Integrand size = 18, antiderivative size = 414 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=-\frac {\sqrt {3} b \sqrt [3]{c} \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac {\sqrt {3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \left (c d^3+e^3\right )}-\frac {a+b \text {arctanh}\left (c x^3\right )}{e (d+e x)}+\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x\right )}{2 \left (c d^3-e^3\right )}-\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3-e^3\right )}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3+e^3\right )}-\frac {b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac {b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )} \] Output:
-1/2*3^(1/2)*b*c^(1/3)*arctan(1/3*(1-2*c^(1/3)*x)*3^(1/2))/(c^(2/3)*d^2+c^ (1/3)*d*e+e^2)-3^(1/2)*b*c^(1/3)*(c^(1/3)*d+e)*arctan(1/3*(1+2*c^(1/3)*x)* 3^(1/2))/(2*c*d^3+2*e^3)-(a+b*arctanh(c*x^3))/e/(e*x+d)+b*c^(1/3)*(c^(1/3) *d-e)*ln(1-c^(1/3)*x)/(2*c*d^3+2*e^3)+b*c^(1/3)*(c^(1/3)*d+e)*ln(1+c^(1/3) *x)/(2*c*d^3-2*e^3)-3*b*c*d^2*e^2*ln(e*x+d)/(c^2*d^6-e^6)-b*c^(1/3)*(c^(1/ 3)*d+e)*ln(1-c^(1/3)*x+c^(2/3)*x^2)/(4*c*d^3-4*e^3)-b*c^(1/3)*(c^(1/3)*d-e )*ln(1+c^(1/3)*x+c^(2/3)*x^2)/(4*c*d^3+4*e^3)-1/2*b*c*d^2*ln(-c*x^3+1)/e/( c*d^3+e^3)+1/2*b*c*d^2*ln(c*x^3+1)/e/(c*d^3-e^3)
Time = 0.34 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.29 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=\frac {1}{4} \left (-\frac {4 a}{e (d+e x)}+\frac {2 \sqrt {3} b \sqrt [3]{c} \arctan \left (\frac {-1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{c^{2/3} d^2+\sqrt [3]{c} d e+e^2}-\frac {2 \sqrt {3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \arctan \left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{c d^3+e^3}-\frac {4 b \text {arctanh}\left (c x^3\right )}{e (d+e x)}+\frac {2 b \sqrt [3]{c} \left (c^{5/3} d^5-c^{4/3} d^4 e+c d^3 e^2+\sqrt [3]{c} d e^4-e^5\right ) \log \left (1-\sqrt [3]{c} x\right )}{-c^2 d^6 e+e^7}-\frac {2 b \sqrt [3]{c} \left (c^{5/3} d^5+c^{4/3} d^4 e+c d^3 e^2+\sqrt [3]{c} d e^4+e^5\right ) \log \left (1+\sqrt [3]{c} x\right )}{-c^2 d^6 e+e^7}-\frac {12 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac {b \sqrt [3]{c} \left (2 c^{5/3} d^5-c^{4/3} d^4 e-c d^3 e^2-\sqrt [3]{c} d e^4-e^5\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{c^2 d^6 e-e^7}+\frac {b \sqrt [3]{c} \left (2 c^{5/3} d^5+c^{4/3} d^4 e-c d^3 e^2-\sqrt [3]{c} d e^4+e^5\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{-c^2 d^6 e+e^7}+\frac {2 b c d^2 e^2 \log \left (1-c^2 x^6\right )}{c^2 d^6-e^6}\right ) \] Input:
Integrate[(a + b*ArcTanh[c*x^3])/(d + e*x)^2,x]
Output:
((-4*a)/(e*(d + e*x)) + (2*Sqrt[3]*b*c^(1/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqr t[3]])/(c^(2/3)*d^2 + c^(1/3)*d*e + e^2) - (2*Sqrt[3]*b*c^(1/3)*(c^(1/3)*d + e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(c*d^3 + e^3) - (4*b*ArcTanh[c*x^ 3])/(e*(d + e*x)) + (2*b*c^(1/3)*(c^(5/3)*d^5 - c^(4/3)*d^4*e + c*d^3*e^2 + c^(1/3)*d*e^4 - e^5)*Log[1 - c^(1/3)*x])/(-(c^2*d^6*e) + e^7) - (2*b*c^( 1/3)*(c^(5/3)*d^5 + c^(4/3)*d^4*e + c*d^3*e^2 + c^(1/3)*d*e^4 + e^5)*Log[1 + c^(1/3)*x])/(-(c^2*d^6*e) + e^7) - (12*b*c*d^2*e^2*Log[d + e*x])/(c^2*d ^6 - e^6) + (b*c^(1/3)*(2*c^(5/3)*d^5 - c^(4/3)*d^4*e - c*d^3*e^2 - c^(1/3 )*d*e^4 - e^5)*Log[1 - c^(1/3)*x + c^(2/3)*x^2])/(c^2*d^6*e - e^7) + (b*c^ (1/3)*(2*c^(5/3)*d^5 + c^(4/3)*d^4*e - c*d^3*e^2 - c^(1/3)*d*e^4 + e^5)*Lo g[1 + c^(1/3)*x + c^(2/3)*x^2])/(-(c^2*d^6*e) + e^7) + (2*b*c*d^2*e^2*Log[ 1 - c^2*x^6])/(c^2*d^6 - e^6))/4
Time = 1.06 (sec) , antiderivative size = 410, 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 = {6486, 7276, 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.
| \(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 6486 |
| \(\displaystyle \frac {3 b c \int \frac {x^2}{(d+e x) \left (1-c^2 x^6\right )}dx}{e}-\frac {a+b \text {arctanh}\left (c x^3\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 b c \int \left (\frac {d^2 e^4}{\left (e^3-c d^3\right ) \left (c d^3+e^3\right ) (d+e x)}+\frac {-x e^2+d e-c d^2 x^2}{2 \left (c d^3+e^3\right ) \left (c x^3-1\right )}+\frac {-x e^2+d e+c d^2 x^2}{2 \left (c d^3-e^3\right ) \left (c x^3+1\right )}\right )dx}{e}-\frac {a+b \text {arctanh}\left (c x^3\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b c \left (-\frac {e \arctan \left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right ) \left (\sqrt [3]{c} d+e\right )}{2 \sqrt {3} c^{2/3} \left (c d^3+e^3\right )}-\frac {e \arctan \left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \sqrt {3} c^{2/3} \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac {e \left (\sqrt [3]{c} d+e\right ) \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{12 c^{2/3} \left (c d^3-e^3\right )}-\frac {e \left (\sqrt [3]{c} d-e\right ) \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{12 c^{2/3} \left (c d^3+e^3\right )}+\frac {e \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{6 c^{2/3} \left (c d^3+e^3\right )}+\frac {e \left (\sqrt [3]{c} d+e\right ) \log \left (\sqrt [3]{c} x+1\right )}{6 c^{2/3} \left (c d^3-e^3\right )}-\frac {d^2 e^3 \log (d+e x)}{c^2 d^6-e^6}-\frac {d^2 \log \left (1-c x^3\right )}{6 \left (c d^3+e^3\right )}+\frac {d^2 \log \left (c x^3+1\right )}{6 \left (c d^3-e^3\right )}\right )}{e}-\frac {a+b \text {arctanh}\left (c x^3\right )}{e (d+e x)}\) |
Input:
Int[(a + b*ArcTanh[c*x^3])/(d + e*x)^2,x]
Output:
-((a + b*ArcTanh[c*x^3])/(e*(d + e*x))) + (3*b*c*(-1/2*(e*ArcTan[(1 - 2*c^ (1/3)*x)/Sqrt[3]])/(Sqrt[3]*c^(2/3)*(c^(2/3)*d^2 + c^(1/3)*d*e + e^2)) - ( e*(c^(1/3)*d + e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(2*Sqrt[3]*c^(2/3)*(c *d^3 + e^3)) + ((c^(1/3)*d - e)*e*Log[1 - c^(1/3)*x])/(6*c^(2/3)*(c*d^3 + e^3)) + (e*(c^(1/3)*d + e)*Log[1 + c^(1/3)*x])/(6*c^(2/3)*(c*d^3 - e^3)) - (d^2*e^3*Log[d + e*x])/(c^2*d^6 - e^6) - (e*(c^(1/3)*d + e)*Log[1 - c^(1/ 3)*x + c^(2/3)*x^2])/(12*c^(2/3)*(c*d^3 - e^3)) - ((c^(1/3)*d - e)*e*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(12*c^(2/3)*(c*d^3 + e^3)) - (d^2*Log[1 - c*x ^3])/(6*(c*d^3 + e^3)) + (d^2*Log[1 + c*x^3])/(6*(c*d^3 - e^3))))/e
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcTanh[c*x^n])/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 - c^2*x^(2 *n))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.81 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.11
| method | result | size |
| default | \(-\frac {a}{\left (e x +d \right ) e}+b \left (-\frac {\operatorname {arctanh}\left (c \,x^{3}\right )}{\left (e x +d \right ) e}+\frac {3 c \left (\frac {d e \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-e^{2} \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )-\frac {d^{2} \ln \left (c \,x^{3}-1\right )}{3}}{2 c \,d^{3}+2 e^{3}}-\frac {d^{2} e^{3} \ln \left (e x +d \right )}{\left (c \,d^{3}+e^{3}\right ) \left (c \,d^{3}-e^{3}\right )}+\frac {d e \left (\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-e^{2} \left (-\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {d^{2} \ln \left (c \,x^{3}+1\right )}{3}}{2 c \,d^{3}-2 e^{3}}\right )}{e}\right )\) | \(461\) |
| parts | \(-\frac {a}{\left (e x +d \right ) e}+b \left (-\frac {\operatorname {arctanh}\left (c \,x^{3}\right )}{\left (e x +d \right ) e}+\frac {3 c \left (\frac {d e \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-e^{2} \left (\frac {\ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )-\frac {d^{2} \ln \left (c \,x^{3}-1\right )}{3}}{2 c \,d^{3}+2 e^{3}}-\frac {d^{2} e^{3} \ln \left (e x +d \right )}{\left (c \,d^{3}+e^{3}\right ) \left (c \,d^{3}-e^{3}\right )}+\frac {d e \left (\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {2}{3}}}\right )-e^{2} \left (-\frac {\ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{6 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 c \left (\frac {1}{c}\right )^{\frac {1}{3}}}\right )+\frac {d^{2} \ln \left (c \,x^{3}+1\right )}{3}}{2 c \,d^{3}-2 e^{3}}\right )}{e}\right )\) | \(461\) |
Input:
int((a+b*arctanh(c*x^3))/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-a/(e*x+d)/e+b*(-1/(e*x+d)/e*arctanh(c*x^3)+3/e*c*((d*e*(1/3/c/(1/c)^(2/3) *ln(x-(1/c)^(1/3))-1/6/c/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))-1/3 /c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1)))-e^2*(1/3/c /(1/c)^(1/3)*ln(x-(1/c)^(1/3))-1/6/c/(1/c)^(1/3)*ln(x^2+(1/c)^(1/3)*x+(1/c )^(2/3))+1/3*3^(1/2)/c/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1)) )-1/3*d^2*ln(c*x^3-1))/(2*c*d^3+2*e^3)-d^2*e^3/(c*d^3+e^3)/(c*d^3-e^3)*ln( e*x+d)+(d*e*(1/3/c/(1/c)^(2/3)*ln(x+(1/c)^(1/3))-1/6/c/(1/c)^(2/3)*ln(x^2- (1/c)^(1/3)*x+(1/c)^(2/3))+1/3/c/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2 /(1/c)^(1/3)*x-1)))-e^2*(-1/3/c/(1/c)^(1/3)*ln(x+(1/c)^(1/3))+1/6/c/(1/c)^ (1/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+1/3*3^(1/2)/c/(1/c)^(1/3)*arctan(1 /3*3^(1/2)*(2/(1/c)^(1/3)*x-1)))+1/3*d^2*ln(c*x^3+1))/(2*c*d^3-2*e^3)))
Result contains complex when optimal does not.
Time = 137.99 (sec) , antiderivative size = 12984, normalized size of antiderivative = 31.36 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=\text {Timed out} \] Input:
integrate((a+b*atanh(c*x**3))/(e*x+d)**2,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=-\frac {1}{4} \, {\left ({\left (\frac {12 \, d^{2} e^{2} \log \left (e x + d\right )}{c^{2} d^{6} - e^{6}} + \frac {2 \, \sqrt {3} {\left (c d e + c^{\frac {2}{3}} e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x + c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{{\left (c^{2} d^{3} e + c e^{4}\right )} c^{\frac {1}{3}}} - \frac {2 \, \sqrt {3} {\left (c d e - c^{\frac {2}{3}} e^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, c^{\frac {2}{3}} x - c^{\frac {1}{3}}\right )}}{3 \, c^{\frac {1}{3}}}\right )}{{\left (c^{2} d^{3} e - c e^{4}\right )} c^{\frac {1}{3}}} + \frac {{\left (2 \, c d^{2} + c^{\frac {2}{3}} d e - c^{\frac {1}{3}} e^{2}\right )} \log \left (c^{\frac {2}{3}} x^{2} + c^{\frac {1}{3}} x + 1\right )}{c^{2} d^{3} e + c e^{4}} - \frac {{\left (2 \, c d^{2} - c^{\frac {2}{3}} d e - c^{\frac {1}{3}} e^{2}\right )} \log \left (c^{\frac {2}{3}} x^{2} - c^{\frac {1}{3}} x + 1\right )}{c^{2} d^{3} e - c e^{4}} - \frac {2 \, {\left (c d^{2} + c^{\frac {2}{3}} d e + c^{\frac {1}{3}} e^{2}\right )} \log \left (\frac {c^{\frac {1}{3}} x + 1}{c^{\frac {1}{3}}}\right )}{c^{2} d^{3} e - c e^{4}} + \frac {2 \, {\left (c d^{2} - c^{\frac {2}{3}} d e + c^{\frac {1}{3}} e^{2}\right )} \log \left (\frac {c^{\frac {1}{3}} x - 1}{c^{\frac {1}{3}}}\right )}{c^{2} d^{3} e + c e^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x^{3}\right )}{e^{2} x + d e}\right )} b - \frac {a}{e^{2} x + d e} \] Input:
integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="maxima")
Output:
-1/4*((12*d^2*e^2*log(e*x + d)/(c^2*d^6 - e^6) + 2*sqrt(3)*(c*d*e + c^(2/3 )*e^2)*arctan(1/3*sqrt(3)*(2*c^(2/3)*x + c^(1/3))/c^(1/3))/((c^2*d^3*e + c *e^4)*c^(1/3)) - 2*sqrt(3)*(c*d*e - c^(2/3)*e^2)*arctan(1/3*sqrt(3)*(2*c^( 2/3)*x - c^(1/3))/c^(1/3))/((c^2*d^3*e - c*e^4)*c^(1/3)) + (2*c*d^2 + c^(2 /3)*d*e - c^(1/3)*e^2)*log(c^(2/3)*x^2 + c^(1/3)*x + 1)/(c^2*d^3*e + c*e^4 ) - (2*c*d^2 - c^(2/3)*d*e - c^(1/3)*e^2)*log(c^(2/3)*x^2 - c^(1/3)*x + 1) /(c^2*d^3*e - c*e^4) - 2*(c*d^2 + c^(2/3)*d*e + c^(1/3)*e^2)*log((c^(1/3)* x + 1)/c^(1/3))/(c^2*d^3*e - c*e^4) + 2*(c*d^2 - c^(2/3)*d*e + c^(1/3)*e^2 )*log((c^(1/3)*x - 1)/c^(1/3))/(c^2*d^3*e + c*e^4))*c + 4*arctanh(c*x^3)/( e^2*x + d*e))*b - a/(e^2*x + d*e)
Time = 15.72 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.34 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="giac")
Output:
-3*b*c*d^2*e^2*log(e*x + d)/(c^2*d^6 - e^6) + 1/2*b*c*d^2*log(abs(c*x^3 + 1))/(c*d^3*e - e^4) - 1/2*b*c*d^2*log(abs(-c*x^3 + 1))/(c*d^3*e + e^4) - 1 /2*sqrt(3)*b*c*abs(c)^(2/3)*arctan(1/3*sqrt(3)*c^(1/3)*(2*x + 1/c^(1/3)))/ (c^2*d^2 - c*d*e*abs(c)^(2/3) + e^2*abs(c)^(4/3)) - 1/2*sqrt(3)*b*c*arctan (1/3*sqrt(3)*(2*x + (-1/c)^(1/3))/(-1/c)^(1/3))/(c*d*e + (-c^2)^(2/3)*d^2 - (-c^2)^(1/3)*e^2) + 1/2*(b*c^3*d^3*e^3*(-1/c)^(1/3) - b*c^3*d^4*e^2 - b* c^2*e^6*(-1/c)^(1/3) + b*c^2*d*e^5)*(-1/c)^(1/3)*log(abs(x - (-1/c)^(1/3)) )/(c^3*d^6*e^2 - 2*c^2*d^3*e^5 + c*e^8) + 1/4*((-c^2)^(1/3)*b*c*d - (-c^2) ^(2/3)*b*e)*log(x^2 + x*(-1/c)^(1/3) + (-1/c)^(2/3))/(c^2*d^3 - c*e^3) - 1 /4*(b*c*d*abs(c)^(2/3) - b*e*abs(c)^(4/3))*log(x^2 + x/c^(1/3) + 1/c^(2/3) )/(c^2*d^3 + c*e^3) - 1/2*b*log(-(c*x^3 + 1)/(c*x^3 - 1))/(e^2*x + d*e) + 1/2*(b*c^3*d^4*e^2 - b*c^(8/3)*d^3*e^3 + b*c^2*d*e^5 - b*c^(5/3)*e^6)*log( abs(x - 1/c^(1/3)))/((c^3*d^6*e^2 + 2*c^2*d^3*e^5 + c*e^8)*c^(1/3)) - a/(e ^2*x + d*e)
Time = 3.92 (sec) , antiderivative size = 2638, normalized size of antiderivative = 6.37 \[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=\text {Too large to display} \] Input:
int((a + b*atanh(c*x^3))/(d + e*x)^2,x)
Output:
symsum(log(-(729*b^6*c^14*d*e^2 + 54432*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^6*c^12*e^15*x + 729*b^6 *c^14*e^3*x + 31104*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^6*c^14*d^7*e^8 + 243*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)*b^5*c^15*d^ 5 + 62208*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c* d*e*z - b^3*c, z, k)^6*c^12*d*e^14 - 5832*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^2*b^4*c^14*d^3*e^4 - 1944*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^3*b^3*c^15*d^7*e^2 + 15552*root(8*c*d^3*e^3*z^3 - 8*e^6*z^ 3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^4*b^2*c^14*d^5*e^6 + 10692*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e *z - b^3*c, z, k)^3*b^3*c^13*d*e^8 + 101088*root(8*c*d^3*e^3*z^3 - 8*e^6*z ^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^5*b*c^13*d^3*e^10 + 3888*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e* z - b^3*c, z, k)^5*b*c^15*d^9*e^4 + 12636*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^3*b^3*c^13*e^9*x + 38 880*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^6*c^14*d^6*e^9*x + 116640*root(8*c*d^3*e^3*z^3 - 8*e^6*z^3 - 12*b*c*d^2*e^2*z^2 + 6*b^2*c*d*e*z - b^3*c, z, k)^5*b*c^13*d^2*e^11*x...
\[ \int \frac {a+b \text {arctanh}\left (c x^3\right )}{(d+e x)^2} \, dx=\int \frac {\mathit {atanh} \left (c \,x^{3}\right ) b +a}{\left (e x +d \right )^{2}}d x \] Input:
int((a+b*atanh(c*x^3))/(e*x+d)^2,x)
Output:
int((a+b*atanh(c*x^3))/(e*x+d)^2,x)