3.1.34 \(\int \frac {a+b \tanh ^{-1}(c x^3)}{d+e x} \, dx\) [34]

Optimal. Leaf size=523 \[ \frac {\left (a+b \tanh ^{-1}\left (c x^3\right )\right ) \log (d+e x)}{e}+\frac {b \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left (1+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (-\frac {e \left (\sqrt [3]{-1}+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left ((-1)^{2/3}+\sqrt [3]{c} x\right )}{\sqrt [3]{c} d-(-1)^{2/3} e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (\frac {(-1)^{2/3} e \left (1+\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{c} d+(-1)^{2/3} e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (\frac {\sqrt [3]{-1} e \left (1+(-1)^{2/3} \sqrt [3]{c} x\right )}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right ) \log (d+e x)}{2 e}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e} \]

[Out]

(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))*ln(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*polylog(2,c^(1/3)*(e*x+d)/(c^(1/
3)*d+(-1)^(2/3)*e))/e

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Rubi [A]
time = 0.71, antiderivative size = 523, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {6067, 281, 212, 2463, 266, 2441, 2440, 2438} \begin {gather*} \frac {\log (d+e x) \left (a+b \tanh ^{-1}\left (c x^3\right )\right )}{e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+\sqrt [3]{-1} e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-(-1)^{2/3} e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+(-1)^{2/3} e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {e \left (1-\sqrt [3]{c} x\right )}{\sqrt [3]{c} d+e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt [3]{c} x+1\right )}{\sqrt [3]{c} d-e}\right )}{2 e}+\frac {b \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 )}{2 e}-\frac {b \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 )}{2 e}+\frac {b \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 )}{2 e}-\frac {b \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 )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2463

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 6067

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[Log[d + e*x]*((a + b*ArcT
anh[c*x^n])/e), x] - Dist[b*c*(n/e), Int[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]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx &=\int \left (\frac {a}{d+e x}+\frac {b \tanh ^{-1}\left (c x^3\right )}{d+e x}\right ) \, dx\\ &=\frac {a \log (d+e x)}{e}+b \int \frac {\tanh ^{-1}\left (c x^3\right )}{d+e x} \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 69.94, size = 515, normalized size = 0.98 \begin {gather*} \frac {a \log (d+e x)}{e}+\frac {b \left (2 \tanh ^{-1}\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 )-\text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d-e}\right )+\text {PolyLog}\left (2,\frac {\sqrt [3]{c} (d+e x)}{\sqrt [3]{c} d+e}\right )+\text {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-e-i \sqrt {3} e}\right )-\text {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d+e-i \sqrt {3} e}\right )+\text {PolyLog}\left (2,\frac {2 \sqrt [3]{c} (d+e x)}{2 \sqrt [3]{c} d-e+i \sqrt {3} e}\right )-\text {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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Log[d + e*x])/e + (b*(2*ArcTanh[c*x^3]*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[(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)] + P
olyLog[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)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.18, size = 182, normalized size = 0.35

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x^3\right )}{d+e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^3))/(d + e*x),x)

[Out]

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

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