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

Optimal. Leaf size=93 \[ -\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{b c \log (1-c x)}{2 e (c d+e)}+\frac{b c \log (c x+1)}{2 e (c d-e)} \]

[Out]

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

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Rubi [A]  time = 0.0670172, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5926, 706, 31, 633} \[ -\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{b c \log (1-c x)}{2 e (c d+e)}+\frac{b c \log (c x+1)}{2 e (c d-e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5926

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

Rule 706

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

Rule 31

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

Rule 633

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[e/2 + (c*d)/(2*q),
Int[1/(-q + c*x), x], x] + Dist[e/2 - (c*d)/(2*q), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[-(a*c)]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}+\frac{(b c) \int \frac{1}{(d+e x) \left (1-c^2 x^2\right )} \, dx}{e}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{(b c) \int \frac{-c^2 d+c^2 e x}{1-c^2 x^2} \, dx}{e \left (c^2 d^2-e^2\right )}-\frac{(b c e) \int \frac{1}{d+e x} \, dx}{c^2 d^2-e^2}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{\left (b c^3\right ) \int \frac{1}{-c-c^2 x} \, dx}{2 (c d-e) e}+\frac{\left (b c^3\right ) \int \frac{1}{c-c^2 x} \, dx}{2 e (c d+e)}\\ &=-\frac{a+b \tanh ^{-1}(c x)}{e (d+e x)}-\frac{b c \log (1-c x)}{2 e (c d+e)}+\frac{b c \log (1+c x)}{2 (c d-e) e}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}\\ \end{align*}

Mathematica [A]  time = 0.119348, size = 102, normalized size = 1.1 \[ -\frac{a}{e (d+e x)}-\frac{b c \log (d+e x)}{c^2 d^2-e^2}-\frac{b c \log (1-c x)}{2 e (c d+e)}-\frac{b c \log (c x+1)}{2 e (e-c d)}-\frac{b \tanh ^{-1}(c x)}{e (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.031, size = 114, normalized size = 1.2 \begin{align*} -{\frac{ac}{ \left ( cxe+cd \right ) e}}-{\frac{bc{\it Artanh} \left ( cx \right ) }{ \left ( cxe+cd \right ) e}}-{\frac{bc\ln \left ( cxe+cd \right ) }{ \left ( cd+e \right ) \left ( cd-e \right ) }}-{\frac{bc\ln \left ( cx-1 \right ) }{e \left ( 2\,cd+2\,e \right ) }}+{\frac{bc\ln \left ( cx+1 \right ) }{e \left ( 2\,cd-2\,e \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.967194, size = 134, normalized size = 1.44 \begin{align*} \frac{1}{2} \,{\left (c{\left (\frac{\log \left (c x + 1\right )}{c d e - e^{2}} - \frac{\log \left (c x - 1\right )}{c d e + e^{2}} - \frac{2 \, \log \left (e x + d\right )}{c^{2} d^{2} - e^{2}}\right )} - \frac{2 \, \operatorname{artanh}\left (c x\right )}{e^{2} x + d e}\right )} b - \frac{a}{e^{2} x + d e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.91065, size = 387, normalized size = 4.16 \begin{align*} -\frac{2 \, a c^{2} d^{2} - 2 \, a e^{2} -{\left (b c^{2} d^{2} + b c d e +{\left (b c^{2} d e + b c e^{2}\right )} x\right )} \log \left (c x + 1\right ) +{\left (b c^{2} d^{2} - b c d e +{\left (b c^{2} d e - b c e^{2}\right )} x\right )} \log \left (c x - 1\right ) + 2 \,{\left (b c e^{2} x + b c d e\right )} \log \left (e x + d\right ) +{\left (b c^{2} d^{2} - b e^{2}\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{2 \,{\left (c^{2} d^{3} e - d e^{3} +{\left (c^{2} d^{2} e^{2} - e^{4}\right )} x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*c^2*d^2 - 2*a*e^2 - (b*c^2*d^2 + b*c*d*e + (b*c^2*d*e + b*c*e^2)*x)*log(c*x + 1) + (b*c^2*d^2 - b*c*
d*e + (b*c^2*d*e - b*c*e^2)*x)*log(c*x - 1) + 2*(b*c*e^2*x + b*c*d*e)*log(e*x + d) + (b*c^2*d^2 - b*e^2)*log(-
(c*x + 1)/(c*x - 1)))/(c^2*d^3*e - d*e^3 + (c^2*d^2*e^2 - e^4)*x)

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Sympy [A]  time = 8.62908, size = 782, normalized size = 8.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*x/(d**2 + d*e*x), Eq(c, 0)), ((a*x + b*x*atanh(c*x) + b*log(x - 1/c)/c + b*atanh(c*x)/c)/d**2, Eq
(e, 0)), (-2*a*d/(4*d**2*e + 4*d*e**2*x) + 2*a*e*x/(4*d**2*e + 4*d*e**2*x) + 2*b*d*atanh(e*x/d)/(4*d**2*e + 4*
d*e**2*x) + b*d/(4*d**2*e + 4*d*e**2*x) - 2*b*e*x*atanh(e*x/d)/(4*d**2*e + 4*d*e**2*x) - b*e*x/(4*d**2*e + 4*d
*e**2*x), Eq(c, -e/d)), (-2*a*d/(4*d**2*e + 4*d*e**2*x) + 2*a*e*x/(4*d**2*e + 4*d*e**2*x) - 2*b*d*atanh(e*x/d)
/(4*d**2*e + 4*d*e**2*x) - b*d/(4*d**2*e + 4*d*e**2*x) + 2*b*e*x*atanh(e*x/d)/(4*d**2*e + 4*d*e**2*x) + b*e*x/
(4*d**2*e + 4*d*e**2*x), Eq(c, e/d)), (zoo*(a*x + b*x*atanh(c*x) + b*log(x - 1/c)/c + b*atanh(c*x)/c), Eq(d, -
e*x)), (-a*c**2*d**2/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) + a*e**2/(c**2*d**3*e + c**2*d**2*e**2
*x - d*e**3 - e**4*x) + b*c**2*d*e*x*atanh(c*x)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) + b*c*d*e*l
og(x - 1/c)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) - b*c*d*e*log(d/e + x)/(c**2*d**3*e + c**2*d**2
*e**2*x - d*e**3 - e**4*x) + b*c*d*e*atanh(c*x)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) + b*c*e**2*
x*log(x - 1/c)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) - b*c*e**2*x*log(d/e + x)/(c**2*d**3*e + c**
2*d**2*e**2*x - d*e**3 - e**4*x) + b*c*e**2*x*atanh(c*x)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x) +
b*e**2*atanh(c*x)/(c**2*d**3*e + c**2*d**2*e**2*x - d*e**3 - e**4*x), True))

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Giac [A]  time = 1.1734, size = 176, normalized size = 1.89 \begin{align*} \frac{1}{2} \,{\left (c{\left (\frac{\log \left ({\left | c - \frac{c d}{x e + d} + \frac{e}{x e + d} \right |}\right )}{c d e^{3} - e^{4}} - \frac{\log \left ({\left | c - \frac{c d}{x e + d} - \frac{e}{x e + d} \right |}\right )}{c d e^{3} + e^{4}}\right )} e^{2} - \frac{e^{\left (-1\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{x e + d}\right )} b - \frac{a e^{\left (-1\right )}}{x e + d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(log(abs(c - c*d/(x*e + d) + e/(x*e + d)))/(c*d*e^3 - e^4) - log(abs(c - c*d/(x*e + d) - e/(x*e + d)))/
(c*d*e^3 + e^4))*e^2 - e^(-1)*log(-(c*x + 1)/(c*x - 1))/(x*e + d))*b - a*e^(-1)/(x*e + d)