Integrand size = 16, antiderivative size = 146 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {b^2 x^2}{6 c^2}-\frac {b^2 \text {arctanh}\left (c x^2\right )}{6 c^3}+\frac {b x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{6 c}+\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{6 c^3}+\frac {1}{6} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {b \left (a+b \text {arctanh}\left (c x^2\right )\right ) \log \left (\frac {2}{1-c x^2}\right )}{3 c^3}-\frac {b^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{6 c^3} \]
1/6*b^2*x^2/c^2-1/6*b^2*arctanh(c*x^2)/c^3+1/6*b*x^4*(a+b*arctanh(c*x^2))/ c+1/6*(a+b*arctanh(c*x^2))^2/c^3+1/6*x^6*(a+b*arctanh(c*x^2))^2-1/3*b*(a+b *arctanh(c*x^2))*ln(2/(-c*x^2+1))/c^3-1/6*b^2*polylog(2,1-2/(-c*x^2+1))/c^ 3
Time = 0.23 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.90 \[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\frac {b^2 c x^2+a b c^2 x^4+a^2 c^3 x^6+b^2 \left (-1+c^3 x^6\right ) \text {arctanh}\left (c x^2\right )^2+b \text {arctanh}\left (c x^2\right ) \left (-b+b c^2 x^4+2 a c^3 x^6-2 b \log \left (1+e^{-2 \text {arctanh}\left (c x^2\right )}\right )\right )+a b \log \left (-1+c^2 x^4\right )+b^2 \operatorname {PolyLog}\left (2,-e^{-2 \text {arctanh}\left (c x^2\right )}\right )}{6 c^3} \]
(b^2*c*x^2 + a*b*c^2*x^4 + a^2*c^3*x^6 + b^2*(-1 + c^3*x^6)*ArcTanh[c*x^2] ^2 + b*ArcTanh[c*x^2]*(-b + b*c^2*x^4 + 2*a*c^3*x^6 - 2*b*Log[1 + E^(-2*Ar cTanh[c*x^2])]) + a*b*Log[-1 + c^2*x^4] + b^2*PolyLog[2, -E^(-2*ArcTanh[c* x^2])])/(6*c^3)
Time = 0.92 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6454, 6452, 6542, 6452, 262, 219, 6546, 6470, 2849, 2752}
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.
\(\displaystyle \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 6454 |
\(\displaystyle \frac {1}{2} \int x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2dx^2\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \int \frac {x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{1-c^2 x^4}dx^2\right )\) |
\(\Big \downarrow \) 6542 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{1-c^2 x^4}dx^2}{c^2}-\frac {\int x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )dx^2}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{1-c^2 x^4}dx^2}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \int \frac {x^4}{1-c^2 x^4}dx^2}{c^2}\right )\right )\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{1-c^2 x^4}dx^2}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\int \frac {1}{1-c^2 x^4}dx^2}{c^2}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\int \frac {x^2 \left (a+b \text {arctanh}\left (c x^2\right )\right )}{1-c^2 x^4}dx^2}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^3}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6546 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\int \frac {a+b \text {arctanh}\left (c x^2\right )}{1-c x^2}dx^2}{c}-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^3}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 6470 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^2}\right ) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{c}-b \int \frac {\log \left (\frac {2}{1-c x^2}\right )}{1-c^2 x^4}dx^2}{c}-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^3}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 2849 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {b \int \frac {\log \left (\frac {2}{1-c x^2}\right )}{1-\frac {2}{1-c x^2}}d\frac {1}{1-c x^2}}{c}+\frac {\log \left (\frac {2}{1-c x^2}\right ) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{c}}{c}-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^3}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
\(\Big \downarrow \) 2752 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} x^6 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2-\frac {2}{3} b c \left (\frac {\frac {\frac {\log \left (\frac {2}{1-c x^2}\right ) \left (a+b \text {arctanh}\left (c x^2\right )\right )}{c}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x^2}\right )}{2 c}}{c}-\frac {\left (a+b \text {arctanh}\left (c x^2\right )\right )^2}{2 b c^2}}{c^2}-\frac {\frac {1}{2} x^4 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{2} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^3}-\frac {x^2}{c^2}\right )}{c^2}\right )\right )\) |
((x^6*(a + b*ArcTanh[c*x^2])^2)/3 - (2*b*c*(-(((x^4*(a + b*ArcTanh[c*x^2]) )/2 - (b*c*(-(x^2/c^2) + ArcTanh[c*x^2]/c^3))/2)/c^2) + (-1/2*(a + b*ArcTa nh[c*x^2])^2/(b*c^2) + (((a + b*ArcTanh[c*x^2])*Log[2/(1 - c*x^2)])/c + (b *PolyLog[2, 1 - 2/(1 - c*x^2)])/(2*c))/c)/c^2))/3)/2
3.1.65.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)), x_Symbol ] :> Simp[(-(a + b*ArcTanh[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c *(p/e) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2/(1 + e*(x/d))]/(1 - c^2*x^ 2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2 , 0]
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]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*e*(p + 1)), x] + Simp[1/ (c*d) Int[(a + b*ArcTanh[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(379\) vs. \(2(132)=264\).
Time = 1.35 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.60
method | result | size |
risch | \(\frac {b^{2} x^{4} \ln \left (c \,x^{2}+1\right )}{12 c}+\frac {b^{2} x^{6} \ln \left (c \,x^{2}+1\right )^{2}}{24}-\frac {b^{2} \ln \left (c \,x^{2}+1\right )}{12 c^{3}}+\frac {b^{2} \ln \left (c \,x^{2}+1\right )^{2}}{24 c^{3}}-\frac {b^{2} \operatorname {dilog}\left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{6 c^{3}}-\frac {2 b^{2} \ln \left (c \,x^{2}-1\right )}{9 c^{3}}-\frac {b^{2} x^{4} \ln \left (-c \,x^{2}+1\right )}{12 c}-\frac {a b \,x^{6} \ln \left (-c \,x^{2}+1\right )}{6}+\frac {a b \ln \left (c \,x^{2}-1\right )}{6 c^{3}}-\frac {17 b^{2}}{108 c^{3}}+\frac {b^{2} x^{6} \ln \left (-c \,x^{2}+1\right )^{2}}{24}+\frac {11 b^{2} \ln \left (-c \,x^{2}+1\right )}{36 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right )^{2}}{24 c^{3}}+\frac {a b \,x^{4}}{6 c}+\frac {b^{2} x^{2}}{6 c^{2}}+\frac {a^{2} x^{6}}{6}+\frac {b a \,x^{6} \ln \left (c \,x^{2}+1\right )}{6}+\frac {b a \ln \left (c \,x^{2}+1\right )}{6 c^{3}}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right ) x^{6}}{12}-\frac {b^{2} \ln \left (-c \,x^{2}+1\right ) \ln \left (c \,x^{2}+1\right )}{12 c^{3}}+\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (c \,x^{2}+1\right )}{6 c^{3}}-\frac {b^{2} \ln \left (\frac {1}{2}-\frac {c \,x^{2}}{2}\right ) \ln \left (\frac {c \,x^{2}}{2}+\frac {1}{2}\right )}{6 c^{3}}\) | \(380\) |
default | \(\text {Expression too large to display}\) | \(710\) |
parts | \(\text {Expression too large to display}\) | \(710\) |
1/12*b^2/c*x^4*ln(c*x^2+1)+1/24*b^2*x^6*ln(c*x^2+1)^2-1/12*b^2/c^3*ln(c*x^ 2+1)+1/24/c^3*b^2*ln(c*x^2+1)^2-1/6*b^2/c^3*dilog(1/2*c*x^2+1/2)-2/9*b^2/c ^3*ln(c*x^2-1)-1/12*b^2/c*x^4*ln(-c*x^2+1)-1/6*a*b*x^6*ln(-c*x^2+1)+1/6*a* b/c^3*ln(c*x^2-1)-17/108/c^3*b^2+1/24*b^2*x^6*ln(-c*x^2+1)^2+11/36/c^3*b^2 *ln(-c*x^2+1)-1/24/c^3*b^2*ln(-c*x^2+1)^2+1/6/c*a*b*x^4+1/6*b^2*x^2/c^2+1/ 6*a^2*x^6+1/6*b*a*x^6*ln(c*x^2+1)+1/6*b*a/c^3*ln(c*x^2+1)-1/12*b^2*ln(-c*x ^2+1)*ln(c*x^2+1)*x^6-1/12*b^2/c^3*ln(-c*x^2+1)*ln(c*x^2+1)+1/6*b^2/c^3*ln (1/2-1/2*c*x^2)*ln(c*x^2+1)-1/6*b^2/c^3*ln(1/2-1/2*c*x^2)*ln(1/2*c*x^2+1/2 )
\[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
\[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int x^{5} \left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}\, dx \]
\[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
1/6*a^2*x^6 + 1/6*(2*x^6*arctanh(c*x^2) + (x^4/c^2 + log(c^2*x^4 - 1)/c^4) *c)*a*b + 1/432*(18*x^6*log(-c*x^2 + 1)^2 - 2*c^4*(2*(c^2*x^6 + 3*x^2)/c^6 - 3*log(c*x^2 + 1)/c^7 + 3*log(c*x^2 - 1)/c^7) + 3*c^3*(x^4/c^4 + log(c^2 *x^4 - 1)/c^6) + 1296*c^3*integrate(1/9*x^7*log(c*x^2 + 1)/(c^4*x^4 - c^2) , x) - 9*c^2*(2*x^2/c^4 - log(c*x^2 + 1)/c^5 + log(c*x^2 - 1)/c^5) - 6*c*( (2*c^2*x^6 + 3*c*x^4 + 6*x^2)/c^3 + 6*log(c*x^2 - 1)/c^4)*log(-c*x^2 + 1) + 648*c*integrate(1/9*x^3*log(c*x^2 + 1)/(c^4*x^4 - c^2), x) + 6*(3*c^3*x^ 6*log(c*x^2 + 1)^2 + (2*c^3*x^6 - 3*c^2*x^4 + 6*c*x^2 - 6*(c^3*x^6 + 1)*lo g(c*x^2 + 1))*log(-c*x^2 + 1))/c^3 + (4*c^3*x^6 + 15*c^2*x^4 + 66*c*x^2 + 18*log(c*x^2 - 1)^2 + 66*log(c*x^2 - 1))/c^3 - 18*log(9*c^4*x^4 - 9*c^2)/c ^3 + 648*integrate(1/9*x*log(c*x^2 + 1)/(c^4*x^4 - c^2), x))*b^2
\[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2} x^{5} \,d x } \]
Timed out. \[ \int x^5 \left (a+b \text {arctanh}\left (c x^2\right )\right )^2 \, dx=\int x^5\,{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2 \,d x \]