3.1.26 \(\int \frac {a+b \tanh ^{-1}(c x^2)}{d+e x} \, dx\) [26]

Optimal. Leaf size=325 \[ \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right ) \log (d+e x)}{e}-\frac {b \log \left (\frac {e \left (1-\sqrt {-c} x\right )}{\sqrt {-c} d+e}\right ) \log (d+e x)}{2 e}-\frac {b \log \left (-\frac {e \left (1+\sqrt {-c} x\right )}{\sqrt {-c} d-e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (\frac {e \left (1-\sqrt {c} x\right )}{\sqrt {c} d+e}\right ) \log (d+e x)}{2 e}+\frac {b \log \left (-\frac {e \left (1+\sqrt {c} x\right )}{\sqrt {c} d-e}\right ) \log (d+e x)}{2 e}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d-e}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )}{2 e}-\frac {b \text {PolyLog}\left (2,\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d+e}\right )}{2 e}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )}{2 e} \]

[Out]

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

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Rubi [A]
time = 0.41, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 19, 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^2\right )\right )}{e}-\frac {b \text {Li}_2\left (\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d-e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )}{2 e}-\frac {b \text {Li}_2\left (\frac {\sqrt {-c} (d+e x)}{\sqrt {-c} d+e}\right )}{2 e}+\frac {b \text {Li}_2\left (\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (\frac {e \left (1-\sqrt {-c} x\right )}{\sqrt {-c} d+e}\right )}{2 e}-\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt {-c} x+1\right )}{\sqrt {-c} d-e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (\frac {e \left (1-\sqrt {c} x\right )}{\sqrt {c} d+e}\right )}{2 e}+\frac {b \log (d+e x) \log \left (-\frac {e \left (\sqrt {c} x+1\right )}{\sqrt {c} d-e}\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a + b*ArcTanh[c*x^2])*Log[d + e*x])/e - (b*Log[(e*(1 - Sqrt[-c]*x))/(Sqrt[-c]*d + e)]*Log[d + e*x])/(2*e) -
(b*Log[-((e*(1 + Sqrt[-c]*x))/(Sqrt[-c]*d - e))]*Log[d + e*x])/(2*e) + (b*Log[(e*(1 - Sqrt[c]*x))/(Sqrt[c]*d +
e)]*Log[d + e*x])/(2*e) + (b*Log[-((e*(1 + Sqrt[c]*x))/(Sqrt[c]*d - e))]*Log[d + e*x])/(2*e) - (b*PolyLog[2,
(Sqrt[-c]*(d + e*x))/(Sqrt[-c]*d - e)])/(2*e) + (b*PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d - e)])/(2*e) - (b
*PolyLog[2, (Sqrt[-c]*(d + e*x))/(Sqrt[-c]*d + e)])/(2*e) + (b*PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + 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^2\right )}{d+e x} \, dx &=\int \left (\frac {a}{d+e x}+\frac {b \tanh ^{-1}\left (c x^2\right )}{d+e x}\right ) \, dx\\ &=\frac {a \log (d+e x)}{e}+b \int \frac {\tanh ^{-1}\left (c x^2\right )}{d+e x} \, dx\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 13.63, size = 285, normalized size = 0.88 \begin {gather*} \frac {a \log (d+e x)}{e}+\frac {b \left (2 \tanh ^{-1}\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 )+\text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-e}\right )-\text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d-i e}\right )-\text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+i e}\right )+\text {PolyLog}\left (2,\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+e}\right )\right )}{2 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(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))]*
Log[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)]
+ PolyLog[2, (Sqrt[c]*(d + e*x))/(Sqrt[c]*d + e)]))/(2*e)

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Maple [A]
time = 0.14, size = 362, normalized size = 1.11

method	result	size
default	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \ln \left (e x +d \right ) \arctanh \left (c \,x^{2}\right )}{e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +d c}{e \sqrt {c}+d c}\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -d c}{e \sqrt {c}-d c}\right )}{2 e}+\frac {b \dilog \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +d c}{e \sqrt {c}+d c}\right )}{2 e}+\frac {b \dilog \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -d c}{e \sqrt {c}-d c}\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +d c}{e \sqrt {-c}+d c}\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -d c}{e \sqrt {-c}-d c}\right )}{2 e}-\frac {b \dilog \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +d c}{e \sqrt {-c}+d c}\right )}{2 e}-\frac {b \dilog \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -d c}{e \sqrt {-c}-d c}\right )}{2 e}\)	\(362\)
risch	\(\frac {a \ln \left (e x +d \right )}{e}-\frac {b \ln \left (e x +d \right ) \ln \left (-c \,x^{2}+1\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +d c}{e \sqrt {c}+d c}\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -d c}{e \sqrt {c}-d c}\right )}{2 e}+\frac {b \dilog \left (\frac {e \sqrt {c}-\left (e x +d \right ) c +d c}{e \sqrt {c}+d c}\right )}{2 e}+\frac {b \dilog \left (\frac {e \sqrt {c}+\left (e x +d \right ) c -d c}{e \sqrt {c}-d c}\right )}{2 e}+\frac {b \ln \left (e x +d \right ) \ln \left (c \,x^{2}+1\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +d c}{e \sqrt {-c}+d c}\right )}{2 e}-\frac {b \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -d c}{e \sqrt {-c}-d c}\right )}{2 e}-\frac {b \dilog \left (\frac {e \sqrt {-c}-\left (e x +d \right ) c +d c}{e \sqrt {-c}+d c}\right )}{2 e}-\frac {b \dilog \left (\frac {e \sqrt {-c}+\left (e x +d \right ) c -d c}{e \sqrt {-c}-d c}\right )}{2 e}\)	\(386\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2))/(e*x+d),x, algorithm="maxima")

[Out]

a*e^(-1)*log(x*e + d) + 1/2*b*integrate((log(c*x^2 + 1) - log(-c*x^2 + 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^2))/(e*x+d),x, algorithm="fricas")

[Out]

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

[Out]

integrate((b*arctanh(c*x^2) + 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^2\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^2))/(d + e*x),x)

[Out]

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

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