Integrand size = 14, antiderivative size = 63 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=-\frac {2 b c}{3 x}-\frac {1}{3} b c^{3/2} \arctan \left (\sqrt {c} x\right )+\frac {1}{3} b c^{3/2} \text {arctanh}\left (\sqrt {c} x\right )-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3} \] Output:
-2/3*b*c/x-1/3*b*c^(3/2)*arctan(c^(1/2)*x)+1/3*b*c^(3/2)*arctanh(c^(1/2)*x )-1/3*(a+b*arctanh(c*x^2))/x^3
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.44 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {2 b c}{3 x}-\frac {1}{3} b c^{3/2} \arctan \left (\sqrt {c} x\right )-\frac {b \text {arctanh}\left (c x^2\right )}{3 x^3}-\frac {1}{6} b c^{3/2} \log \left (1-\sqrt {c} x\right )+\frac {1}{6} b c^{3/2} \log \left (1+\sqrt {c} x\right ) \] Input:
Integrate[(a + b*ArcTanh[c*x^2])/x^4,x]
Output:
-1/3*a/x^3 - (2*b*c)/(3*x) - (b*c^(3/2)*ArcTan[Sqrt[c]*x])/3 - (b*ArcTanh[ c*x^2])/(3*x^3) - (b*c^(3/2)*Log[1 - Sqrt[c]*x])/6 + (b*c^(3/2)*Log[1 + Sq rt[c]*x])/6
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6452, 847, 827, 216, 219}
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.
\(\displaystyle \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {2}{3} b c \int \frac {1}{x^2 \left (1-c^2 x^4\right )}dx-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 847 |
\(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {x^2}{1-c^2 x^4}dx-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {2}{3} b c \left (c^2 \left (\frac {\int \frac {1}{1-c x^2}dx}{2 c}-\frac {\int \frac {1}{c x^2+1}dx}{2 c}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2}{3} b c \left (c^2 \left (\frac {\int \frac {1}{1-c x^2}dx}{2 c}-\frac {\arctan \left (\sqrt {c} x\right )}{2 c^{3/2}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {2}{3} b c \left (c^2 \left (\frac {\text {arctanh}\left (\sqrt {c} x\right )}{2 c^{3/2}}-\frac {\arctan \left (\sqrt {c} x\right )}{2 c^{3/2}}\right )-\frac {1}{x}\right )-\frac {a+b \text {arctanh}\left (c x^2\right )}{3 x^3}\) |
Input:
Int[(a + b*ArcTanh[c*x^2])/x^4,x]
Output:
(2*b*c*(-x^(-1) + c^2*(-1/2*ArcTan[Sqrt[c]*x]/c^(3/2) + ArcTanh[Sqrt[c]*x] /(2*c^(3/2)))))/3 - (a + b*ArcTanh[c*x^2])/(3*x^3)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p , x]
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 = 51, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {a}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{2}\right )}{3 x^{3}}-\frac {b \,c^{\frac {3}{2}} \arctan \left (\sqrt {c}\, x \right )}{3}-\frac {2 b c}{3 x}+\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\sqrt {c}\, x \right )}{3}\) | \(51\) |
parts | \(-\frac {a}{3 x^{3}}-\frac {b \,\operatorname {arctanh}\left (c \,x^{2}\right )}{3 x^{3}}-\frac {b \,c^{\frac {3}{2}} \arctan \left (\sqrt {c}\, x \right )}{3}-\frac {2 b c}{3 x}+\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\sqrt {c}\, x \right )}{3}\) | \(51\) |
risch | \(-\frac {b \ln \left (c \,x^{2}+1\right )}{6 x^{3}}-\frac {a}{3 x^{3}}+\frac {b \ln \left (-c \,x^{2}+1\right )}{6 x^{3}}-\frac {2 b c}{3 x}+\frac {b \,c^{\frac {3}{2}} \operatorname {arctanh}\left (\sqrt {c}\, x \right )}{3}-\frac {b \,c^{\frac {3}{2}} \arctan \left (\sqrt {c}\, x \right )}{3}\) | \(68\) |
Input:
int((a+b*arctanh(c*x^2))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/3*a/x^3-1/3*b/x^3*arctanh(c*x^2)-1/3*b*c^(3/2)*arctan(c^(1/2)*x)-2/3*b* c/x+1/3*b*c^(3/2)*arctanh(c^(1/2)*x)
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (47) = 94\).
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.87 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=\left [-\frac {2 \, b c^{\frac {3}{2}} x^{3} \arctan \left (\sqrt {c} x\right ) - b c^{\frac {3}{2}} x^{3} \log \left (\frac {c x^{2} + 2 \, \sqrt {c} x + 1}{c x^{2} - 1}\right ) + 4 \, b c x^{2} + b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{6 \, x^{3}}, -\frac {2 \, b \sqrt {-c} c x^{3} \arctan \left (\sqrt {-c} x\right ) - b \sqrt {-c} c x^{3} \log \left (\frac {c x^{2} - 2 \, \sqrt {-c} x - 1}{c x^{2} + 1}\right ) + 4 \, b c x^{2} + b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right ) + 2 \, a}{6 \, x^{3}}\right ] \] Input:
integrate((a+b*arctanh(c*x^2))/x^4,x, algorithm="fricas")
Output:
[-1/6*(2*b*c^(3/2)*x^3*arctan(sqrt(c)*x) - b*c^(3/2)*x^3*log((c*x^2 + 2*sq rt(c)*x + 1)/(c*x^2 - 1)) + 4*b*c*x^2 + b*log(-(c*x^2 + 1)/(c*x^2 - 1)) + 2*a)/x^3, -1/6*(2*b*sqrt(-c)*c*x^3*arctan(sqrt(-c)*x) - b*sqrt(-c)*c*x^3*l og((c*x^2 - 2*sqrt(-c)*x - 1)/(c*x^2 + 1)) + 4*b*c*x^2 + b*log(-(c*x^2 + 1 )/(c*x^2 - 1)) + 2*a)/x^3]
Leaf count of result is larger than twice the leaf count of optimal. 1904 vs. \(2 (58) = 116\).
Time = 5.30 (sec) , antiderivative size = 1904, normalized size of antiderivative = 30.22 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=\text {Too large to display} \] Input:
integrate((a+b*atanh(c*x**2))/x**4,x)
Output:
Piecewise((-a/(3*x**3), Eq(c, 0)), (-(a - oo*b)/(3*x**3), Eq(c, -1/x**2)), (-(a + oo*b)/(3*x**3), Eq(c, x**(-2))), (-a*c*x**4*sqrt(-1/c)/(3*c*x**7*s qrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqrt(1/c)/c) + a*c*x**4*sqrt(1/c)/(3*c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*s qrt(-1/c)/c + 3*x**3*sqrt(1/c)/c) + a*sqrt(-1/c)/(3*c**2*x**7*sqrt(-1/c) - 3*c**2*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c) + 3*x**3*sqrt(1/c)) - a*sqrt(1/ c)/(3*c**2*x**7*sqrt(-1/c) - 3*c**2*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c) + 3 *x**3*sqrt(1/c)) + b*c**3*x**7*sqrt(-1/c)*sqrt(1/c)*log(x + sqrt(-1/c))/(3 *c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqr t(1/c)/c) - b*c**3*x**7*sqrt(-1/c)*sqrt(1/c)*log(x - sqrt(1/c))/(3*c*x**7* sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqrt(1/c)/c ) - b*c**3*x**7*sqrt(-1/c)*sqrt(1/c)*atanh(c*x**2)/(3*c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqrt(1/c)/c) - b*c**2*x* *7*log(x - sqrt(-1/c))/(3*c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3* sqrt(-1/c)/c + 3*x**3*sqrt(1/c)/c) + b*c**2*x**7*log(x - sqrt(1/c))/(3*c*x **7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqrt(1/ c)/c) + b*c**2*x**7*atanh(c*x**2)/(3*c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c ) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sqrt(1/c)/c) - 2*b*c**2*x**6*sqrt(-1/c)/( 3*c*x**7*sqrt(-1/c) - 3*c*x**7*sqrt(1/c) - 3*x**3*sqrt(-1/c)/c + 3*x**3*sq rt(1/c)/c) + 2*b*c**2*x**6*sqrt(1/c)/(3*c*x**7*sqrt(-1/c) - 3*c*x**7*sq...
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=-\frac {1}{6} \, {\left ({\left (2 \, \sqrt {c} \arctan \left (\sqrt {c} x\right ) + \sqrt {c} \log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right ) + \frac {4}{x}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x^{2}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \] Input:
integrate((a+b*arctanh(c*x^2))/x^4,x, algorithm="maxima")
Output:
-1/6*((2*sqrt(c)*arctan(sqrt(c)*x) + sqrt(c)*log((c*x - sqrt(c))/(c*x + sq rt(c))) + 4/x)*c + 2*arctanh(c*x^2)/x^3)*b - 1/3*a/x^3
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (47) = 94\).
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.51 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=-\frac {b c^{3} \arctan \left (x \sqrt {{\left | c \right |}}\right )}{3 \, {\left | c \right |}^{\frac {3}{2}}} + \frac {b c^{3} \log \left ({\left | x + \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{6 \, {\left | c \right |}^{\frac {3}{2}}} - \frac {b c^{3} \log \left ({\left | x - \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right )}{6 \, {\left | c \right |}^{\frac {3}{2}}} - \frac {b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{6 \, x^{3}} - \frac {2 \, b c x^{2} + a}{3 \, x^{3}} \] Input:
integrate((a+b*arctanh(c*x^2))/x^4,x, algorithm="giac")
Output:
-1/3*b*c^3*arctan(x*sqrt(abs(c)))/abs(c)^(3/2) + 1/6*b*c^3*log(abs(x + 1/s qrt(abs(c))))/abs(c)^(3/2) - 1/6*b*c^3*log(abs(x - 1/sqrt(abs(c))))/abs(c) ^(3/2) - 1/6*b*log(-(c*x^2 + 1)/(c*x^2 - 1))/x^3 - 1/3*(2*b*c*x^2 + a)/x^3
Time = 3.85 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=\frac {b\,\ln \left (1-c\,x^2\right )}{6\,x^3}-\frac {b\,c^{3/2}\,\mathrm {atan}\left (\sqrt {c}\,x\right )}{3}-\frac {b\,\ln \left (c\,x^2+1\right )}{6\,x^3}-\frac {2\,b\,c\,x^2+a}{3\,x^3}-\frac {b\,c^{3/2}\,\mathrm {atan}\left (\sqrt {c}\,x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{3} \] Input:
int((a + b*atanh(c*x^2))/x^4,x)
Output:
(b*log(1 - c*x^2))/(6*x^3) - (b*c^(3/2)*atan(c^(1/2)*x))/3 - (b*c^(3/2)*at an(c^(1/2)*x*1i)*1i)/3 - (b*log(c*x^2 + 1))/(6*x^3) - (a + 2*b*c*x^2)/(3*x ^3)
Time = 0.17 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.41 \[ \int \frac {a+b \text {arctanh}\left (c x^2\right )}{x^4} \, dx=\frac {-2 \sqrt {c}\, \mathit {atan} \left (\frac {c x}{\sqrt {c}}\right ) b c \,x^{3}-2 \sqrt {c}\, \mathit {atanh} \left (c \,x^{2}\right ) b c \,x^{3}-2 \mathit {atanh} \left (c \,x^{2}\right ) b -2 \sqrt {c}\, \mathrm {log}\left (\sqrt {c}\, x -1\right ) b c \,x^{3}+\sqrt {c}\, \mathrm {log}\left (c \,x^{2}+1\right ) b c \,x^{3}-2 a -4 b c \,x^{2}}{6 x^{3}} \] Input:
int((a+b*atanh(c*x^2))/x^4,x)
Output:
( - 2*sqrt(c)*atan((c*x)/sqrt(c))*b*c*x**3 - 2*sqrt(c)*atanh(c*x**2)*b*c*x **3 - 2*atanh(c*x**2)*b - 2*sqrt(c)*log(sqrt(c)*x - 1)*b*c*x**3 + sqrt(c)* log(c*x**2 + 1)*b*c*x**3 - 2*a - 4*b*c*x**2)/(6*x**3)