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

Optimal. Leaf size=71 \[ \frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b d^2 (c x+1)^2}{6 c}+\frac{4 b d^2 \log (1-c x)}{3 c}+\frac{2}{3} b d^2 x \]

[Out]

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

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Rubi [A]  time = 0.0425078, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {5926, 627, 43} \[ \frac{d^2 (c x+1)^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 c}+\frac{b d^2 (c x+1)^2}{6 c}+\frac{4 b d^2 \log (1-c x)}{3 c}+\frac{2}{3} b d^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 627

Int[((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)^(m + p)*(a/d + (c*x)/e)^
p, x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && I
ntegerQ[m + p]))

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.10513, size = 92, normalized size = 1.3 \[ \frac{d^2 \left (2 a c^3 x^3+6 a c^2 x^2+6 a c x+b c^2 x^2+b \log \left (1-c^2 x^2\right )+2 b c x \left (c^2 x^2+3 c x+3\right ) \tanh ^{-1}(c x)+6 b c x+6 b \log (1-c x)\right )}{6 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 121, normalized size = 1.7 \begin{align*}{\frac{{c}^{2}{x}^{3}a{d}^{2}}{3}}+c{x}^{2}a{d}^{2}+ax{d}^{2}+{\frac{{d}^{2}a}{3\,c}}+{\frac{{c}^{2}{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{3}}{3}}+c{d}^{2}b{\it Artanh} \left ( cx \right ){x}^{2}+{d}^{2}b{\it Artanh} \left ( cx \right ) x+{\frac{{d}^{2}b{\it Artanh} \left ( cx \right ) }{3\,c}}+{\frac{c{d}^{2}b{x}^{2}}{6}}+b{d}^{2}x+{\frac{4\,{d}^{2}b\ln \left ( cx-1 \right ) }{3\,c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.959171, size = 198, normalized size = 2.79 \begin{align*} \frac{1}{3} \, a c^{2} d^{2} x^{3} + \frac{1}{6} \,{\left (2 \, x^{3} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{x^{2}}{c^{2}} + \frac{\log \left (c^{2} x^{2} - 1\right )}{c^{4}}\right )}\right )} b c^{2} d^{2} + a c d^{2} x^{2} + \frac{1}{2} \,{\left (2 \, x^{2} \operatorname{artanh}\left (c x\right ) + c{\left (\frac{2 \, x}{c^{2}} - \frac{\log \left (c x + 1\right )}{c^{3}} + \frac{\log \left (c x - 1\right )}{c^{3}}\right )}\right )} b c d^{2} + a d^{2} x + \frac{{\left (2 \, c x \operatorname{artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*c^2*d^2*x^3 + 1/6*(2*x^3*arctanh(c*x) + c*(x^2/c^2 + log(c^2*x^2 - 1)/c^4))*b*c^2*d^2 + a*c*d^2*x^2 + 1/
2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*b*c*d^2 + a*d^2*x + 1/2*(2*c*x*arct
anh(c*x) + log(-c^2*x^2 + 1))*b*d^2/c

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Fricas [A]  time = 2.03953, size = 255, normalized size = 3.59 \begin{align*} \frac{2 \, a c^{3} d^{2} x^{3} +{\left (6 \, a + b\right )} c^{2} d^{2} x^{2} + 6 \,{\left (a + b\right )} c d^{2} x + b d^{2} \log \left (c x + 1\right ) + 7 \, b d^{2} \log \left (c x - 1\right ) +{\left (b c^{3} d^{2} x^{3} + 3 \, b c^{2} d^{2} x^{2} + 3 \, b c d^{2} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right )}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*a*c^3*d^2*x^3 + (6*a + b)*c^2*d^2*x^2 + 6*(a + b)*c*d^2*x + b*d^2*log(c*x + 1) + 7*b*d^2*log(c*x - 1) +
(b*c^3*d^2*x^3 + 3*b*c^2*d^2*x^2 + 3*b*c*d^2*x)*log(-(c*x + 1)/(c*x - 1)))/c

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Sympy [A]  time = 1.54826, size = 131, normalized size = 1.85 \begin{align*} \begin{cases} \frac{a c^{2} d^{2} x^{3}}{3} + a c d^{2} x^{2} + a d^{2} x + \frac{b c^{2} d^{2} x^{3} \operatorname{atanh}{\left (c x \right )}}{3} + b c d^{2} x^{2} \operatorname{atanh}{\left (c x \right )} + \frac{b c d^{2} x^{2}}{6} + b d^{2} x \operatorname{atanh}{\left (c x \right )} + b d^{2} x + \frac{4 b d^{2} \log{\left (x - \frac{1}{c} \right )}}{3 c} + \frac{b d^{2} \operatorname{atanh}{\left (c x \right )}}{3 c} & \text{for}\: c \neq 0 \\a d^{2} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*c**2*d**2*x**3/3 + a*c*d**2*x**2 + a*d**2*x + b*c**2*d**2*x**3*atanh(c*x)/3 + b*c*d**2*x**2*atanh
(c*x) + b*c*d**2*x**2/6 + b*d**2*x*atanh(c*x) + b*d**2*x + 4*b*d**2*log(x - 1/c)/(3*c) + b*d**2*atanh(c*x)/(3*
c), Ne(c, 0)), (a*d**2*x, True))

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Giac [A]  time = 1.21788, size = 163, normalized size = 2.3 \begin{align*} \frac{1}{3} \, a c^{2} d^{2} x^{3} + \frac{1}{6} \,{\left (6 \, a c d^{2} + b c d^{2}\right )} x^{2} + \frac{b d^{2} \log \left (c x + 1\right )}{6 \, c} + \frac{7 \, b d^{2} \log \left (c x - 1\right )}{6 \, c} +{\left (a d^{2} + b d^{2}\right )} x + \frac{1}{6} \,{\left (b c^{2} d^{2} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b d^{2} x\right )} \log \left (-\frac{c x + 1}{c x - 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*c^2*d^2*x^3 + 1/6*(6*a*c*d^2 + b*c*d^2)*x^2 + 1/6*b*d^2*log(c*x + 1)/c + 7/6*b*d^2*log(c*x - 1)/c + (a*d
^2 + b*d^2)*x + 1/6*(b*c^2*d^2*x^3 + 3*b*c*d^2*x^2 + 3*b*d^2*x)*log(-(c*x + 1)/(c*x - 1))