\(\int \frac {a+b \text {arctanh}(c x^3)}{d+e x} \, dx\) [35]

Optimal result

Integrand size = 18, antiderivative size = 523 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right ) \log (d+e x)}{e}+\frac {b \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left (1+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (-\frac {e \left (\sqrt [3]{-1}+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left ((-1)^{2/3}+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (\frac {(-1)^{2/3} e \left (1+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (\frac {\sqrt [3]{-1} e \left (1+(-1)^{2/3} \sqrt [3]{c} x\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 e}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e} \] Output:

(a+b*arctanh(c*x^3))*ln(e*x+d)/e+1/2*b*ln(e*(1-c^(1/3)*x)/(c^(1/3)*d+e))*l 
n(e*x+d)/e-1/2*b*ln(-e*(1+c^(1/3)*x)/(c^(1/3)*d-e))*ln(e*x+d)/e+1/2*b*ln(- 
e*((-1)^(1/3)+c^(1/3)*x)/(c^(1/3)*d-(-1)^(1/3)*e))*ln(e*x+d)/e-1/2*b*ln(-e 
*((-1)^(2/3)+c^(1/3)*x)/(c^(1/3)*d-(-1)^(2/3)*e))*ln(e*x+d)/e+1/2*b*ln((-1 
)^(2/3)*e*(1+(-1)^(1/3)*c^(1/3)*x)/(c^(1/3)*d+(-1)^(2/3)*e))*ln(e*x+d)/e-1 
/2*b*ln((-1)^(1/3)*e*(1+(-1)^(2/3)*c^(1/3)*x)/(c^(1/3)*d+(-1)^(1/3)*e))*ln 
(e*x+d)/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d-e))/e+1/2*b*polylog(2 
,c^(1/3)*(e*x+d)/(c^(1/3)*d+e))/e+1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3) 
*d-(-1)^(1/3)*e))/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d+(-1)^(1/3)* 
e))/e-1/2*b*polylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d-(-1)^(2/3)*e))/e+1/2*b*po 
lylog(2,c^(1/3)*(e*x+d)/(c^(1/3)*d+(-1)^(2/3)*e))/e
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.78 (sec) , antiderivative size = 515, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (2 \text {arctanh}\left (c x^3\right ) \log (d+e x)-\log \left (\frac {e \left (1-i \sqrt {3}-2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+e-i \sqrt {3} e}\right ) \log (d+e x)+\log \left (\frac {e \left (-i+\sqrt {3}-2 i \sqrt [3]{c} x\right )}{2 i \sqrt [3]{c} d+\left (-i+\sqrt {3}\right ) e}\right ) \log (d+e x)+\log \left (\frac {e \left (i+\sqrt {3}+2 i \sqrt [3]{c} x\right )}{-2 i \sqrt [3]{c} d+\left (i+\sqrt {3}\right ) e}\right ) \log (d+e x)-\log \left (-\frac {e \left (1+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-e}\right ) \log (d+e x)-\log \left (-\frac {e \left (-1-i \sqrt {3}+2 \sqrt [3]{c} x\right )}{2 \sqrt [3]{c} d+e+i \sqrt {3} e}\right ) \log (d+e x)+\log (d+e x) \log \left (\frac {e-\sqrt [3]{c} e x}{\sqrt [3]{c} d+e}\right )-\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )+\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-e-i \sqrt {3} e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+e-i \sqrt {3} e}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-e+i \sqrt {3} e}\right )-\operatorname {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+e+i \sqrt {3} e}\right )\right )}{2 e} \] Input:

Integrate[(a + b*ArcTanh[c*x^3])/(d + e*x),x]
 

Output:

(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^3]*Log[d + e*x] - Log[(e*(1 - I*Sqr 
t[3] - 2*c^(1/3)*x))/(2*c^(1/3)*d + e - I*Sqrt[3]*e)]*Log[d + e*x] + Log[( 
e*(-I + Sqrt[3] - (2*I)*c^(1/3)*x))/((2*I)*c^(1/3)*d + (-I + Sqrt[3])*e)]* 
Log[d + e*x] + Log[(e*(I + Sqrt[3] + (2*I)*c^(1/3)*x))/((-2*I)*c^(1/3)*d + 
 (I + Sqrt[3])*e)]*Log[d + e*x] - Log[-((e*(1 + c^(1/3)*x))/(c^(1/3)*d - e 
))]*Log[d + e*x] - Log[-((e*(-1 - I*Sqrt[3] + 2*c^(1/3)*x))/(2*c^(1/3)*d + 
 e + I*Sqrt[3]*e))]*Log[d + e*x] + Log[d + e*x]*Log[(e - c^(1/3)*e*x)/(c^( 
1/3)*d + e)] - PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - e)] + PolyLog[2 
, (c^(1/3)*(d + e*x))/(c^(1/3)*d + e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/ 
(2*c^(1/3)*d - e - I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^( 
1/3)*d + e - I*Sqrt[3]*e)] + PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d 
 - e + I*Sqrt[3]*e)] - PolyLog[2, (2*c^(1/3)*(d + e*x))/(2*c^(1/3)*d + e + 
 I*Sqrt[3]*e)]))/(2*e)
 

Rubi [A] (verified)

Time = 1.24 (sec) , antiderivative size = 519, normalized size of antiderivative = 0.99, 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.

\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx\)

\(\Big \downarrow \) 6482

\(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{e}-\frac {3 b c \int \frac {x^2 \log (d+e x)}{1-c^2 x^6}dx}{e}\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{e}-\frac {3 b c \int \left (\frac {c \log (d+e x) x^2}{2 \left (c-c^2 x^3\right )}+\frac {c \log (d+e x) x^2}{2 \left (c^2 x^3+c\right )}\right )dx}{e}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log (d+e x) \left (a+b \text {arctanh}\left (c x^3\right )\right )}{e}-\frac {3 b c \left (\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{6 c}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{6 c}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{6 c}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{6 c}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{6 c}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{6 c}-\frac {\log (d+e x) \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right )}{6 c}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )}{6 c}-\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+\sqrt [3]{-1}\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{6 c}+\frac {\log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+(-1)^{2/3}\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{6 c}-\frac {\log (d+e x) \log \left (\frac {(-1)^{2/3} e \left (\sqrt [3]{-1} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{6 c}+\frac {\log (d+e x) \log \left (\frac {\sqrt [3]{-1} e \left ((-1)^{2/3} \sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{6 c}\right )}{e}\)

Input:

Int[(a + b*ArcTanh[c*x^3])/(d + e*x),x]
 

Output:

((a + b*ArcTanh[c*x^3])*Log[d + e*x])/e - (3*b*c*(-1/6*(Log[(e*(1 - c^(1/3 
)*x))/(c^(1/3)*d + e)]*Log[d + e*x])/c + (Log[-((e*(1 + c^(1/3)*x))/(c^(1/ 
3)*d - e))]*Log[d + e*x])/(6*c) - (Log[-((e*((-1)^(1/3) + c^(1/3)*x))/(c^( 
1/3)*d - (-1)^(1/3)*e))]*Log[d + e*x])/(6*c) + (Log[-((e*((-1)^(2/3) + c^( 
1/3)*x))/(c^(1/3)*d - (-1)^(2/3)*e))]*Log[d + e*x])/(6*c) - (Log[((-1)^(2/ 
3)*e*(1 + (-1)^(1/3)*c^(1/3)*x))/(c^(1/3)*d + (-1)^(2/3)*e)]*Log[d + e*x]) 
/(6*c) + (Log[((-1)^(1/3)*e*(1 + (-1)^(2/3)*c^(1/3)*x))/(c^(1/3)*d + (-1)^ 
(1/3)*e)]*Log[d + e*x])/(6*c) + PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d 
- e)]/(6*c) - PolyLog[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d + e)]/(6*c) - Poly 
Log[2, (c^(1/3)*(d + e*x))/(c^(1/3)*d - (-1)^(1/3)*e)]/(6*c) + PolyLog[2, 
(c^(1/3)*(d + e*x))/(c^(1/3)*d + (-1)^(1/3)*e)]/(6*c) + PolyLog[2, (c^(1/3 
)*(d + e*x))/(c^(1/3)*d - (-1)^(2/3)*e)]/(6*c) - PolyLog[2, (c^(1/3)*(d + 
e*x))/(c^(1/3)*d + (-1)^(2/3)*e)]/(6*c)))/e
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

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]
 

rule 6482

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.33 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.35

method	result	size
default	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \operatorname {arctanh}\left (c \,x^{3}\right )}{e}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}-e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}+e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}\)	\(182\)
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \operatorname {arctanh}\left (c \,x^{3}\right )}{e}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}-e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}+e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}\)	\(182\)
risch	\(\frac {a \ln \left (e x +d \right )}{e}-\frac {b \ln \left (e x +d \right ) \ln \left (-c \,x^{3}+1\right )}{2 e}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}-e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (c \,x^{3}+1\right )}{2 e}-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\textit {\_Z}^{3} c -3 \textit {\_Z}^{2} c d +3 \textit {\_Z} c \,d^{2}-c \,d^{3}+e^{3}\right )}{\sum }\left (\ln \left (e x +d \right ) \ln \left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-e x +\textit {\_R1} -d}{\textit {\_R1}}\right )\right )\right )}{2 e}\)	\(206\)

Input:

int((a+b*arctanh(c*x^3))/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

a*ln(e*x+d)/e+b*ln(e*x+d)/e*arctanh(c*x^3)+1/2*b/e*sum(ln(e*x+d)*ln((-e*x+ 
_R1-d)/_R1)+dilog((-e*x+_R1-d)/_R1),_R1=RootOf(_Z^3*c-3*_Z^2*c*d+3*_Z*c*d^ 
2-c*d^3-e^3))-1/2*b/e*sum(ln(e*x+d)*ln((-e*x+_R1-d)/_R1)+dilog((-e*x+_R1-d 
)/_R1),_R1=RootOf(_Z^3*c-3*_Z^2*c*d+3*_Z*c*d^2-c*d^3+e^3))
 

Fricas [F]

\[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^3))/(e*x+d),x, algorithm="fricas")
 

Output:

integral((b*arctanh(c*x^3) + a)/(e*x + d), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\text {Timed out} \] Input:

integrate((a+b*atanh(c*x**3))/(e*x+d),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^3))/(e*x+d),x, algorithm="maxima")
 

Output:

1/2*b*integrate((log(c*x^3 + 1) - log(-c*x^3 + 1))/(e*x + d), x) + a*log(e 
*x + d)/e
 

Giac [F]

\[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\int { \frac {b \operatorname {artanh}\left (c x^{3}\right ) + a}{e x + d} \,d x } \] Input:

integrate((a+b*arctanh(c*x^3))/(e*x+d),x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^3) + a)/(e*x + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\int \frac {a+b\,\mathrm {atanh}\left (c\,x^3\right )}{d+e\,x} \,d x \] Input:

int((a + b*atanh(c*x^3))/(d + e*x),x)
 

Output:

int((a + b*atanh(c*x^3))/(d + e*x), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{d+e x} \, dx=\frac {\left (\int \frac {\mathit {atanh} \left (c \,x^{3}\right )}{e x +d}d x \right ) b e +\mathrm {log}\left (e x +d \right ) a}{e} \] Input:

int((a+b*atanh(c*x^3))/(e*x+d),x)
                                                                                    
                                                                                    
 

Output:

(int(atanh(c*x**3)/(d + e*x),x)*b*e + log(d + e*x)*a)/e