\(\int \frac {(a+b \text {arctanh}(c x^3))^2}{x^7} \, dx\) [122]

Optimal result

Integrand size = 16, antiderivative size = 88 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=-\frac {b c \left (a+b \text {arctanh}\left (c x^3\right )\right )}{3 x^3}+\frac {1}{6} c^2 \left (a+b \text {arctanh}\left (c x^3\right )\right )^2-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1-c^2 x^6\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\frac {1}{6} \left (-\frac {a^2}{x^6}-\frac {2 a b c}{x^3}-\frac {2 b \left (a+b c x^3\right ) \text {arctanh}\left (c x^3\right )}{x^6}+\frac {b^2 \left (-1+c^2 x^6\right ) \text {arctanh}\left (c x^3\right )^2}{x^6}+6 b^2 c^2 \log (x)-b (a+b) c^2 \log \left (1-c x^3\right )+(a-b) b c^2 \log \left (1+c x^3\right )\right ) \] Input:

Integrate[(a + b*ArcTanh[c*x^3])^2/x^7,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6454, 6452, 6544, 6452, 243, 47, 14, 16, 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.

\(\displaystyle \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx\)

\(\Big \downarrow \) 6454

\(\displaystyle \frac {1}{3} \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^9}dx^3\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{3} \left (b c \int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^6 \left (1-c^2 x^6\right )}dx^3-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 6544

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3+\int \frac {a+b \text {arctanh}\left (c x^3\right )}{x^6}dx^3\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 6452

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3+b c \int \frac {1}{x^3 \left (1-c^2 x^6\right )}dx^3-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3+\frac {1}{2} b c \int \frac {1}{x^3 \left (1-c^2 x^6\right )}dx^6-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^6}dx^6+\int \frac {1}{x^3}dx^6\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^6}dx^6+\log \left (x^6\right )\right )-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (b c \left (c^2 \int \frac {a+b \text {arctanh}\left (c x^3\right )}{1-c^2 x^6}dx^3-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}+\frac {1}{2} b c \left (\log \left (x^6\right )-\log \left (1-c^2 x^6\right )\right )\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

\(\Big \downarrow \) 6510

\(\displaystyle \frac {1}{3} \left (b c \left (\frac {c \left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 b}-\frac {a+b \text {arctanh}\left (c x^3\right )}{x^3}+\frac {1}{2} b c \left (\log \left (x^6\right )-\log \left (1-c^2 x^6\right )\right )\right )-\frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{2 x^6}\right )\)

Input:

Int[(a + b*ArcTanh[c*x^3])^2/x^7,x]
 

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 6452

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]
 

rule 6454

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]]
 

rule 6510

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]
 

rule 6544

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( 
e_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x 
], x] - Simp[e/(d*f^2)   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] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.66 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.73

method	result	size
parallelrisch	\(\frac {b^{2} \operatorname {arctanh}\left (c \,x^{3}\right )^{2} x^{6} c^{2}+6 b^{2} c^{2} \ln \left (x \right ) x^{6}-2 \ln \left (c \,x^{3}-1\right ) b^{2} c^{2} x^{6}+2 x^{6} \operatorname {arctanh}\left (c \,x^{3}\right ) a b \,c^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) b^{2} c^{2} x^{6}-a^{2} c^{2} x^{6}-2 b^{2} \operatorname {arctanh}\left (c \,x^{3}\right ) x^{3} c -2 a b c \,x^{3}-b^{2} \operatorname {arctanh}\left (c \,x^{3}\right )^{2}-2 \,\operatorname {arctanh}\left (c \,x^{3}\right ) a b -a^{2}}{6 x^{6}}\)	\(152\)
risch	\(\frac {b^{2} \left (c^{2} x^{6}-1\right ) \ln \left (c \,x^{3}+1\right )^{2}}{24 x^{6}}-\frac {b \left (b \,c^{2} \ln \left (-c \,x^{3}+1\right ) x^{6}+2 b c \,x^{3}-b \ln \left (-c \,x^{3}+1\right )+2 a \right ) \ln \left (c \,x^{3}+1\right )}{12 x^{6}}+\frac {b^{2} c^{2} x^{6} \ln \left (-c \,x^{3}+1\right )^{2}+24 b^{2} c^{2} \ln \left (x \right ) x^{6}+4 b \,c^{2} \ln \left (c \,x^{3}+1\right ) x^{6} a -4 \ln \left (c \,x^{3}+1\right ) b^{2} c^{2} x^{6}-4 b \,c^{2} \ln \left (c \,x^{3}-1\right ) x^{6} a -4 \ln \left (c \,x^{3}-1\right ) b^{2} c^{2} x^{6}+4 b^{2} c \,x^{3} \ln \left (-c \,x^{3}+1\right )-8 a b c \,x^{3}-b^{2} \ln \left (-c \,x^{3}+1\right )^{2}+4 b \ln \left (-c \,x^{3}+1\right ) a -4 a^{2}}{24 x^{6}}\)	\(257\)
default	\(\text {Expression too large to display}\)	\(2954\)
parts	\(\text {Expression too large to display}\)	\(2954\)

Input:

int((a+b*arctanh(c*x^3))^2/x^7,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\frac {24 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 4 \, {\left (a b - b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} + 1\right ) - 4 \, {\left (a b + b^{2}\right )} c^{2} x^{6} \log \left (c x^{3} - 1\right ) - 8 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} - b^{2}\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )^{2} - 4 \, a^{2} - 4 \, {\left (b^{2} c x^{3} + a b\right )} \log \left (-\frac {c x^{3} + 1}{c x^{3} - 1}\right )}{24 \, x^{6}} \] Input:

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="fricas")
 

Output:

1/24*(24*b^2*c^2*x^6*log(x) + 4*(a*b - b^2)*c^2*x^6*log(c*x^3 + 1) - 4*(a* 
b + b^2)*c^2*x^6*log(c*x^3 - 1) - 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^2 - 4*(b^2*c*x^3 + a*b)*log(-(c*x^3 + 1)/( 
c*x^3 - 1)))/x^6
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\text {Timed out} \] Input:

integrate((a+b*atanh(c*x**3))**2/x**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (80) = 160\).

Time = 0.03 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.99 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\frac {1}{6} \, {\left ({\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{24} \, {\left ({\left (2 \, {\left (\log \left (c x^{3} - 1\right ) - 2\right )} \log \left (c x^{3} + 1\right ) - \log \left (c x^{3} + 1\right )^{2} - \log \left (c x^{3} - 1\right )^{2} - 4 \, \log \left (c x^{3} - 1\right ) + 24 \, \log \left (x\right )\right )} c^{2} + 4 \, {\left (c \log \left (c x^{3} + 1\right ) - c \log \left (c x^{3} - 1\right ) - \frac {2}{x^{3}}\right )} c \operatorname {artanh}\left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \operatorname {artanh}\left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \] Input:

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="maxima")
 

Output:

1/6*((c*log(c*x^3 + 1) - c*log(c*x^3 - 1) - 2/x^3)*c - 2*arctanh(c*x^3)/x^ 
6)*a*b + 1/24*((2*(log(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) + 24*log(x))*c^2 + 4*(c*log(c*x^3 + 1 
) - c*log(c*x^3 - 1) - 2/x^3)*c*arctanh(c*x^3))*b^2 - 1/6*b^2*arctanh(c*x^ 
3)^2/x^6 - 1/6*a^2/x^6
 

Giac [F]

\[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x^{3}\right ) + a\right )}^{2}}{x^{7}} \,d x } \] Input:

integrate((a+b*arctanh(c*x^3))^2/x^7,x, algorithm="giac")
 

Output:

integrate((b*arctanh(c*x^3) + a)^2/x^7, x)
 

Mupad [B] (verification not implemented)

Time = 4.27 (sec) , antiderivative size = 278, normalized size of antiderivative = 3.16 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\frac {b^2\,c^2\,{\ln \left (c\,x^3+1\right )}^2}{24}-\frac {b^2\,c^2\,\ln \left (c\,x^3-1\right )}{6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a^2}{6\,x^6}+\frac {b^2\,c^2\,{\ln \left (1-c\,x^3\right )}^2}{24}-\frac {b^2\,{\ln \left (c\,x^3+1\right )}^2}{24\,x^6}-\frac {b^2\,{\ln \left (1-c\,x^3\right )}^2}{24\,x^6}+b^2\,c^2\,\ln \left (x\right )-\frac {a\,b\,c^2\,\ln \left (c\,x^3-1\right )}{6}+\frac {a\,b\,c^2\,\ln \left (c\,x^3+1\right )}{6}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,\ln \left (c\,x^3+1\right )}{6\,x^6}+\frac {a\,b\,\ln \left (1-c\,x^3\right )}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12}-\frac {b^2\,c\,\ln \left (c\,x^3+1\right )}{6\,x^3}+\frac {b^2\,c\,\ln \left (1-c\,x^3\right )}{6\,x^3}+\frac {b^2\,\ln \left (c\,x^3+1\right )\,\ln \left (1-c\,x^3\right )}{12\,x^6} \] Input:

int((a + b*atanh(c*x^3))^2/x^7,x)
 

Output:

(b^2*c^2*log(c*x^3 + 1)^2)/24 - (b^2*c^2*log(c*x^3 - 1))/6 - (b^2*c^2*log( 
c*x^3 + 1))/6 - a^2/(6*x^6) + (b^2*c^2*log(1 - c*x^3)^2)/24 - (b^2*log(c*x 
^3 + 1)^2)/(24*x^6) - (b^2*log(1 - c*x^3)^2)/(24*x^6) + b^2*c^2*log(x) - ( 
a*b*c^2*log(c*x^3 - 1))/6 + (a*b*c^2*log(c*x^3 + 1))/6 - (a*b*c)/(3*x^3) - 
 (a*b*log(c*x^3 + 1))/(6*x^6) + (a*b*log(1 - c*x^3))/(6*x^6) - (b^2*c^2*lo 
g(c*x^3 + 1)*log(1 - c*x^3))/12 - (b^2*c*log(c*x^3 + 1))/(6*x^3) + (b^2*c* 
log(1 - c*x^3))/(6*x^3) + (b^2*log(c*x^3 + 1)*log(1 - c*x^3))/(12*x^6)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.92 \[ \int \frac {\left (a+b \text {arctanh}\left (c x^3\right )\right )^2}{x^7} \, dx=\frac {\mathit {atanh} \left (c \,x^{3}\right )^{2} b^{2} c^{2} x^{6}-\mathit {atanh} \left (c \,x^{3}\right )^{2} b^{2}+2 \mathit {atanh} \left (c \,x^{3}\right ) a b \,c^{2} x^{6}-2 \mathit {atanh} \left (c \,x^{3}\right ) a b +2 \mathit {atanh} \left (c \,x^{3}\right ) b^{2} c^{2} x^{6}-2 \mathit {atanh} \left (c \,x^{3}\right ) b^{2} c \,x^{3}-2 \,\mathrm {log}\left (c^{\frac {2}{3}} x^{2}-c^{\frac {1}{3}} x +1\right ) b^{2} c^{2} x^{6}-2 \,\mathrm {log}\left (c^{\frac {2}{3}} x +c^{\frac {1}{3}}\right ) b^{2} c^{2} x^{6}+6 \,\mathrm {log}\left (x \right ) b^{2} c^{2} x^{6}-a^{2}-2 a b c \,x^{3}}{6 x^{6}} \] Input:

int((a+b*atanh(c*x^3))^2/x^7,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**2*x**6 - 2* 
atanh(c*x**3)*b**2*c*x**3 - 2*log(c**(2/3)*x**2 - c**(1/3)*x + 1)*b**2*c** 
2*x**6 - 2*log(c**(2/3)*x + c**(1/3))*b**2*c**2*x**6 + 6*log(x)*b**2*c**2* 
x**6 - a**2 - 2*a*b*c*x**3)/(6*x**6)