3.2.68 \(\int \frac {a+b \tanh ^{-1}(\frac {c}{x^2})}{x^2} \, dx\) [168]

Optimal. Leaf size=46 \[ \frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} \]

[Out]

(-a-b*arctanh(c/x^2))/x+b*arctan(x/c^(1/2))/c^(1/2)+b*arctanh(x/c^(1/2))/c^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {6037, 269, 218, 212, 209} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}+\frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^2,x]

[Out]

(b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (a + b*ArcTanh[c/x^2])/x + (b*ArcTanh[x/Sqrt[c]])/Sqrt[c]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 6037

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] - Dist[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
]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x^2} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}-(2 b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^4} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}-(2 b c) \int \frac {1}{-c^2+x^4} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}+b \int \frac {1}{c-x^2} \, dx+b \int \frac {1}{c+x^2} \, dx\\ &=\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 72, normalized size = 1.57 \begin {gather*} -\frac {a}{x}+\frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{x}-\frac {b \log \left (\sqrt {c}-x\right )}{2 \sqrt {c}}+\frac {b \log \left (\sqrt {c}+x\right )}{2 \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(a/x) + (b*ArcTan[x/Sqrt[c]])/Sqrt[c] - (b*ArcTanh[c/x^2])/x - (b*Log[Sqrt[c] - x])/(2*Sqrt[c]) + (b*Log[Sqrt
[c] + x])/(2*Sqrt[c])

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Maple [A]
time = 0.12, size = 47, normalized size = 1.02

method	result	size
derivativedivides	\(-\frac {a}{x}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{x}-\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}+\frac {b \arctanh \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}\)	\(47\)
default	\(-\frac {a}{x}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{x}-\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}+\frac {b \arctanh \left (\frac {\sqrt {c}}{x}\right )}{\sqrt {c}}\)	\(47\)
risch	\(-\frac {b \ln \left (x^{2}+c \right )}{2 x}-\frac {-i \pi b c \,\mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+2 i \pi b c \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}-2 i \pi b c -i \pi b c \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{3}+i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b c \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{3}+i \pi b c \,\mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )-2 \sqrt {-c}\, b \ln \left (c -x \sqrt {-c}\right ) x +2 \sqrt {-c}\, b \ln \left (-x \sqrt {-c}-c \right ) x -2 \sqrt {c}\, b \ln \left (-\sqrt {c}-x \right ) x +2 \sqrt {c}\, b \ln \left (-x +\sqrt {c}\right ) x -2 \ln \left (-x^{2}+c \right ) b c +4 a c}{4 c x}\)	\(373\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/x-b/x*arctanh(c/x^2)-b/c^(1/2)*arctan(1/x*c^(1/2))+b/c^(1/2)*arctanh(1/x*c^(1/2))

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Maxima [A]
time = 0.46, size = 57, normalized size = 1.24 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}} - \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x}\right )} b - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*(2*arctan(x/sqrt(c))/c^(3/2) - log((x - sqrt(c))/(x + sqrt(c)))/c^(3/2)) - 2*arctanh(c/x^2)/x)*b - a/x

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Fricas [A] Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (38) = 76\).
time = 0.37, size = 159, normalized size = 3.46 \begin {gather*} \left [\frac {2 \, b \sqrt {c} x \arctan \left (\frac {x}{\sqrt {c}}\right ) + b \sqrt {c} x \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) - b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) - 2 \, a c}{2 \, c x}, -\frac {2 \, b \sqrt {-c} x \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x \log \left (\frac {x^{2} - 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{2 \, c x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*b*sqrt(c)*x*arctan(x/sqrt(c)) + b*sqrt(c)*x*log((x^2 + 2*sqrt(c)*x + c)/(x^2 - c)) - b*c*log((x^2 + c)
/(x^2 - c)) - 2*a*c)/(c*x), -1/2*(2*b*sqrt(-c)*x*arctan(sqrt(-c)*x/c) + b*sqrt(-c)*x*log((x^2 - 2*sqrt(-c)*x -
c)/(x^2 + c)) + b*c*log((x^2 + c)/(x^2 - c)) + 2*a*c)/(c*x)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 886 vs. \(2 (42) = 84\).
time = 3.66, size = 886, normalized size = 19.26 \begin {gather*} \begin {cases} - \frac {a}{x} & \text {for}\: c = 0 \\- \frac {a - \infty b}{x} & \text {for}\: c = - x^{2} \\- \frac {a + \infty b}{x} & \text {for}\: c = x^{2} \\\frac {2 a c^{\frac {7}{2}} \sqrt {- c}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 a c^{\frac {3}{2}} x^{4} \sqrt {- c}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{\frac {7}{2}} x \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c^{\frac {7}{2}} x \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{\frac {7}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c^{\frac {3}{2}} x^{5} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{\frac {3}{2}} x^{5} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c^{\frac {3}{2}} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{3} x \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{3} x \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {b c^{3} x \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {2 b c^{3} x \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c x^{5} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c x^{5} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} + \frac {b c x^{5} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} - \frac {2 b c x^{5} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 2 c^{\frac {7}{2}} x \sqrt {- c} + 2 c^{\frac {3}{2}} x^{5} \sqrt {- c}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a/x, Eq(c, 0)), (-(a - oo*b)/x, Eq(c, -x**2)), (-(a + oo*b)/x, Eq(c, x**2)), (2*a*c**(7/2)*sqrt(-c
)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - 2*a*c**(3/2)*x**4*sqrt(-c)/(-2*c**(7/2)*x*sqrt(-c) + 2
*c**(3/2)*x**5*sqrt(-c)) - b*c**(7/2)*x*log(x - sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c))
+ b*c**(7/2)*x*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + 2*b*c**(7/2)*sqrt(-c)*a
tanh(c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + b*c**(3/2)*x**5*log(x - sqrt(-c))/(-2*c**(7
/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b*c**(3/2)*x**5*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**
(3/2)*x**5*sqrt(-c)) - 2*b*c**(3/2)*x**4*sqrt(-c)*atanh(c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt
(-c)) + 2*b*c**3*x*sqrt(-c)*log(-sqrt(c) + x)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b*c**3*x*s
qrt(-c)*log(x - sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - b*c**3*x*sqrt(-c)*log(x + sqrt
(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) + 2*b*c**3*x*sqrt(-c)*atanh(c/x**2)/(-2*c**(7/2)*x*s
qrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - 2*b*c*x**5*sqrt(-c)*log(-sqrt(c) + x)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/
2)*x**5*sqrt(-c)) + b*c*x**5*sqrt(-c)*log(x - sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) +
b*c*x**5*sqrt(-c)*log(x + sqrt(-c))/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)) - 2*b*c*x**5*sqrt(-c)*
atanh(c/x**2)/(-2*c**(7/2)*x*sqrt(-c) + 2*c**(3/2)*x**5*sqrt(-c)), True))

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Giac [A]
time = 0.43, size = 62, normalized size = 1.35 \begin {gather*} -b c {\left (\frac {\arctan \left (\frac {x}{\sqrt {-c}}\right )}{\sqrt {-c} c} - \frac {\arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {3}{2}}}\right )} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{2 \, x} - \frac {a}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*c*(arctan(x/sqrt(-c))/(sqrt(-c)*c) - arctan(x/sqrt(c))/c^(3/2)) - 1/2*b*log((x^2 + c)/(x^2 - c))/x - a/x

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Mupad [B]
time = 0.96, size = 59, normalized size = 1.28 \begin {gather*} \frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {a}{x}-\frac {b\,\ln \left (x^2+c\right )}{2\,x}+\frac {b\,\ln \left (x^2-c\right )}{2\,x}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{\sqrt {c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))/x^2,x)

[Out]

(b*atan(x/c^(1/2)))/c^(1/2) - a/x - (b*atan((x*1i)/c^(1/2))*1i)/c^(1/2) - (b*log(c + x^2))/(2*x) + (b*log(x^2
- c))/(2*x)

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