d] + e*Log[(f*(Sqrt[-1 - c] - Sqrt[d]*Sqrt[x]))/( Sqrt[d]*e + Sqrt[-1 - c]*f)]*Log[e + f*Sqrt[x]] - e*Log[(f*(Sqrt[1 - c] - Sqrt[d]*Sqrt[x]))/(Sqrt[d]*e + Sqrt[1 - c]*f)]*Log[e + f*Sqrt[x]] + e*Log[ (f*(Sqrt[-1 - c] + Sqrt[d]*Sqrt[x]))/(-(Sqrt[d]*e) + Sqrt[-1 - c]*f)]*Log[ e + f*Sqrt[x]] - e*Log[(f*(Sqrt[1 - c] + Sqrt[d]*Sqrt[x]))/(-(Sqrt[d]*e) + Sqrt[1 - c]*f)]*Log[e + f*Sqrt[x]] - f*Sqrt[x]*Log[(-1 + c + d*x)/(c + d* x)] + e*Log[e + f*Sqrt[x]]*Log[(-1 + c + d*x)/(c + d*x)] + f*Sqrt[x]*Log[( 1 + c + d*x)/(c + d*x)] - e*Log[e + f*Sqrt[x]]*Log[(1 + c + d*x)/(c + d*x) ] + e*PolyLog[2, (Sqrt[d]*(e + f*Sqrt[x]))/(Sqrt[d]*e - Sqrt[-1 - c]*f)] + e*PolyLog[2, (Sqrt[d]*(e + f*Sqrt[x]))/(Sqrt[d]*e + Sqrt[-1 - c]*f)] - e* PolyLog[2, (Sqrt[d]*(e + f*Sqrt[x]))/(Sqrt[d]*e - Sqrt[1 - c]*f)] - e*Poly Log[2, (Sqrt[d]*(e + f*Sqrt[x]))/(Sqrt[d]*e + Sqrt[1 - c]*f)]))/f^2
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.
| \(\displaystyle \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle 2 \int \frac {\sqrt {x} \left (a+b \coth ^{-1}(c+d x)\right )}{e+f \sqrt {x}}d\sqrt {x}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {\sqrt {x} a}{e+f \sqrt {x}}+\frac {b \sqrt {x} \coth ^{-1}(c+d x)}{e+f \sqrt {x}}\right )d\sqrt {x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {b e \int \frac {\coth ^{-1}(c+d x)}{e+f \sqrt {x}}d\sqrt {x}}{f}-\frac {a e \log \left (e+f \sqrt {x}\right )}{f^2}+\frac {a \sqrt {x}}{f}+\frac {b \sqrt {c+1} \arctan \left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {c+1}}\right )}{\sqrt {d} f}-\frac {b \sqrt {1-c} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {x}}{\sqrt {1-c}}\right )}{\sqrt {d} f}+\frac {b \sqrt {x} \coth ^{-1}(c+d x)}{f}\right )\) |
Input:
Int[(a + b*ArcCoth[c + d*x])/(e + f*Sqrt[x]),x]
Output:
$Aborted
Time = 0.58 (sec) , antiderivative size = 673, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(\frac {2 a \sqrt {x}}{f}-\frac {2 a e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+2 b \left (\frac {\operatorname {arccoth}\left (d x +c \right ) \sqrt {x}}{f}-\frac {\operatorname {arccoth}\left (d x +c \right ) e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+\frac {2 d \left (f^{2} \left (\frac {\left (c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}+d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}+d \,f^{2}}}+\frac {\left (-c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}-d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}-d \,f^{2}}}\right )+e \,f^{2} \left (\frac {\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )\right )}{2 d}+\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )}{2 d}}{2 f^{2}}+\frac {-\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )\right )}{2 d}-\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )}{2 d}}{2 f^{2}}\right )\right )}{f^{2}}\right )\) | \(673\) |
| default | \(\frac {2 a \sqrt {x}}{f}-\frac {2 a e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+2 b \left (\frac {\operatorname {arccoth}\left (d x +c \right ) \sqrt {x}}{f}-\frac {\operatorname {arccoth}\left (d x +c \right ) e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+\frac {2 d \left (f^{2} \left (\frac {\left (c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}+d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}+d \,f^{2}}}+\frac {\left (-c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}-d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}-d \,f^{2}}}\right )+e \,f^{2} \left (\frac {\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )\right )}{2 d}+\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )}{2 d}}{2 f^{2}}+\frac {-\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )\right )}{2 d}-\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )}{2 d}}{2 f^{2}}\right )\right )}{f^{2}}\right )\) | \(673\) |
| parts | \(a \left (\frac {2 \sqrt {x}}{f}-\frac {e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+\frac {e \ln \left (f \sqrt {x}-e \right )}{f^{2}}-\frac {e \ln \left (f^{2} x -e^{2}\right )}{f^{2}}\right )+b \left (\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) \sqrt {x}}{f}-\frac {2 \,\operatorname {arccoth}\left (d x +c \right ) e \ln \left (e +f \sqrt {x}\right )}{f^{2}}+\frac {4 d \left (f^{2} \left (\frac {\left (c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}+d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}+d \,f^{2}}}+\frac {\left (-c +1\right ) \arctan \left (\frac {-2 d e +2 \left (e +f \sqrt {x}\right ) d}{2 \sqrt {c d \,f^{2}-d \,f^{2}}}\right )}{2 d \sqrt {c d \,f^{2}-d \,f^{2}}}\right )-e \,f^{2} \left (\frac {\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )\right )}{2 d}+\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}+d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}+d \,f^{2}}}\right )}{2 d}}{2 f^{2}}+\frac {-\frac {\ln \left (e +f \sqrt {x}\right ) \left (\ln \left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\ln \left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )\right )}{2 d}-\frac {\operatorname {dilog}\left (\frac {d e -\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )+\operatorname {dilog}\left (\frac {-d e +\left (e +f \sqrt {x}\right ) d +\sqrt {-c d \,f^{2}-d \,f^{2}}}{-d e +\sqrt {-c d \,f^{2}-d \,f^{2}}}\right )}{2 d}}{2 f^{2}}\right )\right )}{f^{2}}\right )\) | \(708\) |
Input:
int((a+b*arccoth(d*x+c))/(e+f*x^(1/2)),x,method=_RETURNVERBOSE)
Output:
2*a*x^(1/2)/f-2*a*e/f^2*ln(e+f*x^(1/2))+2*b*(arccoth(d*x+c)/f*x^(1/2)-arcc oth(d*x+c)*e/f^2*ln(e+f*x^(1/2))+2*d/f^2*(f^2*(1/2*(c+1)/d/(c*d*f^2+d*f^2) ^(1/2)*arctan(1/2*(-2*d*e+2*(e+f*x^(1/2))*d)/(c*d*f^2+d*f^2)^(1/2))+1/2*(- c+1)/d/(c*d*f^2-d*f^2)^(1/2)*arctan(1/2*(-2*d*e+2*(e+f*x^(1/2))*d)/(c*d*f^ 2-d*f^2)^(1/2)))+e*f^2*(1/2/f^2*(1/2*ln(e+f*x^(1/2))*(ln((d*e-(e+f*x^(1/2) )*d+(-c*d*f^2-d*f^2)^(1/2))/(d*e+(-c*d*f^2-d*f^2)^(1/2)))+ln((-d*e+(e+f*x^ (1/2))*d+(-c*d*f^2-d*f^2)^(1/2))/(-d*e+(-c*d*f^2-d*f^2)^(1/2))))/d+1/2*(di log((d*e-(e+f*x^(1/2))*d+(-c*d*f^2-d*f^2)^(1/2))/(d*e+(-c*d*f^2-d*f^2)^(1/ 2)))+dilog((-d*e+(e+f*x^(1/2))*d+(-c*d*f^2-d*f^2)^(1/2))/(-d*e+(-c*d*f^2-d *f^2)^(1/2))))/d)+1/2/f^2*(-1/2*ln(e+f*x^(1/2))*(ln((d*e-(e+f*x^(1/2))*d+( -c*d*f^2+d*f^2)^(1/2))/(d*e+(-c*d*f^2+d*f^2)^(1/2)))+ln((-d*e+(e+f*x^(1/2) )*d+(-c*d*f^2+d*f^2)^(1/2))/(-d*e+(-c*d*f^2+d*f^2)^(1/2))))/d-1/2*(dilog(( d*e-(e+f*x^(1/2))*d+(-c*d*f^2+d*f^2)^(1/2))/(d*e+(-c*d*f^2+d*f^2)^(1/2)))+ dilog((-d*e+(e+f*x^(1/2))*d+(-c*d*f^2+d*f^2)^(1/2))/(-d*e+(-c*d*f^2+d*f^2) ^(1/2))))/d))))
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{f \sqrt {x} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(e+f*x^(1/2)),x, algorithm="fricas")
Output:
integral(-(b*e*arccoth(d*x + c) + a*e - (b*f*arccoth(d*x + c) + a*f)*sqrt( x))/(f^2*x - e^2), x)
Timed out. \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\text {Timed out} \] Input:
integrate((a+b*acoth(d*x+c))/(e+f*x**(1/2)),x)
Output:
Timed out
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{f \sqrt {x} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(e+f*x^(1/2)),x, algorithm="maxima")
Output:
-2*a*(e*log(f*sqrt(x) + e)/f^2 - sqrt(x)/f) + b*integrate(1/2*log(1/(d*x + c) + 1)/(f*sqrt(x) + e), x) - b*integrate(1/2*log(-1/(d*x + c) + 1)/(f*sq rt(x) + e), x)
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\int { \frac {b \operatorname {arcoth}\left (d x + c\right ) + a}{f \sqrt {x} + e} \,d x } \] Input:
integrate((a+b*arccoth(d*x+c))/(e+f*x^(1/2)),x, algorithm="giac")
Output:
integrate((b*arccoth(d*x + c) + a)/(f*sqrt(x) + e), x)
Timed out. \[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\int \frac {a+b\,\mathrm {acoth}\left (c+d\,x\right )}{e+f\,\sqrt {x}} \,d x \] Input:
int((a + b*acoth(c + d*x))/(e + f*x^(1/2)),x)
Output:
int((a + b*acoth(c + d*x))/(e + f*x^(1/2)), x)
\[ \int \frac {a+b \coth ^{-1}(c+d x)}{e+f \sqrt {x}} \, dx=\frac {2 \sqrt {x}\, a f +\left (\int \frac {\mathit {acoth} \left (d x +c \right )}{\sqrt {x}\, f +e}d x \right ) b \,f^{2}-2 \,\mathrm {log}\left (\sqrt {x}\, f +e \right ) a e}{f^{2}} \] Input:
int((a+b*acoth(d*x+c))/(e+f*x^(1/2)),x)
Output:
(2*sqrt(x)*a*f + int(acoth(c + d*x)/(sqrt(x)*f + e),x)*b*f**2 - 2*log(sqrt (x)*f + e)*a*e)/f**2