Integrand size = 18, antiderivative size = 325 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {b \log \left (\frac {e \left (1-\sqrt {-c} x\right )}{\sqrt {-c} d+e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left (1+\sqrt {-c} x\right )}{\sqrt {-c} d-e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (\frac {e \left (1-\sqrt {c} x\right )}{\sqrt {c} d+e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (-\frac {e \left (1+\sqrt {c} x\right )}{\sqrt {c} d-e}\right ) \log (d+e x)}{2 e}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d-e}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d+e}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )}{2 e} \] Output:
(a+b*arctanh(c*x^2))*ln(e*x+d)/e-1/2*b*ln(e*(1-(-c)^(1/2)*x)/((-c)^(1/2)*d +e))*ln(e*x+d)/e-1/2*b*ln(-e*(1+(-c)^(1/2)*x)/((-c)^(1/2)*d-e))*ln(e*x+d)/ e+1/2*b*ln(e*(1-c^(1/2)*x)/(c^(1/2)*d+e))*ln(e*x+d)/e+1/2*b*ln(-e*(1+c^(1/ 2)*x)/(c^(1/2)*d-e))*ln(e*x+d)/e-1/2*b*polylog(2,(-c)^(1/2)*(e*x+d)/((-c)^ (1/2)*d-e))/e+1/2*b*polylog(2,c^(1/2)*(e*x+d)/(c^(1/2)*d-e))/e-1/2*b*polyl og(2,(-c)^(1/2)*(e*x+d)/((-c)^(1/2)*d+e))/e+1/2*b*polylog(2,c^(1/2)*(e*x+d )/(c^(1/2)*d+e))/e
Result contains complex when optimal does not.
Time = 17.33 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.88 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \text {arctanh}\left (c x^2\right ) \log (d+e x)-\log \left (\frac {e \left (i-\sqrt {c} x\right )}{\sqrt {c} d+i e}\right ) \log (d+e x)-\log \left (-\frac {e \left (i+\sqrt {c} x\right )}{\sqrt {c} d-i e}\right ) \log (d+e x)+\log \left (-\frac {e \left (1+\sqrt {c} x\right )}{\sqrt {c} d-e}\right ) \log (d+e x)+\log (d+e x) \log \left (\frac {e-\sqrt {c} e x}{\sqrt {c} d+e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+i e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )\right )}{2 e} \] Input:
Integrate[(a + b*ArcTanh[c*x^2])/(d + e*x),x]
Output:
(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^2]*Log[d + e*x] - Log[(e*(I - Sqrt[ c]*x))/(Sqrt[c]*d + I*e)]*Log[d + e*x] - Log[-((e*(I + Sqrt[c]*x))/(Sqrt[c ]*d - I*e))]*Log[d + e*x] + Log[-((e*(1 + Sqrt[c]*x))/(Sqrt[c]*d - e))]*Lo g[d + e*x] + Log[d + e*x]*Log[(e - Sqrt[c]*e*x)/(Sqrt[c]*d + e)] + PolyLog [2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/ (Sqrt[c]*d - I*e)] - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + I*e)] + P olyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + e)]))/(2*e)
Time = 0.80 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6482, 2863, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 6482 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{e}-\frac {2 b c \int \frac {x \log (d+e x)}{1-c^2 x^4}dx}{e}\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{e}-\frac {2 b c \int \left (\frac {c x \log (d+e x)}{2 \left (c-c^2 x^2\right )}+\frac {c x \log (d+e x)}{2 \left (c^2 x^2+c\right )}\right )dx}{e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{e}-\frac {2 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d-e}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )}{4 c}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d+e}\right )}{4 c}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )}{4 c}+\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt {-c} x\right )}{\sqrt {-c} d+e}\right )}{4 c}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt {-c} x+1\right )}{\sqrt {-c} d-e}\right )}{4 c}-\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt {c} x\right )}{\sqrt {c} d+e}\right )}{4 c}-\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt {c} x+1\right )}{\sqrt {c} d-e}\right )}{4 c}\right )}{e}\) |
Input:
Int[(a + b*ArcTanh[c*x^2])/(d + e*x),x]
Output:
((a + b*ArcTanh[c*x^2])*Log[d + e*x])/e - (2*b*c*((Log[(e*(1 - Sqrt[-c]*x) )/(Sqrt[-c]*d + e)]*Log[d + e*x])/(4*c) + (Log[-((e*(1 + Sqrt[-c]*x))/(Sqr t[-c]*d - e))]*Log[d + e*x])/(4*c) - (Log[(e*(1 - Sqrt[c]*x))/(Sqrt[c]*d + e)]*Log[d + e*x])/(4*c) - (Log[-((e*(1 + Sqrt[c]*x))/(Sqrt[c]*d - e))]*Lo g[d + e*x])/(4*c) + PolyLog[2, (Sqrt[-c]*(d + e*x))/(Sqrt[-c]*d - e)]/(4*c ) - PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - e)]/(4*c) + PolyLog[2, (Sq rt[-c]*(d + e*x))/(Sqrt[-c]*d + e)]/(4*c) - PolyLog[2, (Sqrt[c]*(d + e*x)) /(Sqrt[c]*d + e)]/(4*c)))/e
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) ^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c , d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))/((d_.) + (e_.)*(x_)), x_Symbol ] :> Simp[Log[d + e*x]*((a + b*ArcTanh[c*x^n])/e), x] - Simp[b*c*(n/e) In t[x^(n - 1)*(Log[d + e*x]/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IntegerQ[n]
Time = 0.28 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(\frac {a \ln \left (e x +d \right )}{e}+b \left (\frac {\ln \left (e x +d \right ) \operatorname {arctanh}\left (c \,x^{2}\right )}{e}-\frac {2 c \left (-\frac {e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )+\ln \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )}{2 c}\right )}{2}+\frac {e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )+\ln \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )}{2 c}\right )}{2}\right )}{e^{3}}\right )\) | \(347\) |
| parts | \(\frac {a \ln \left (e x +d \right )}{e}+b \left (\frac {\ln \left (e x +d \right ) \operatorname {arctanh}\left (c \,x^{2}\right )}{e}-\frac {2 c \left (-\frac {e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )+\ln \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )}{2 c}\right )}{2}+\frac {e^{2} \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )+\ln \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )\right )}{2 c}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )}{2 c}\right )}{2}\right )}{e^{3}}\right )\) | \(347\) |
| risch | \(\frac {a \ln \left (e x +d \right )}{e}-\frac {b \ln \left (e x +d \right ) \ln \left (-c \,x^{2}+1\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {e \sqrt {c}-\left (e x +d \right ) c +c d}{e \sqrt {c}+c d}\right )}{2 e}+\frac {b \operatorname {dilog}\left (\frac {e \sqrt {c}+\left (e x +d \right ) c -c d}{e \sqrt {c}-c d}\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (c \,x^{2}+1\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )}{2 e}-\frac {b \operatorname {dilog}\left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +c d}{e \sqrt {-c}+c d}\right )}{2 e}-\frac {b \operatorname {dilog}\left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -c d}{e \sqrt {-c}-c d}\right )}{2 e}\) | \(386\) |
Input:
int((a+b*arctanh(c*x^2))/(e*x+d),x,method=_RETURNVERBOSE)
Output:
a*ln(e*x+d)/e+b*(ln(e*x+d)/e*arctanh(c*x^2)-2/e^3*c*(-1/2*e^2*(1/2*ln(e*x+ d)*(ln((e*c^(1/2)-(e*x+d)*c+c*d)/(e*c^(1/2)+c*d))+ln((e*c^(1/2)+(e*x+d)*c- c*d)/(e*c^(1/2)-c*d)))/c+1/2*(dilog((e*c^(1/2)-(e*x+d)*c+c*d)/(e*c^(1/2)+c *d))+dilog((e*c^(1/2)+(e*x+d)*c-c*d)/(e*c^(1/2)-c*d)))/c)+1/2*e^2*(1/2*ln( e*x+d)*(ln((e*(-c)^(1/2)-(e*x+d)*c+c*d)/(e*(-c)^(1/2)+c*d))+ln((e*(-c)^(1/ 2)+(e*x+d)*c-c*d)/(e*(-c)^(1/2)-c*d)))/c+1/2*(dilog((e*(-c)^(1/2)-(e*x+d)* c+c*d)/(e*(-c)^(1/2)+c*d))+dilog((e*(-c)^(1/2)+(e*x+d)*c-c*d)/(e*(-c)^(1/2 )-c*d)))/c)))
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x^2))/(e*x+d),x, algorithm="fricas")
Output:
integral((b*arctanh(c*x^2) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(c*x**2))/(e*x+d),x)
Output:
Timed out
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x^2))/(e*x+d),x, algorithm="maxima")
Output:
1/2*b*integrate((log(c*x^2 + 1) - log(-c*x^2 + 1))/(e*x + d), x) + a*log(e *x + d)/e
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{2}\right ) + a}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x^2))/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x^2) + a)/(e*x + d), x)
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^2\right )}{d+e\,x} \,d x \] Input:
int((a + b*atanh(c*x^2))/(d + e*x),x)
Output:
int((a + b*atanh(c*x^2))/(d + e*x), x)
\[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (c \,x^{2}\right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:
int((a+b*atanh(c*x^2))/(e*x+d),x)
Output:
(int(atanh(c*x**2)/(d + e*x),x)*b*e + log(d + e*x)*a)/e