Integrand size = 16, antiderivative size = 91 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {a b x^3}{3 c}+\frac {b^2 x^3 \text {arctanh}\left (c x^3\right )}{3 c}-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 c^2}+\frac {1}{6} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2+\frac {b^2 \log \left (1-c^2 x^6\right )}{6 c^2} \] Output:
1/3*a*b*x^3/c+1/3*b^2*x^3*arctanh(c*x^3)/c-1/6*(a+b*arctanh(c*x^3))^2/c^2+ 1/6*x^6*(a+b*arctanh(c*x^3))^2+1/6*b^2*ln(-c^2*x^6+1)/c^2
Time = 0.04 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.16 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {2 a b c x^3+a^2 c^2 x^6+2 b c x^3 \left (b+a c x^3\right ) \text {arctanh}\left (c x^3\right )+b^2 \left (-1+c^2 x^6\right ) \text {arctanh}\left (c x^3\right )^2+b (a+b) \log \left (1-c x^3\right )-a b \log \left (1+c x^3\right )+b^2 \log \left (1+c x^3\right )}{6 c^2} \] Input:
Integrate[x^5*(a + b*ArcTanh[c*x^3])^2,x]
Output:
(2*a*b*c*x^3 + a^2*c^2*x^6 + 2*b*c*x^3*(b + a*c*x^3)*ArcTanh[c*x^3] + b^2* (-1 + c^2*x^6)*ArcTanh[c*x^3]^2 + b*(a + b)*Log[1 - c*x^3] - a*b*Log[1 + c *x^3] + b^2*Log[1 + c*x^3])/(6*c^2)
Time = 0.52 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.01, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6454, 6452, 6542, 2009, 6510}
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.
\(\displaystyle \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle \frac {1}{3} \int x^3 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2dx^3\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \int \frac {x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )}{1-c^2 x^6}dx^3\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {\int \left (a+b \text {arctanh}\left (c x^3\right )\right )dx^3}{c^2}\right )\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3}{c^2}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {1}{3} \left (\frac {1}{2} x^6 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-b c \left (\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b c^3}-\frac {a x^3+b x^3 \text {arctanh}\left (c x^3\right )+\frac {b \log \left (1-c^2 x^6\right )}{2 c}}{c^2}\right )\right )\) |
Input:
Int[x^5*(a + b*ArcTanh[c*x^3])^2,x]
Output:
((x^6*(a + b*ArcTanh[c*x^3])^2)/2 - b*c*((a + b*ArcTanh[c*x^3])^2/(2*b*c^3 ) - (a*x^3 + b*x^3*ArcTanh[c*x^3] + (b*Log[1 - c^2*x^6])/(2*c))/c^2))/3
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[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]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*ArcTanh[c*x])^p, x ], x, x^n], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 1] && IntegerQ[Simpl ify[(m + 1)/n]]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcTanh[c* x])^p, x], x] - Simp[d*(f^2/e) Int[(f*x)^(m - 2)*((a + b*ArcTanh[c*x])^p/ (d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
Time = 1.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(\frac {b^{2} \operatorname {arctanh}\left (c \,x^{3}\right )^{2} x^{6} c^{2}+2 x^{6} \operatorname {arctanh}\left (c \,x^{3}\right ) a b \,c^{2}+a^{2} c^{2} x^{6}+2 b^{2} \operatorname {arctanh}\left (c \,x^{3}\right ) x^{3} c +2 a b c \,x^{3}-b^{2} \operatorname {arctanh}\left (c \,x^{3}\right )^{2}+2 \ln \left (c \,x^{3}-1\right ) b^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) a b +2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) b^{2}}{6 c^{2}}\) | \(121\) |
risch | \(\frac {b^{2} \left (c^{2} x^{6}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{24 c^{2}}+\frac {b \left (-2 b \,x^{6} \ln \left (-c \,x^{3}+1\right ) a \,c^{2}+4 a^{2} c^{2} x^{6}+4 a b c \,x^{3}+2 b \ln \left (-c \,x^{3}+1\right ) a +b^{2}\right ) \ln \left (c \,x^{3}+1\right )}{24 a \,c^{2}}+\frac {b^{2} x^{6} \ln \left (-c \,x^{3}+1\right )^{2}}{24}-\frac {a b \,x^{6} \ln \left (-c \,x^{3}+1\right )}{6}+\frac {x^{6} a^{2}}{6}-\frac {b^{2} x^{3} \ln \left (-c \,x^{3}+1\right )}{6 c}+\frac {a b \,x^{3}}{3 c}-\frac {b^{2} \ln \left (-c \,x^{3}+1\right )^{2}}{24 c^{2}}+\frac {a \ln \left (-c \,x^{3}+1\right ) b}{6 c^{2}}+\frac {b^{2} \ln \left (-c \,x^{3}+1\right )}{6 c^{2}}-\frac {a \ln \left (-c \,x^{3}-1\right ) b}{6 c^{2}}+\frac {\ln \left (-c \,x^{3}-1\right ) b^{2}}{6 c^{2}}-\frac {\ln \left (-c \,x^{3}-1\right ) b^{3}}{24 a \,c^{2}}+\frac {b^{2}}{6 c^{2}}\) | \(287\) |
default | \(\text {Expression too large to display}\) | \(2963\) |
parts | \(\text {Expression too large to display}\) | \(2963\) |
Input:
int(x^5*(a+b*arctanh(c*x^3))^2,x,method=_RETURNVERBOSE)
Output:
1/6*(b^2*arctanh(c*x^3)^2*x^6*c^2+2*x^6*arctanh(c*x^3)*a*b*c^2+a^2*c^2*x^6 +2*b^2*arctanh(c*x^3)*x^3*c+2*a*b*c*x^3-b^2*arctanh(c*x^3)^2+2*ln(c*x^3-1) *b^2-2*arctanh(c*x^3)*a*b+2*arctanh(c*x^3)*b^2)/c^2
Time = 0.09 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.52 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {4 \, a^{2} c^{2} x^{6} + 8 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, {\left (a b - b^{2}\right )} \log \left (c x^{3} + 1\right ) + 4 \, {\left (a b + b^{2}\right )} \log \left (c x^{3} - 1\right ) + 4 \, {\left (a b c^{2} x^{6} + b^{2} c x^{3}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, c^{2}} \] Input:
integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="fricas")
Output:
1/24*(4*a^2*c^2*x^6 + 8*a*b*c*x^3 + (b^2*c^2*x^6 - b^2)*log(-(c*x^3 + 1)/( c*x^3 - 1))^2 - 4*(a*b - b^2)*log(c*x^3 + 1) + 4*(a*b + b^2)*log(c*x^3 - 1 ) + 4*(a*b*c^2*x^6 + b^2*c*x^3)*log(-(c*x^3 + 1)/(c*x^3 - 1)))/c^2
Timed out. \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate(x**5*(a+b*atanh(c*x**3))**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (81) = 162\).
Time = 0.04 (sec) , antiderivative size = 186, normalized size of antiderivative = 2.04 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {1}{6} \, b^{2} x^{6} \operatorname {artanh}\left (c x^{3}\right )^{2} + \frac {1}{6} \, a^{2} x^{6} + \frac {1}{6} \, {\left (2 \, x^{6} \operatorname {artanh}\left (c x^{3}\right ) + c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )}\right )} a b + \frac {1}{24} \, {\left (4 \, c {\left (\frac {2 \, x^{3}}{c^{2}} - \frac {\log \left (c x^{3} + 1\right )}{c^{3}} + \frac {\log \left (c x^{3} - 1\right )}{c^{3}}\right )} \operatorname {artanh}\left (c x^{3}\right ) - \frac {2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right )}{c^{2}}\right )} b^{2} \] Input:
integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="maxima")
Output:
1/6*b^2*x^6*arctanh(c*x^3)^2 + 1/6*a^2*x^6 + 1/6*(2*x^6*arctanh(c*x^3) + c *(2*x^3/c^2 - log(c*x^3 + 1)/c^3 + log(c*x^3 - 1)/c^3))*a*b + 1/24*(4*c*(2 *x^3/c^2 - log(c*x^3 + 1)/c^3 + log(c*x^3 - 1)/c^3)*arctanh(c*x^3) - (2*(l og(c*x^3 - 1) - 2)*log(c*x^3 + 1) - log(c*x^3 + 1)^2 - log(c*x^3 - 1)^2 - 4*log(c*x^3 - 1))/c^2)*b^2
Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (81) = 162\).
Time = 0.16 (sec) , antiderivative size = 361, normalized size of antiderivative = 3.97 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {1}{6} \, {\left (\frac {{\left (c x^{3} + 1\right )} b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2}}{{\left (c x^{3} - 1\right )} {\left (\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}\right )}} + \frac {2 \, {\left (\frac {2 \, {\left (c x^{3} + 1\right )} a b}{c x^{3} - 1} + \frac {{\left (c x^{3} + 1\right )} b^{2}}{c x^{3} - 1} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}} + \frac {4 \, {\left (\frac {{\left (c x^{3} + 1\right )} a^{2}}{c x^{3} - 1} + \frac {{\left (c x^{3} + 1\right )} a b}{c x^{3} - 1} - a b\right )}}{\frac {{\left (c x^{3} + 1\right )}^{2} c^{3}}{{\left (c x^{3} - 1\right )}^{2}} - \frac {2 \, {\left (c x^{3} + 1\right )} c^{3}}{c x^{3} - 1} + c^{3}} - \frac {2 \, b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1} + 1\right )}{c^{3}} + \frac {2 \, b^{2} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{c^{3}}\right )} c \] Input:
integrate(x^5*(a+b*arctanh(c*x^3))^2,x, algorithm="giac")
Output:
1/6*((c*x^3 + 1)*b^2*log(-(c*x^3 + 1)/(c*x^3 - 1))^2/((c*x^3 - 1)*((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3 + 1)*c^3/(c*x^3 - 1) + c^3)) + 2*(2*(c *x^3 + 1)*a*b/(c*x^3 - 1) + (c*x^3 + 1)*b^2/(c*x^3 - 1) - b^2)*log(-(c*x^3 + 1)/(c*x^3 - 1))/((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3 + 1)*c^3/(c *x^3 - 1) + c^3) + 4*((c*x^3 + 1)*a^2/(c*x^3 - 1) + (c*x^3 + 1)*a*b/(c*x^3 - 1) - a*b)/((c*x^3 + 1)^2*c^3/(c*x^3 - 1)^2 - 2*(c*x^3 + 1)*c^3/(c*x^3 - 1) + c^3) - 2*b^2*log(-(c*x^3 + 1)/(c*x^3 - 1) + 1)/c^3 + 2*b^2*log(-(c*x ^3 + 1)/(c*x^3 - 1))/c^3)*c
Time = 4.17 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.02 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {a^2\,x^6}{6}+\frac {b^2\,\ln \left (c\,x^3-1\right )}{6\,c^2}+\frac {b^2\,\ln \left (c\,x^3+1\right )}{6\,c^2}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{24\,c^2}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{24\,c^2}+\frac {b^2\,x^6\,{\ln \left (c\,x^3+1\right )}^2}{24}+\frac {b^2\,x^6\,{\ln \left (1-c\,x^3\right )}^2}{24}+\frac {b^2\,x^3\,\ln \left (c\,x^3+1\right )}{6\,c}-\frac {b^2\,x^3\,\ln \left (1-c\,x^3\right )}{6\,c}+\frac {a\,b\,\ln \left (c\,x^3-1\right )}{6\,c^2}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{6\,c^2}+\frac {a\,b\,x^6\,\ln \left (c\,x^3+1\right )}{6}-\frac {a\,b\,x^6\,\ln \left (1-c\,x^3\right )}{6}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12\,c^2}+\frac {a\,b\,x^3}{3\,c}-\frac {b^2\,x^6\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12} \] Input:
int(x^5*(a + b*atanh(c*x^3))^2,x)
Output:
(a^2*x^6)/6 + (b^2*log(c*x^3 - 1))/(6*c^2) + (b^2*log(c*x^3 + 1))/(6*c^2) - (b^2*log(c*x^3 + 1)^2)/(24*c^2) - (b^2*log(1 - c*x^3)^2)/(24*c^2) + (b^2 *x^6*log(c*x^3 + 1)^2)/24 + (b^2*x^6*log(1 - c*x^3)^2)/24 + (b^2*x^3*log(c *x^3 + 1))/(6*c) - (b^2*x^3*log(1 - c*x^3))/(6*c) + (a*b*log(c*x^3 - 1))/( 6*c^2) - (a*b*log(c*x^3 + 1))/(6*c^2) + (a*b*x^6*log(c*x^3 + 1))/6 - (a*b* x^6*log(1 - c*x^3))/6 + (b^2*log(c*x^3 + 1)*log(1 - c*x^3))/(12*c^2) + (a* b*x^3)/(3*c) - (b^2*x^6*log(c*x^3 + 1)*log(1 - c*x^3))/12
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.57 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2 \, dx=\frac {\mathit {atanh} \left (c \,x^{3}\right )^{2} b^{2} c^{2} x^{6}-\mathit {atanh} \left (c \,x^{3}\right )^{2} b^{2}+2 \mathit {atanh} \left (c \,x^{3}\right ) a b \,c^{2} x^{6}-2 \mathit {atanh} \left (c \,x^{3}\right ) a b +2 \mathit {atanh} \left (c \,x^{3}\right ) b^{2} c \,x^{3}-2 \mathit {atanh} \left (c \,x^{3}\right ) b^{2}+2 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b^{2}+2 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b^{2}+a^{2} c^{2} x^{6}+2 a b c \,x^{3}}{6 c^{2}} \] Input:
int(x^5*(a+b*atanh(c*x^3))^2,x)
Output:
(atanh(c*x**3)**2*b**2*c**2*x**6 - atanh(c*x**3)**2*b**2 + 2*atanh(c*x**3) *a*b*c**2*x**6 - 2*atanh(c*x**3)*a*b + 2*atanh(c*x**3)*b**2*c*x**3 - 2*ata nh(c*x**3)*b**2 + 2*log(c**(2/3)*x**2 - c**(1/3)*x + 1)*b**2 + 2*log(c**(2 /3)*x + c**(1/3))*b**2 + a**2*c**2*x**6 + 2*a*b*c*x**3)/(6*c**2)