3.1.35 \(\int \frac {a+b \tanh ^{-1}(c x^3)}{(d+e x)^2} \, dx\) [35]

Optimal. Leaf size=414 \[ -\frac {\sqrt {3} b \sqrt [3]{c} \text {ArcTan}\left (\frac {1-2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac {\sqrt {3} b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \left (c d^3+e^3\right )}-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1-\sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}+\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1+\sqrt [3]{c} x\right )}{2 \left (c d^3-e^3\right )}-\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (1-\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3-e^3\right )}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (1+\sqrt [3]{c} x+c^{2/3} x^2\right )}{4 \left (c d^3+e^3\right )}-\frac {b c d^2 \log \left (1-c x^3\right )}{2 e \left (c d^3+e^3\right )}+\frac {b c d^2 \log \left (1+c x^3\right )}{2 e \left (c d^3-e^3\right )} \]

[Out]

(-a-b*arctanh(c*x^3))/e/(e*x+d)+1/2*b*c^(1/3)*(c^(1/3)*d-e)*ln(1-c^(1/3)*x)/(c*d^3+e^3)+1/2*b*c^(1/3)*(c^(1/3)
*d+e)*ln(1+c^(1/3)*x)/(c*d^3-e^3)-3*b*c*d^2*e^2*ln(e*x+d)/(c^2*d^6-e^6)-1/4*b*c^(1/3)*(c^(1/3)*d+e)*ln(1-c^(1/
3)*x+c^(2/3)*x^2)/(c*d^3-e^3)-1/4*b*c^(1/3)*(c^(1/3)*d-e)*ln(1+c^(1/3)*x+c^(2/3)*x^2)/(c*d^3+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)-1/2*b*c^(1/3)*arctan(1/3*(1-2*c^(1/3)*x)*3^(1
/2))*3^(1/2)/(c^(2/3)*d^2+c^(1/3)*d*e+e^2)-1/2*b*c^(1/3)*(c^(1/3)*d+e)*arctan(1/3*(1+2*c^(1/3)*x)*3^(1/2))*3^(
1/2)/(c*d^3+e^3)

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Rubi [A]
time = 0.52, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6071, 6857, 1885, 1875, 31, 648, 631, 210, 642, 266, 1874} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac {\sqrt {3} b \sqrt [3]{c} \text {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} \text {ArcTan}\left (\frac {2 \sqrt [3]{c} x+1}{\sqrt {3}}\right ) \left (\sqrt [3]{c} d+e\right )}{2 \left (c d^3+e^3\right )}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (c^{2/3} x^2-\sqrt [3]{c} x+1\right )}{4 \left (c d^3-e^3\right )}-\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right ) \log \left (c^{2/3} x^2+\sqrt [3]{c} x+1\right )}{4 \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\right )}{2 \left (c d^3+e^3\right )}+\frac {b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right ) \log \left (\sqrt [3]{c} x+1\right )}{2 \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 (c x^3+1\right )}{2 e \left (c d^3-e^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^3])/(d + e*x)^2,x]

[Out]

-1/2*(Sqrt[3]*b*c^(1/3)*ArcTan[(1 - 2*c^(1/3)*x)/Sqrt[3]])/(c^(2/3)*d^2 + c^(1/3)*d*e + e^2) - (Sqrt[3]*b*c^(1
/3)*(c^(1/3)*d + e)*ArcTan[(1 + 2*c^(1/3)*x)/Sqrt[3]])/(2*(c*d^3 + e^3)) - (a + b*ArcTanh[c*x^3])/(e*(d + e*x)
) + (b*c^(1/3)*(c^(1/3)*d - e)*Log[1 - c^(1/3)*x])/(2*(c*d^3 + e^3)) + (b*c^(1/3)*(c^(1/3)*d + e)*Log[1 + c^(1
/3)*x])/(2*(c*d^3 - e^3)) - (3*b*c*d^2*e^2*Log[d + e*x])/(c^2*d^6 - e^6) - (b*c^(1/3)*(c^(1/3)*d + e)*Log[1 -
c^(1/3)*x + c^(2/3)*x^2])/(4*(c*d^3 - e^3)) - (b*c^(1/3)*(c^(1/3)*d - e)*Log[1 + c^(1/3)*x + c^(2/3)*x^2])/(4*
(c*d^3 + e^3)) - (b*c*d^2*Log[1 - c*x^3])/(2*e*(c*d^3 + e^3)) + (b*c*d^2*Log[1 + c*x^3])/(2*e*(c*d^3 - e^3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1874

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[a/b, 3]], s = Denominator[R
t[a/b, 3]]}, Dist[(-r)*((B*r - A*s)/(3*a*s)), Int[1/(r + s*x), x], x] + Dist[r/(3*a*s), Int[(r*(B*r + 2*A*s) +
s*(B*r - A*s)*x)/(r^2 - r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && PosQ[
a/b]

Rule 1875

Int[((A_) + (B_.)*(x_))/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 3]], s = Denominator[
Rt[-a/b, 3]]}, Dist[r*((B*r + A*s)/(3*a*s)), Int[1/(r - s*x), x], x] - Dist[r/(3*a*s), Int[(r*(B*r - 2*A*s) -
s*(B*r + A*s)*x)/(r^2 + r*s*x + s^2*x^2), x], x]] /; FreeQ[{a, b, A, B}, x] && NeQ[a*B^3 - b*A^3, 0] && NegQ[a
/b]

Rule 1885

Int[(P2_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> With[{A = Coeff[P2, x, 0], B = Coeff[P2, x, 1], C = Coeff[P2, x,
2]}, Int[(A + B*x)/(a + b*x^3), x] + Dist[C, Int[x^2/(a + b*x^3), x], x] /; EqQ[a*B^3 - b*A^3, 0] ||  !Ration
alQ[a/b]] /; FreeQ[{a, b}, x] && PolyQ[P2, x, 2]

Rule 6071

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] - Dist[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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{(d+e x)^2} \, dx &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {b \int \frac {3 c x^2}{(d+e x) \left (1-c^2 x^6\right )} \, dx}{e}\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {(3 b c) \int \frac {x^2}{(d+e x) \left (1-c^2 x^6\right )} \, dx}{e}\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}+\frac {(3 b c) \int \left (\frac {d^2 e^4}{\left (-c d^3+e^3\right ) \left (c d^3+e^3\right ) (d+e x)}+\frac {d e-e^2 x-c d^2 x^2}{2 \left (c d^3+e^3\right ) \left (-1+c x^3\right )}+\frac {d e-e^2 x+c d^2 x^2}{2 \left (c d^3-e^3\right ) \left (1+c x^3\right )}\right ) \, dx}{e}\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac {(3 b c) \int \frac {d e-e^2 x+c d^2 x^2}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac {(3 b c) \int \frac {d e-e^2 x-c d^2 x^2}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}+\frac {(3 b c) \int \frac {d e-e^2 x}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac {\left (3 b c^2 d^2\right ) \int \frac {x^2}{1+c x^3} \, dx}{2 e \left (c d^3-e^3\right )}+\frac {(3 b c) \int \frac {d e-e^2 x}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}-\frac {\left (3 b c^2 d^2\right ) \int \frac {x^2}{-1+c x^3} \, dx}{2 e \left (c d^3+e^3\right )}\\ &=-\frac {a+b \tanh ^{-1}\left (c x^3\right )}{e (d+e x)}-\frac {3 b c d^2 e^2 \log (d+e x)}{c^2 d^6-e^6}-\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 )}+\frac {\left (b c^{2/3}\right ) \int \frac {2 \sqrt [3]{c} d e-e^2+\sqrt [3]{c} \left (-\sqrt [3]{c} d e-e^2\right ) x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c d^3-e^3\right )}+\frac {\left (b c^{2/3} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac {1}{1+\sqrt [3]{c} x} \, dx}{2 \left (c d^3-e^3\right )}+\frac {\left (b c^{2/3}\right ) \int \frac {-2 \sqrt [3]{c} d e-e^2-\sqrt [3]{c} \left (\sqrt [3]{c} d e-e^2\right ) x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{2 e \left (c d^3+e^3\right )}-\frac {\left (b c \left (d-\frac {e}{\sqrt [3]{c}}\right )\right ) \int \frac {1}{1-\sqrt [3]{c} x} \, dx}{2 \left (c d^3+e^3\right )}\\ &=-\frac {a+b \tanh ^{-1}\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 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 )}+\frac {\left (3 b c^{2/3}\right ) \int \frac {1}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}-\frac {\left (b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac {-\sqrt [3]{c}+2 c^{2/3} x}{1-\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3-e^3\right )}-\frac {\left (b \sqrt [3]{c} \left (\sqrt [3]{c} d-e\right )\right ) \int \frac {\sqrt [3]{c}+2 c^{2/3} x}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3+e^3\right )}-\frac {\left (3 b c^{2/3} \left (\sqrt [3]{c} d+e\right )\right ) \int \frac {1}{1+\sqrt [3]{c} x+c^{2/3} x^2} \, dx}{4 \left (c d^3+e^3\right )}\\ &=-\frac {a+b \tanh ^{-1}\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 )}+\frac {\left (3 b \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-2 \sqrt [3]{c} x\right )}{2 \left (c^{2/3} d^2+\sqrt [3]{c} d e+e^2\right )}+\frac {\left (3 b \sqrt [3]{c} \left (\sqrt [3]{c} d+e\right )\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{c} x\right )}{2 \left (c d^3+e^3\right )}\\ &=-\frac {\sqrt {3} b \sqrt [3]{c} \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{2 \left (c d^3+e^3\right )}-\frac {a+b \tanh ^{-1}\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 )}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 534, normalized size = 1.29 \begin {gather*} \frac {1}{4} \left (-\frac {4 a}{e (d+e x)}+\frac {2 \sqrt {3} b \sqrt [3]{c} \text {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 ) \text {ArcTan}\left (\frac {1+2 \sqrt [3]{c} x}{\sqrt {3}}\right )}{c d^3+e^3}-\frac {4 b \tanh ^{-1}\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^3])/(d + e*x)^2,x]

[Out]

((-4*a)/(e*(d + e*x)) + (2*Sqrt[3]*b*c^(1/3)*ArcTan[(-1 + 2*c^(1/3)*x)/Sqrt[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)*L
og[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)*Log[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

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Maple [A]
time = 0.30, size = 591, normalized size = 1.43

method	result	size
default	\(-\frac {a}{\left (e x +d \right ) e}-\frac {b \arctanh \left (c \,x^{3}\right )}{\left (e x +d \right ) e}+\frac {b d \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{\left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{2 \left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}+\frac {b e \ln \left (x +\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{\left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e \ln \left (x^{2}-\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{2 \left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{\left (2 c \,d^{3}-2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b c \,d^{2} \ln \left (c \,x^{3}+1\right )}{e \left (2 c \,d^{3}-2 e^{3}\right )}+\frac {b d \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{\left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{2 \left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b d \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{\left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {2}{3}}}-\frac {b e \ln \left (x -\left (\frac {1}{c}\right )^{\frac {1}{3}}\right )}{\left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}+\frac {b e \ln \left (x^{2}+\left (\frac {1}{c}\right )^{\frac {1}{3}} x +\left (\frac {1}{c}\right )^{\frac {2}{3}}\right )}{2 \left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b e \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {1}{c}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{\left (2 c \,d^{3}+2 e^{3}\right ) \left (\frac {1}{c}\right )^{\frac {1}{3}}}-\frac {b c \,d^{2} \ln \left (c \,x^{3}-1\right )}{e \left (2 c \,d^{3}+2 e^{3}\right )}-\frac {3 b \,e^{2} c \,d^{2} \ln \left (e x +d \right )}{\left (c \,d^{3}+e^{3}\right ) \left (c \,d^{3}-e^{3}\right )}\)	\(591\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^3))/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

-a/(e*x+d)/e-b/(e*x+d)/e*arctanh(c*x^3)+b/(2*c*d^3-2*e^3)*d/(1/c)^(2/3)*ln(x+(1/c)^(1/3))-1/2*b/(2*c*d^3-2*e^3
)*d/(1/c)^(2/3)*ln(x^2-(1/c)^(1/3)*x+(1/c)^(2/3))+b/(2*c*d^3-2*e^3)*d/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(
2/(1/c)^(1/3)*x-1))+b*e/(2*c*d^3-2*e^3)/(1/c)^(1/3)*ln(x+(1/c)^(1/3))-1/2*b*e/(2*c*d^3-2*e^3)/(1/c)^(1/3)*ln(x
^2-(1/c)^(1/3)*x+(1/c)^(2/3))-b*e/(2*c*d^3-2*e^3)*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x-1))+
b/e*c/(2*c*d^3-2*e^3)*d^2*ln(c*x^3+1)+b/(2*c*d^3+2*e^3)*d/(1/c)^(2/3)*ln(x-(1/c)^(1/3))-1/2*b/(2*c*d^3+2*e^3)*
d/(1/c)^(2/3)*ln(x^2+(1/c)^(1/3)*x+(1/c)^(2/3))-b/(2*c*d^3+2*e^3)*d/(1/c)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/
(1/c)^(1/3)*x+1))-b*e/(2*c*d^3+2*e^3)/(1/c)^(1/3)*ln(x-(1/c)^(1/3))+1/2*b*e/(2*c*d^3+2*e^3)/(1/c)^(1/3)*ln(x^2
+(1/c)^(1/3)*x+(1/c)^(2/3))-b*e/(2*c*d^3+2*e^3)*3^(1/2)/(1/c)^(1/3)*arctan(1/3*3^(1/2)*(2/(1/c)^(1/3)*x+1))-b/
e*c/(2*c*d^3+2*e^3)*d^2*ln(c*x^3-1)-3*b*e^2*c*d^2/(c*d^3+e^3)/(c*d^3-e^3)*ln(e*x+d)

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Maxima [A]
time = 0.47, size = 393, normalized size = 0.95 \begin {gather*} -\frac {1}{4} \, {\left ({\left (\frac {12 \, d^{2} e^{2} \log \left (x e + 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 )}{x e^{2} + d e}\right )} b - \frac {a}{x e^{2} + d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="maxima")

[Out]

-1/4*((12*d^2*e^2*log(x*e + 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)/(x*e^2 + d*e))*b - a/(x*e^2 + d*e)

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Fricas [C] Result contains complex when optimal does not.
time = 84.38, size = 36636, normalized size = 88.49 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="fricas")

[Out]

-1/16*(16*a*c^2*d^6 - 16*a*cosh(1)^6 - 96*a*cosh(1)^5*sinh(1) - 240*a*cosh(1)^4*sinh(1)^2 - 320*a*cosh(1)^3*si
nh(1)^3 - 240*a*cosh(1)^2*sinh(1)^4 - 96*a*cosh(1)*sinh(1)^5 - 16*a*sinh(1)^6 + 2*(c^2*d^6*x*cosh(1)^2 + c^2*d
^7*cosh(1) - x*cosh(1)^8 - x*sinh(1)^8 - d*cosh(1)^7 - (8*x*cosh(1) + d)*sinh(1)^7 - 7*(4*x*cosh(1)^2 + d*cosh
(1))*sinh(1)^6 - 7*(8*x*cosh(1)^3 + 3*d*cosh(1)^2)*sinh(1)^5 - 35*(2*x*cosh(1)^4 + d*cosh(1)^3)*sinh(1)^4 - 7*
(8*x*cosh(1)^5 + 5*d*cosh(1)^4)*sinh(1)^3 + (c^2*d^6*x - 28*x*cosh(1)^6 - 21*d*cosh(1)^5)*sinh(1)^2 + (2*c^2*d
^6*x*cosh(1) + c^2*d^7 - 8*x*cosh(1)^7 - 7*d*cosh(1)^6)*sinh(1))*((-I*sqrt(3) + 1)*((b*c*d^2*cosh(1) - b*c*d^2
*sinh(1))^2/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - (b^2*c*d*cosh(1)^2
- 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)
^2 + sinh(1)^3))/(-1/16*(cosh(1)^2 - sinh(1)^2)^3*b^3*c/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*s
inh(1)^2 + sinh(1)^3)^2 - 1/8*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))^3/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) +
3*cosh(1)*sinh(1)^2 + sinh(1)^3)^3 + 3/16*(b^2*c*d*cosh(1)^2 - 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)
*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^
3)^2 - 1/16*(b^3*c*cosh(1)^3 - 3*b^3*c*cosh(1)^2*sinh(1) + 3*b^3*c*cosh(1)*sinh(1)^2 - b^3*c*sinh(1)^3)/(c*d^3
+ cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3))^(1/3) + 4*(-1/16*(cosh(1)^2 - sinh(1)^2
)^3*b^3*c/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/8*(b*c*d^2*cosh(1)
- b*c*d^2*sinh(1))^3/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^3 + 3/16*(b^
2*c*d*cosh(1)^2 - 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 +
cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/16*(b^3*c*cosh(1)^3 - 3*b^3*c*cosh(1)
^2*sinh(1) + 3*b^3*c*cosh(1)*sinh(1)^2 - b^3*c*sinh(1)^3)/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)
*sinh(1)^2 + sinh(1)^3))^(1/3)*(I*sqrt(3) + 1) + 4*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^3 + 3*
cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3))*log(b^2*c*x*cosh(1) + b^2*c*x*sinh(1) - 2*b^2*c*d - 1/16
*(c*d^3*cosh(1)^2 + cosh(1)^5 + 10*cosh(1)^2*sinh(1)^3 + 5*cosh(1)*sinh(1)^4 + sinh(1)^5 + (c*d^3 + 10*cosh(1)
^3)*sinh(1)^2 + (2*c*d^3*cosh(1) + 5*cosh(1)^4)*sinh(1))*((-I*sqrt(3) + 1)*((b*c*d^2*cosh(1) - b*c*d^2*sinh(1)
)^2/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - (b^2*c*d*cosh(1)^2 - 2*b^2
*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sin
h(1)^3))/(-1/16*(cosh(1)^2 - sinh(1)^2)^3*b^3*c/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2
+ sinh(1)^3)^2 - 1/8*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))^3/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(
1)*sinh(1)^2 + sinh(1)^3)^3 + 3/16*(b^2*c*d*cosh(1)^2 - 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)*(b*c*d^
2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1
/16*(b^3*c*cosh(1)^3 - 3*b^3*c*cosh(1)^2*sinh(1) + 3*b^3*c*cosh(1)*sinh(1)^2 - b^3*c*sinh(1)^3)/(c*d^3 + cosh(
1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3))^(1/3) + 4*(-1/16*(cosh(1)^2 - sinh(1)^2)^3*b^3*
c/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/8*(b*c*d^2*cosh(1) - b*c*d
^2*sinh(1))^3/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^3 + 3/16*(b^2*c*d*co
sh(1)^2 - 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^
3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/16*(b^3*c*cosh(1)^3 - 3*b^3*c*cosh(1)^2*sinh(
1) + 3*b^3*c*cosh(1)*sinh(1)^2 - b^3*c*sinh(1)^3)/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)
^2 + sinh(1)^3))^(1/3)*(I*sqrt(3) + 1) + 4*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^3 + 3*cosh(1)^
2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3))^2 + 3/4*(b*c*d^2*cosh(1) + b*c*d^2*sinh(1))*((-I*sqrt(3) + 1)*((
b*c*d^2*cosh(1) - b*c*d^2*sinh(1))^2/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^
3)^2 - (b^2*c*d*cosh(1)^2 - 2*b^2*c*d*cosh(1)*sinh(1) + b^2*c*d*sinh(1)^2)/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*si
nh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3))/(-1/16*(cosh(1)^2 - sinh(1)^2)^3*b^3*c/(c*d^3 + cosh(1)^3 + 3*cosh(1
)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/8*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))^3/(c*d^3 + cosh(1)^
3 + 3*cosh(1)^2*sinh(1) + 3*cosh(1)*sinh(1)^2 + sinh(1)^3)^3 + 3/16*(b^2*c*d*cosh(1)^2 - 2*b^2*c*d*cosh(1)*sin
h(1) + b^2*c*d*sinh(1)^2)*(b*c*d^2*cosh(1) - b*c*d^2*sinh(1))/(c*d^3 + cosh(1)^3 + 3*cosh(1)^2*sinh(1) + 3*cos
h(1)*sinh(1)^2 + sinh(1)^3)^2 - 1/16*(b^3*c*cosh(1)^3 - 3*b^3*c*cosh(1)^2*sinh(1) + 3*b^3*c*cosh(1)*sinh(1)^2
- b^3*c*sinh(1)^3)/(c*d^3 + cosh(1)^3 + 3*cosh(...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**3))/(e*x+d)**2,x)

[Out]

Timed out

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Giac [A]
time = 14.89, size = 554, normalized size = 1.34 \begin {gather*} -\frac {3 \, b c d^{2} e^{2} \log \left (e x + d\right )}{c^{2} d^{6} - e^{6}} - \frac {b c d^{2} \log \left ({\left | c x^{3} - 1 \right |}\right )}{2 \, {\left (c d^{3} e + e^{4}\right )}} + \frac {b c d^{2} \log \left ({\left | -c x^{3} - 1 \right |}\right )}{2 \, {\left (c d^{3} e - e^{4}\right )}} - \frac {\sqrt {3} b c {\left | c \right |}^{\frac {2}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} c^{\frac {1}{3}} {\left (2 \, x + \frac {1}{c^{\frac {1}{3}}}\right )}\right )}{2 \, {\left (c^{2} d^{2} - c d e {\left | c \right |}^{\frac {2}{3}} + e^{2} {\left | c \right |}^{\frac {4}{3}}\right )}} - \frac {\sqrt {3} b c \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {1}{c}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {1}{c}\right )^{\frac {1}{3}}}\right )}{2 \, {\left (c d e + \left (-c^{2}\right )^{\frac {2}{3}} d^{2} - \left (-c^{2}\right )^{\frac {1}{3}} e^{2}\right )}} + \frac {{\left (b c^{3} d^{3} e^{3} \left (-\frac {1}{c}\right )^{\frac {1}{3}} - b c^{3} d^{4} e^{2} - b c^{2} e^{6} \left (-\frac {1}{c}\right )^{\frac {1}{3}} + b c^{2} d e^{5}\right )} \left (-\frac {1}{c}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {1}{c}\right )^{\frac {1}{3}} \right |}\right )}{2 \, {\left (c^{3} d^{6} e^{2} - 2 \, c^{2} d^{3} e^{5} + c e^{8}\right )}} + \frac {{\left (\left (-c^{2}\right )^{\frac {1}{3}} b c d - \left (-c^{2}\right )^{\frac {2}{3}} b e\right )} \log \left (x^{2} + x \left (-\frac {1}{c}\right )^{\frac {1}{3}} + \left (-\frac {1}{c}\right )^{\frac {2}{3}}\right )}{4 \, {\left (c^{2} d^{3} - c e^{3}\right )}} - \frac {{\left (b c d {\left | c \right |}^{\frac {2}{3}} - b e {\left | c \right |}^{\frac {4}{3}}\right )} \log \left (x^{2} + \frac {x}{c^{\frac {1}{3}}} + \frac {1}{c^{\frac {2}{3}}}\right )}{4 \, {\left (c^{2} d^{3} + c e^{3}\right )}} - \frac {b \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{2 \, {\left (e^{2} x + d e\right )}} + \frac {{\left (b c^{3} d^{4} e^{2} - b c^{\frac {8}{3}} d^{3} e^{3} + b c^{2} d e^{5} - b c^{\frac {5}{3}} e^{6}\right )} \log \left ({\left | x - \frac {1}{c^{\frac {1}{3}}} \right |}\right )}{2 \, {\left (c^{3} d^{6} e^{2} + 2 \, c^{2} d^{3} e^{5} + c e^{8}\right )} c^{\frac {1}{3}}} - \frac {a}{e^{2} x + d e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^3))/(e*x+d)^2,x, algorithm="giac")

[Out]

-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*lo
g(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*lo
g(-(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)

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Mupad [B]
time = 1.41, size = 2638, normalized size = 6.37 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^3))/(d + e*x)^2,x)

[Out]

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 + 38880*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 + 11664*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^8*e^5*x - 11664*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^2*e^5*x - 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)^3*b^3*c^15*d^6*e^3*x + 38880*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^4*e^7*x + 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^4*e*x)/e^4)*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), k, 1, 3) + 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 + 38880*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 + 11664*ro
ot(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^8*e^5*x - 11664*
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^2*e^5*x - 38
88*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^6*e^3*x +
38880*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^4*e^7
*x + 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^4*e*x
)/e^4)*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), k, 1, 3) - a/(d*e
+ e^2*x) - (b*log(c*x^3 + 1))/(2*(d*e + e^2*x)) + (b*log(1 - c*x^3))/(2*d*e + 2*e^2*x) + (3*b*c*d^2*e^2*log(d
+ e*x))/(e^6 - c^2*d^6)

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