Integrand size = 14, antiderivative size = 54 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {b x^2}{8 c^3}+\frac {b x^6}{24 c}-\frac {b \text {arctanh}\left (c x^2\right )}{8 c^4}+\frac {1}{8} x^8 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \] Output:
1/8*b*x^2/c^3+1/24*b*x^6/c-1/8*b*arctanh(c*x^2)/c^4+1/8*x^8*(a+b*arctanh(c *x^2))
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {b x^2}{8 c^3}+\frac {b x^6}{24 c}+\frac {a x^8}{8}+\frac {1}{8} b x^8 \text {arctanh}\left (c x^2\right )+\frac {b \log \left (1-c x^2\right )}{16 c^4}-\frac {b \log \left (1+c x^2\right )}{16 c^4} \] Input:
Integrate[x^7*(a + b*ArcTanh[c*x^2]),x]
Output:
(b*x^2)/(8*c^3) + (b*x^6)/(24*c) + (a*x^8)/8 + (b*x^8*ArcTanh[c*x^2])/8 + (b*Log[1 - c*x^2])/(16*c^4) - (b*Log[1 + c*x^2])/(16*c^4)
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6452, 807, 254, 2009}
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.
\(\displaystyle \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {1}{8} x^8 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{4} b c \int \frac {x^9}{1-c^2 x^4}dx\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{8} x^8 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{8} b c \int \frac {x^8}{1-c^2 x^4}dx^2\) |
\(\Big \downarrow \) 254 |
\(\displaystyle \frac {1}{8} x^8 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{8} b c \int \left (-\frac {x^4}{c^2}+\frac {1}{c^4 \left (1-c^2 x^4\right )}-\frac {1}{c^4}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{8} x^8 \left (a+b \text {arctanh}\left (c x^2\right )\right )-\frac {1}{8} b c \left (\frac {\text {arctanh}\left (c x^2\right )}{c^5}-\frac {x^2}{c^4}-\frac {x^6}{3 c^2}\right )\) |
Input:
Int[x^7*(a + b*ArcTanh[c*x^2]),x]
Output:
(x^8*(a + b*ArcTanh[c*x^2]))/8 - (b*c*(-(x^2/c^4) - x^6/(3*c^2) + ArcTanh[ c*x^2]/c^5))/8
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
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]
Time = 0.30 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(-\frac {-3 b \,\operatorname {arctanh}\left (c \,x^{2}\right ) x^{8} c^{4}-3 a \,c^{4} x^{8}-b \,c^{3} x^{6}-3 b c \,x^{2}+3 b \,\operatorname {arctanh}\left (c \,x^{2}\right )}{24 c^{4}}\) | \(56\) |
default | \(\frac {a \,x^{8}}{8}+\frac {b \,x^{8} \operatorname {arctanh}\left (c \,x^{2}\right )}{8}+\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}-\frac {b \ln \left (c \,x^{2}+1\right )}{16 c^{4}}+\frac {b \ln \left (c \,x^{2}-1\right )}{16 c^{4}}\) | \(66\) |
parts | \(\frac {a \,x^{8}}{8}+\frac {b \,x^{8} \operatorname {arctanh}\left (c \,x^{2}\right )}{8}+\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}-\frac {b \ln \left (c \,x^{2}+1\right )}{16 c^{4}}+\frac {b \ln \left (c \,x^{2}-1\right )}{16 c^{4}}\) | \(66\) |
risch | \(\frac {x^{8} b \ln \left (c \,x^{2}+1\right )}{16}-\frac {x^{8} b \ln \left (-c \,x^{2}+1\right )}{16}+\frac {a \,x^{8}}{8}+\frac {b \,x^{6}}{24 c}+\frac {b \,x^{2}}{8 c^{3}}+\frac {b \ln \left (c \,x^{2}-1\right )}{16 c^{4}}-\frac {b \ln \left (c \,x^{2}+1\right )}{16 c^{4}}\) | \(83\) |
orering | \(\frac {\left (13 c^{4} x^{8}+14 c^{2} x^{4}-27\right ) \left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )}{48 c^{4}}-\frac {\left (c^{2} x^{4}+3\right ) \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (7 x^{6} \left (a +b \,\operatorname {arctanh}\left (c \,x^{2}\right )\right )+\frac {2 x^{8} b c}{-c^{2} x^{4}+1}\right )}{48 x^{6} c^{4}}\) | \(101\) |
Input:
int(x^7*(a+b*arctanh(c*x^2)),x,method=_RETURNVERBOSE)
Output:
-1/24*(-3*b*arctanh(c*x^2)*x^8*c^4-3*a*c^4*x^8-b*c^3*x^6-3*b*c*x^2+3*b*arc tanh(c*x^2))/c^4
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.19 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {6 \, a c^{4} x^{8} + 2 \, b c^{3} x^{6} + 6 \, b c x^{2} + 3 \, {\left (b c^{4} x^{8} - b\right )} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{48 \, c^{4}} \] Input:
integrate(x^7*(a+b*arctanh(c*x^2)),x, algorithm="fricas")
Output:
1/48*(6*a*c^4*x^8 + 2*b*c^3*x^6 + 6*b*c*x^2 + 3*(b*c^4*x^8 - b)*log(-(c*x^ 2 + 1)/(c*x^2 - 1)))/c^4
Time = 6.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.07 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\begin {cases} \frac {a x^{8}}{8} + \frac {b x^{8} \operatorname {atanh}{\left (c x^{2} \right )}}{8} + \frac {b x^{6}}{24 c} + \frac {b x^{2}}{8 c^{3}} - \frac {b \operatorname {atanh}{\left (c x^{2} \right )}}{8 c^{4}} & \text {for}\: c \neq 0 \\\frac {a x^{8}}{8} & \text {otherwise} \end {cases} \] Input:
integrate(x**7*(a+b*atanh(c*x**2)),x)
Output:
Piecewise((a*x**8/8 + b*x**8*atanh(c*x**2)/8 + b*x**6/(24*c) + b*x**2/(8*c **3) - b*atanh(c*x**2)/(8*c**4), Ne(c, 0)), (a*x**8/8, True))
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.28 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {1}{8} \, a x^{8} + \frac {1}{48} \, {\left (6 \, x^{8} \operatorname {artanh}\left (c x^{2}\right ) + c {\left (\frac {2 \, {\left (c^{2} x^{6} + 3 \, x^{2}\right )}}{c^{4}} - \frac {3 \, \log \left (c x^{2} + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x^{2} - 1\right )}{c^{5}}\right )}\right )} b \] Input:
integrate(x^7*(a+b*arctanh(c*x^2)),x, algorithm="maxima")
Output:
1/8*a*x^8 + 1/48*(6*x^8*arctanh(c*x^2) + c*(2*(c^2*x^6 + 3*x^2)/c^4 - 3*lo g(c*x^2 + 1)/c^5 + 3*log(c*x^2 - 1)/c^5))*b
Time = 0.13 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.44 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {1}{16} \, b x^{8} \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + \frac {1}{8} \, a x^{8} + \frac {b x^{6}}{24 \, c} + \frac {b x^{2}}{8 \, c^{3}} - \frac {b \log \left (c x^{2} + 1\right )}{16 \, c^{4}} + \frac {b \log \left (c x^{2} - 1\right )}{16 \, c^{4}} \] Input:
integrate(x^7*(a+b*arctanh(c*x^2)),x, algorithm="giac")
Output:
1/16*b*x^8*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 1/8*a*x^8 + 1/24*b*x^6/c + 1/8* b*x^2/c^3 - 1/16*b*log(c*x^2 + 1)/c^4 + 1/16*b*log(c*x^2 - 1)/c^4
Time = 3.88 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.28 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {a\,x^8}{8}+\frac {b\,x^2}{8\,c^3}+\frac {b\,x^6}{24\,c}+\frac {b\,x^8\,\ln \left (c\,x^2+1\right )}{16}-\frac {b\,x^8\,\ln \left (1-c\,x^2\right )}{16}+\frac {b\,\mathrm {atan}\left (c\,x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,c^4} \] Input:
int(x^7*(a + b*atanh(c*x^2)),x)
Output:
(a*x^8)/8 + (b*x^2)/(8*c^3) + (b*x^6)/(24*c) + (b*atan(c*x^2*1i)*1i)/(8*c^ 4) + (b*x^8*log(c*x^2 + 1))/16 - (b*x^8*log(1 - c*x^2))/16
Time = 0.17 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int x^7 \left (a+b \text {arctanh}\left (c x^2\right )\right ) \, dx=\frac {3 \mathit {atanh} \left (c \,x^{2}\right ) b \,c^{4} x^{8}-3 \mathit {atanh} \left (c \,x^{2}\right ) b +3 a \,c^{4} x^{8}+b \,c^{3} x^{6}+3 b c \,x^{2}}{24 c^{4}} \] Input:
int(x^7*(a+b*atanh(c*x^2)),x)
Output:
(3*atanh(c*x**2)*b*c**4*x**8 - 3*atanh(c*x**2)*b + 3*a*c**4*x**8 + b*c**3* x**6 + 3*b*c*x**2)/(24*c**4)