Integrand size = 12, antiderivative size = 44 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {\left (a+b x^4\right ) \text {arctanh}\left (a+b x^4\right )}{4 b}+\frac {\log \left (1-\left (a+b x^4\right )^2\right )}{8 b} \] Output:
1/4*(b*x^4+a)*arctanh(b*x^4+a)/b+1/8*ln(1-(b*x^4+a)^2)/b
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {2 \left (a+b x^4\right ) \text {arctanh}\left (a+b x^4\right )+\log \left (1-\left (a+b x^4\right )^2\right )}{8 b} \] Input:
Integrate[x^3*ArcTanh[a + b*x^4],x]
Output:
(2*(a + b*x^4)*ArcTanh[a + b*x^4] + Log[1 - (a + b*x^4)^2])/(8*b)
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.82, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {7266, 6653, 6436, 240}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx\) |
\(\Big \downarrow \) 7266 |
\(\displaystyle \frac {1}{4} \int \text {arctanh}\left (b x^4+a\right )dx^4\) |
\(\Big \downarrow \) 6653 |
\(\displaystyle \frac {\int \text {arctanh}\left (b x^4+a\right )d\left (b x^4+a\right )}{4 b}\) |
\(\Big \downarrow \) 6436 |
\(\displaystyle \frac {\left (a+b x^4\right ) \text {arctanh}\left (a+b x^4\right )-\int \frac {b x^4+a}{1-x^8}d\left (b x^4+a\right )}{4 b}\) |
\(\Big \downarrow \) 240 |
\(\displaystyle \frac {\left (a+b x^4\right ) \text {arctanh}\left (a+b x^4\right )+\frac {1}{2} \log \left (1-x^8\right )}{4 b}\) |
Input:
Int[x^3*ArcTanh[a + b*x^4],x]
Output:
((a + b*x^4)*ArcTanh[a + b*x^4] + Log[1 - x^8]/2)/(4*b)
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcTanh[c*x^n])^p, x] - Simp[b*c*n*p Int[x^n*((a + b*ArcTanh[c*x^n]) ^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[1/d Subst[Int[(a + b*ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d }, x] && IGtQ[p, 0]
Int[(u_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, x]
Time = 0.36 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {\left (b \,x^{4}+a \right ) \operatorname {arctanh}\left (b \,x^{4}+a \right )+\frac {\ln \left (1-\left (b \,x^{4}+a \right )^{2}\right )}{2}}{4 b}\) | \(39\) |
default | \(\frac {\left (b \,x^{4}+a \right ) \operatorname {arctanh}\left (b \,x^{4}+a \right )+\frac {\ln \left (1-\left (b \,x^{4}+a \right )^{2}\right )}{2}}{4 b}\) | \(39\) |
parts | \(\frac {x^{4} \operatorname {arctanh}\left (b \,x^{4}+a \right )}{4}-b \left (\frac {\left (-a -1\right ) \ln \left (b \,x^{4}+a +1\right )}{8 b^{2}}+\frac {\left (a -1\right ) \ln \left (b \,x^{4}+a -1\right )}{8 b^{2}}\right )\) | \(55\) |
parallelrisch | \(-\frac {-\operatorname {arctanh}\left (b \,x^{4}+a \right ) x^{4} b^{2}-\operatorname {arctanh}\left (b \,x^{4}+a \right ) a b -\ln \left (b \,x^{4}+a -1\right ) b -\operatorname {arctanh}\left (b \,x^{4}+a \right ) b}{4 b^{2}}\) | \(58\) |
risch | \(\frac {x^{4} \ln \left (b \,x^{4}+a +1\right )}{8}-\frac {x^{4} \ln \left (-b \,x^{4}-a +1\right )}{8}+\frac {\ln \left (b \,x^{4}+a +1\right ) a}{8 b}-\frac {\ln \left (-b \,x^{4}-a +1\right ) a}{8 b}+\frac {\ln \left (b \,x^{4}+a +1\right )}{8 b}+\frac {\ln \left (-b \,x^{4}-a +1\right )}{8 b}\) | \(97\) |
Input:
int(x^3*arctanh(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4/b*((b*x^4+a)*arctanh(b*x^4+a)+1/2*ln(1-(b*x^4+a)^2))
Time = 0.10 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.34 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {b x^{4} \log \left (-\frac {b x^{4} + a + 1}{b x^{4} + a - 1}\right ) + {\left (a + 1\right )} \log \left (b x^{4} + a + 1\right ) - {\left (a - 1\right )} \log \left (b x^{4} + a - 1\right )}{8 \, b} \] Input:
integrate(x^3*arctanh(b*x^4+a),x, algorithm="fricas")
Output:
1/8*(b*x^4*log(-(b*x^4 + a + 1)/(b*x^4 + a - 1)) + (a + 1)*log(b*x^4 + a + 1) - (a - 1)*log(b*x^4 + a - 1))/b
Time = 0.77 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.36 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\begin {cases} \frac {a \operatorname {atanh}{\left (a + b x^{4} \right )}}{4 b} + \frac {x^{4} \operatorname {atanh}{\left (a + b x^{4} \right )}}{4} + \frac {\log {\left (a + b x^{4} + 1 \right )}}{4 b} - \frac {\operatorname {atanh}{\left (a + b x^{4} \right )}}{4 b} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {atanh}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*atanh(b*x**4+a),x)
Output:
Piecewise((a*atanh(a + b*x**4)/(4*b) + x**4*atanh(a + b*x**4)/4 + log(a + b*x**4 + 1)/(4*b) - atanh(a + b*x**4)/(4*b), Ne(b, 0)), (x**4*atanh(a)/4, True))
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.84 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {2 \, {\left (b x^{4} + a\right )} \operatorname {artanh}\left (b x^{4} + a\right ) + \log \left (-{\left (b x^{4} + a\right )}^{2} + 1\right )}{8 \, b} \] Input:
integrate(x^3*arctanh(b*x^4+a),x, algorithm="maxima")
Output:
1/8*(2*(b*x^4 + a)*arctanh(b*x^4 + a) + log(-(b*x^4 + a)^2 + 1))/b
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (40) = 80\).
Time = 0.12 (sec) , antiderivative size = 223, normalized size of antiderivative = 5.07 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {1}{8} \, {\left ({\left (a + 1\right )} b - {\left (a - 1\right )} b\right )} {\left (\frac {\log \left (\frac {{\left | -b x^{4} - a - 1 \right |}}{{\left | b x^{4} + a - 1 \right |}}\right )}{b^{2}} - \frac {\log \left ({\left | -\frac {b x^{4} + a + 1}{b x^{4} + a - 1} + 1 \right |}\right )}{b^{2}} + \frac {\log \left (-\frac {a - \frac {{\left (\frac {{\left (b x^{4} + a + 1\right )} {\left (a - 1\right )}}{b x^{4} + a - 1} - a - 1\right )} b}{\frac {{\left (b x^{4} + a + 1\right )} b}{b x^{4} + a - 1} - b} + 1}{a - \frac {{\left (\frac {{\left (b x^{4} + a + 1\right )} {\left (a - 1\right )}}{b x^{4} + a - 1} - a - 1\right )} b}{\frac {{\left (b x^{4} + a + 1\right )} b}{b x^{4} + a - 1} - b} - 1}\right )}{b^{2} {\left (\frac {b x^{4} + a + 1}{b x^{4} + a - 1} - 1\right )}}\right )} \] Input:
integrate(x^3*arctanh(b*x^4+a),x, algorithm="giac")
Output:
1/8*((a + 1)*b - (a - 1)*b)*(log(abs(-b*x^4 - a - 1)/abs(b*x^4 + a - 1))/b ^2 - log(abs(-(b*x^4 + a + 1)/(b*x^4 + a - 1) + 1))/b^2 + log(-(a - ((b*x^ 4 + a + 1)*(a - 1)/(b*x^4 + a - 1) - a - 1)*b/((b*x^4 + a + 1)*b/(b*x^4 + a - 1) - b) + 1)/(a - ((b*x^4 + a + 1)*(a - 1)/(b*x^4 + a - 1) - a - 1)*b/ ((b*x^4 + a + 1)*b/(b*x^4 + a - 1) - b) - 1))/(b^2*((b*x^4 + a + 1)/(b*x^4 + a - 1) - 1)))
Time = 3.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.05 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {\ln \left (b\,x^4+a-1\right )}{8\,b}-\frac {x^4\,\ln \left (-b\,x^4-a+1\right )}{8}+\frac {\ln \left (b\,x^4+a+1\right )}{8\,b}+\frac {x^4\,\ln \left (b\,x^4+a+1\right )}{8}-\frac {a\,\ln \left (b\,x^4+a-1\right )}{8\,b}+\frac {a\,\ln \left (b\,x^4+a+1\right )}{8\,b} \] Input:
int(x^3*atanh(a + b*x^4),x)
Output:
log(a + b*x^4 - 1)/(8*b) - (x^4*log(1 - b*x^4 - a))/8 + log(a + b*x^4 + 1) /(8*b) + (x^4*log(a + b*x^4 + 1))/8 - (a*log(a + b*x^4 - 1))/(8*b) + (a*lo g(a + b*x^4 + 1))/(8*b)
Time = 0.22 (sec) , antiderivative size = 377, normalized size of antiderivative = 8.57 \[ \int x^3 \text {arctanh}\left (a+b x^4\right ) \, dx=\frac {2 \mathit {atanh} \left (b \,x^{4}+a \right ) a +2 \mathit {atanh} \left (b \,x^{4}+a \right ) b \,x^{4}+\mathrm {log}\left (\frac {\sqrt {a -1}\, \left (a^{2}-1\right )^{\frac {1}{4}}-b^{\frac {1}{4}} \sqrt {2 \left (a^{2}-1\right )^{\frac {1}{4}} a -2 \left (a^{2}-1\right )^{\frac {1}{4}}-\sqrt {-a +1}\, \sqrt {a^{2}-1}+\sqrt {-a +1}\, a -\sqrt {-a +1}}\, x +\sqrt {b}\, \sqrt {a -1}\, x^{2}}{\sqrt {a -1}}\right )+\mathrm {log}\left (\frac {\sqrt {a -1}\, \left (a^{2}-1\right )^{\frac {1}{4}}-b^{\frac {1}{4}} \sqrt {2 \left (a^{2}-1\right )^{\frac {1}{4}} a -2 \left (a^{2}-1\right )^{\frac {1}{4}}+\sqrt {-a +1}\, \sqrt {a^{2}-1}-\sqrt {-a +1}\, a +\sqrt {-a +1}}\, x +\sqrt {b}\, \sqrt {a -1}\, x^{2}}{\sqrt {a -1}}\right )+\mathrm {log}\left (\frac {\sqrt {a -1}\, \left (a^{2}-1\right )^{\frac {1}{4}}+b^{\frac {1}{4}} \sqrt {2 \left (a^{2}-1\right )^{\frac {1}{4}} a -2 \left (a^{2}-1\right )^{\frac {1}{4}}-\sqrt {-a +1}\, \sqrt {a^{2}-1}+\sqrt {-a +1}\, a -\sqrt {-a +1}}\, x +\sqrt {b}\, \sqrt {a -1}\, x^{2}}{\sqrt {a -1}}\right )+\mathrm {log}\left (\frac {\sqrt {a -1}\, \left (a^{2}-1\right )^{\frac {1}{4}}+b^{\frac {1}{4}} \sqrt {2 \left (a^{2}-1\right )^{\frac {1}{4}} a -2 \left (a^{2}-1\right )^{\frac {1}{4}}+\sqrt {-a +1}\, \sqrt {a^{2}-1}-\sqrt {-a +1}\, a +\sqrt {-a +1}}\, x +\sqrt {b}\, \sqrt {a -1}\, x^{2}}{\sqrt {a -1}}\right )}{8 b} \] Input:
int(x^3*atanh(b*x^4+a),x)
Output:
(2*atanh(a + b*x**4)*a + 2*atanh(a + b*x**4)*b*x**4 + log((sqrt(a - 1)*(a* *2 - 1)**(1/4) - b**(1/4)*sqrt(2*(a**2 - 1)**(1/4)*a - 2*(a**2 - 1)**(1/4) - sqrt( - a + 1)*sqrt(a**2 - 1) + sqrt( - a + 1)*a - sqrt( - a + 1))*x + sqrt(b)*sqrt(a - 1)*x**2)/sqrt(a - 1)) + log((sqrt(a - 1)*(a**2 - 1)**(1/4 ) - b**(1/4)*sqrt(2*(a**2 - 1)**(1/4)*a - 2*(a**2 - 1)**(1/4) + sqrt( - a + 1)*sqrt(a**2 - 1) - sqrt( - a + 1)*a + sqrt( - a + 1))*x + sqrt(b)*sqrt( a - 1)*x**2)/sqrt(a - 1)) + log((sqrt(a - 1)*(a**2 - 1)**(1/4) + b**(1/4)* sqrt(2*(a**2 - 1)**(1/4)*a - 2*(a**2 - 1)**(1/4) - sqrt( - a + 1)*sqrt(a** 2 - 1) + sqrt( - a + 1)*a - sqrt( - a + 1))*x + sqrt(b)*sqrt(a - 1)*x**2)/ sqrt(a - 1)) + log((sqrt(a - 1)*(a**2 - 1)**(1/4) + b**(1/4)*sqrt(2*(a**2 - 1)**(1/4)*a - 2*(a**2 - 1)**(1/4) + sqrt( - a + 1)*sqrt(a**2 - 1) - sqrt ( - a + 1)*a + sqrt( - a + 1))*x + sqrt(b)*sqrt(a - 1)*x**2)/sqrt(a - 1))) /(8*b)