\(\int (a+b x)^2 \coth ^{-1}(a+b x) \, dx\) [97]

Optimal result

Integrand size = 14, antiderivative size = 54 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b} \]

[Out]

1/6*(b*x+a)^2/b+1/3*(b*x+a)^3*arccoth(b*x+a)/b+1/6*ln(1-(b*x+a)^2)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6243, 6038, 272, 45} \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2}{6 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b} \]

[In]

Int[(a + b*x)^2*ArcCoth[a + b*x],x]

[Out]

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

Rule 45

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])

Rule 272

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

Rule 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rule 6243

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[(f*(x/d))^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f,
0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^2 \coth ^{-1}(x) \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \frac {x^3}{1-x^2} \, dx,x,a+b x\right )}{3 b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \frac {x}{1-x} \, dx,x,(a+b x)^2\right )}{6 b} \\ & = \frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}-\frac {\text {Subst}\left (\int \left (-1+\frac {1}{1-x}\right ) \, dx,x,(a+b x)^2\right )}{6 b} \\ & = \frac {(a+b x)^2}{6 b}+\frac {(a+b x)^3 \coth ^{-1}(a+b x)}{3 b}+\frac {\log \left (1-(a+b x)^2\right )}{6 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.78 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {(a+b x)^2+2 (a+b x)^3 \coth ^{-1}(a+b x)+\log \left (1-(a+b x)^2\right )}{6 b} \]

[In]

Integrate[(a + b*x)^2*ArcCoth[a + b*x],x]

[Out]

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

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.89

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\left (b x +a \right )^{2}}{6}+\frac {\ln \left (b x +a -1\right )}{6}+\frac {\ln \left (b x +a +1\right )}{6}}{b}\)	\(48\)
default	\(\frac {\frac {\operatorname {arccoth}\left (b x +a \right ) \left (b x +a \right )^{3}}{3}+\frac {\left (b x +a \right )^{2}}{6}+\frac {\ln \left (b x +a -1\right )}{6}+\frac {\ln \left (b x +a +1\right )}{6}}{b}\)	\(48\)
parts	\(\frac {\operatorname {arccoth}\left (b x +a \right ) b^{2} x^{3}}{3}+\operatorname {arccoth}\left (b x +a \right ) b a \,x^{2}+\operatorname {arccoth}\left (b x +a \right ) a^{2} x +\frac {\operatorname {arccoth}\left (b x +a \right ) a^{3}}{3 b}+\frac {b \,x^{2}}{6}+\frac {a x}{3}+\frac {\ln \left (b x +a -1\right )}{6 b}+\frac {\ln \left (b x +a +1\right )}{6 b}\)	\(87\)
parallelrisch	\(-\frac {-2 b^{4} \operatorname {arccoth}\left (b x +a \right ) x^{3}-6 x^{2} \operatorname {arccoth}\left (b x +a \right ) a \,b^{3}-6 x \,\operatorname {arccoth}\left (b x +a \right ) a^{2} b^{2}-b^{3} x^{2}-2 \,\operatorname {arccoth}\left (b x +a \right ) a^{3} b -2 a \,b^{2} x +5 b \,a^{2}-2 b \ln \left (b x +a -1\right )-2 \,\operatorname {arccoth}\left (b x +a \right ) b -b}{6 b^{2}}\)	\(106\)
risch	\(\frac {\left (b x +a \right )^{3} \ln \left (b x +a +1\right )}{6 b}-\frac {b^{2} \ln \left (b x +a -1\right ) x^{3}}{6}-\frac {b \ln \left (b x +a -1\right ) x^{2} a}{2}-\frac {a^{2} x \ln \left (b x +a -1\right )}{2}-\frac {\ln \left (b x +a -1\right ) a^{3}}{6 b}+\frac {b \,x^{2}}{6}+\frac {a x}{3}+\frac {\ln \left (b^{2} x^{2}+2 a b x +a^{2}-1\right )}{6 b}\)	\(111\)

[In]

int((b*x+a)^2*arccoth(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b*(1/3*arccoth(b*x+a)*(b*x+a)^3+1/6*(b*x+a)^2+1/6*ln(b*x+a-1)+1/6*ln(b*x+a+1))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.59 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} + 2 \, a b x + {\left (a^{3} + 1\right )} \log \left (b x + a + 1\right ) - {\left (a^{3} - 1\right )} \log \left (b x + a - 1\right ) + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x\right )} \log \left (\frac {b x + a + 1}{b x + a - 1}\right )}{6 \, b} \]

[In]

integrate((b*x+a)^2*arccoth(b*x+a),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (39) = 78\).

Time = 0.31 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.80 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\begin {cases} \frac {a^{3} \operatorname {acoth}{\left (a + b x \right )}}{3 b} + a^{2} x \operatorname {acoth}{\left (a + b x \right )} + a b x^{2} \operatorname {acoth}{\left (a + b x \right )} + \frac {a x}{3} + \frac {b^{2} x^{3} \operatorname {acoth}{\left (a + b x \right )}}{3} + \frac {b x^{2}}{6} + \frac {\log {\left (\frac {a}{b} + x + \frac {1}{b} \right )}}{3 b} - \frac {\operatorname {acoth}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\a^{2} x \operatorname {acoth}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((b*x+a)**2*acoth(b*x+a),x)

[Out]

Piecewise((a**3*acoth(a + b*x)/(3*b) + a**2*x*acoth(a + b*x) + a*b*x**2*acoth(a + b*x) + a*x/3 + b**2*x**3*aco
th(a + b*x)/3 + b*x**2/6 + log(a/b + x + 1/b)/(3*b) - acoth(a + b*x)/(3*b), Ne(b, 0)), (a**2*x*acoth(a), True)
)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.50 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{6} \, b {\left (\frac {b x^{2} + 2 \, a x}{b} + \frac {{\left (a^{3} + 1\right )} \log \left (b x + a + 1\right )}{b^{2}} - \frac {{\left (a^{3} - 1\right )} \log \left (b x + a - 1\right )}{b^{2}}\right )} + \frac {1}{3} \, {\left (b^{2} x^{3} + 3 \, a b x^{2} + 3 \, a^{2} x\right )} \operatorname {arcoth}\left (b x + a\right ) \]

[In]

integrate((b*x+a)^2*arccoth(b*x+a),x, algorithm="maxima")

[Out]

1/6*b*((b*x^2 + 2*a*x)/b + (a^3 + 1)*log(b*x + a + 1)/b^2 - (a^3 - 1)*log(b*x + a - 1)/b^2) + 1/3*(b^2*x^3 + 3
*a*b*x^2 + 3*a^2*x)*arccoth(b*x + a)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (48) = 96\).

Time = 0.28 (sec) , antiderivative size = 255, normalized size of antiderivative = 4.72 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {1}{6} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | b x + a + 1 \right |}}{{\left | b x + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | \frac {b x + a + 1}{b x + a - 1} - 1 \right |}\right )}{b^{2}} + \frac {{\left (\frac {3 \, {\left (b x + a + 1\right )}^{2}}{{\left (b x + a - 1\right )}^{2}} + 1\right )} \log \left (-\frac {\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} + 1}{\frac {1}{a - \frac {{\left (\frac {{\left (b x + a + 1\right )} {\left (a - 1\right )}}{b x + a - 1} - a - 1\right )} b}{\frac {{\left (b x + a + 1\right )} b}{b x + a - 1} - b}} - 1}\right )}{b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{3}} + \frac {2 \, {\left (b x + a + 1\right )}}{{\left (b x + a - 1\right )} b^{2} {\left (\frac {b x + a + 1}{b x + a - 1} - 1\right )}^{2}}\right )} \]

[In]

integrate((b*x+a)^2*arccoth(b*x+a),x, algorithm="giac")

[Out]

1/6*((a + 1)*b - (a - 1)*b)*(log(abs(b*x + a + 1)/abs(b*x + a - 1))/b^2 - log(abs((b*x + a + 1)/(b*x + a - 1)
- 1))/b^2 + (3*(b*x + a + 1)^2/(b*x + a - 1)^2 + 1)*log(-(1/(a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)
*b/((b*x + a + 1)*b/(b*x + a - 1) - b)) + 1)/(1/(a - ((b*x + a + 1)*(a - 1)/(b*x + a - 1) - a - 1)*b/((b*x + a
+ 1)*b/(b*x + a - 1) - b)) - 1))/(b^2*((b*x + a + 1)/(b*x + a - 1) - 1)^3) + 2*(b*x + a + 1)/((b*x + a - 1)*b
^2*((b*x + a + 1)/(b*x + a - 1) - 1)^2))

Mupad [B] (verification not implemented)

Time = 4.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.11 \[ \int (a+b x)^2 \coth ^{-1}(a+b x) \, dx=\frac {a\,x}{3}+\ln \left (\frac {1}{a+b\,x}+1\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )+\frac {b\,x^2}{6}-\ln \left (1-\frac {1}{a+b\,x}\right )\,\left (\frac {a^2\,x}{2}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^3}{6}\right )-\frac {\ln \left (a+b\,x-1\right )\,\left (a^3-1\right )}{6\,b}+\frac {\ln \left (a+b\,x+1\right )\,\left (a^3+1\right )}{6\,b} \]

[In]

int(acoth(a + b*x)*(a + b*x)^2,x)

[Out]

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