3.2.69 \(\int \frac {a+b \tanh ^{-1}(\frac {c}{x^2})}{x^4} \, dx\) [169]

Optimal. Leaf size=65 \[ -\frac {2 b}{3 c x}-\frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6037, 269, 331, 304, 209, 212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {2 b}{3 c x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^4,x]

[Out]

(-2*b)/(3*c*x) - (b*ArcTan[x/Sqrt[c]])/(3*c^(3/2)) - (a + b*ArcTanh[c/x^2])/(3*x^3) + (b*ArcTanh[x/Sqrt[c]])/(
3*c^(3/2))

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 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 304

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

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {1}{3} (2 b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^6} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {1}{3} (2 b c) \int \frac {1}{x^2 \left (-c^2+x^4\right )} \, dx\\ &=-\frac {2 b}{3 c x}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {(2 b) \int \frac {x^2}{-c^2+x^4} \, dx}{3 c}\\ &=-\frac {2 b}{3 c x}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \int \frac {1}{c-x^2} \, dx}{3 c}-\frac {b \int \frac {1}{c+x^2} \, dx}{3 c}\\ &=-\frac {2 b}{3 c x}-\frac {b \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}+\frac {b \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 90, normalized size = 1.38 \begin {gather*} -\frac {a}{3 x^3}-\frac {2 b}{3 c x}-\frac {b \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )}{3 c^{3/2}}-\frac {b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{3 x^3}-\frac {b \log \left (\sqrt {c}-x\right )}{6 c^{3/2}}+\frac {b \log \left (\sqrt {c}+x\right )}{6 c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c/x^2])/x^4,x]

[Out]

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

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Maple [A]
time = 0.13, size = 57, normalized size = 0.88

method	result	size
derivativedivides	\(-\frac {a}{3 x^{3}}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}+\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}+\frac {b \arctanh \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}\)	\(57\)
default	\(-\frac {a}{3 x^{3}}-\frac {b \arctanh \left (\frac {c}{x^{2}}\right )}{3 x^{3}}-\frac {2 b}{3 c x}+\frac {b \arctan \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}+\frac {b \arctanh \left (\frac {\sqrt {c}}{x}\right )}{3 c^{\frac {3}{2}}}\)	\(57\)
risch	\(-\frac {b \ln \left (x^{2}+c \right )}{6 x^{3}}-\frac {-2 i \pi b \,c^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )+2 i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b \,c^{2} \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{3}-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{3}+i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b \,c^{2} \mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-2 \sqrt {-c}\, b \ln \left (x \,c^{4} \sqrt {-c}+c^{5}\right ) x^{3}+2 \sqrt {-c}\, b \ln \left (x \,c^{4} \sqrt {-c}-c^{5}\right ) x^{3}-2 \sqrt {c}\, b \ln \left (x +\sqrt {c}\right ) x^{3}+2 \sqrt {c}\, b \ln \left (-\sqrt {c}+x \right ) x^{3}-2 b \ln \left (-x^{2}+c \right ) c^{2}+8 b c \,x^{2}+4 a \,c^{2}}{12 c^{2} x^{3}}\)	\(416\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c/x^2))/x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 64, normalized size = 0.98 \begin {gather*} -\frac {1}{6} \, {\left (c {\left (\frac {2 \, \arctan \left (\frac {x}{\sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {\log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )}{c^{\frac {5}{2}}} + \frac {4}{c^{2} x}\right )} + \frac {2 \, \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))/x^4,x, algorithm="maxima")

[Out]

-1/6*(c*(2*arctan(x/sqrt(c))/c^(5/2) + log((x - sqrt(c))/(x + sqrt(c)))/c^(5/2) + 4/(c^2*x)) + 2*arctanh(c/x^2
)/x^3)*b - 1/3*a/x^3

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Fricas [A] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (49) = 98\).
time = 0.43, size = 189, normalized size = 2.91 \begin {gather*} \left [-\frac {2 \, b \sqrt {c} x^{3} \arctan \left (\frac {x}{\sqrt {c}}\right ) - b \sqrt {c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {c} x + c}{x^{2} - c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}, -\frac {2 \, b \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-c} x}{c}\right ) + b \sqrt {-c} x^{3} \log \left (\frac {x^{2} + 2 \, \sqrt {-c} x - c}{x^{2} + c}\right ) + 4 \, b c x^{2} + b c^{2} \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c^{2}}{6 \, c^{2} x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))/x^4,x, algorithm="fricas")

[Out]

[-1/6*(2*b*sqrt(c)*x^3*arctan(x/sqrt(c)) - b*sqrt(c)*x^3*log((x^2 + 2*sqrt(c)*x + c)/(x^2 - c)) + 4*b*c*x^2 +
b*c^2*log((x^2 + c)/(x^2 - c)) + 2*a*c^2)/(c^2*x^3), -1/6*(2*b*sqrt(-c)*x^3*arctan(sqrt(-c)*x/c) + b*sqrt(-c)*
x^3*log((x^2 + 2*sqrt(-c)*x - c)/(x^2 + c)) + 4*b*c*x^2 + b*c^2*log((x^2 + c)/(x^2 - c)) + 2*a*c^2)/(c^2*x^3)]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (60) = 120\).
time = 4.99, size = 1046, normalized size = 16.09 \begin {gather*} \begin {cases} - \frac {a}{3 x^{3}} & \text {for}\: c = 0 \\- \frac {a - \infty b}{3 x^{3}} & \text {for}\: c = - x^{2} \\- \frac {a + \infty b}{3 x^{3}} & \text {for}\: c = x^{2} \\\frac {2 a c^{\frac {17}{2}} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 a c^{\frac {13}{2}} x^{4} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{\frac {17}{2}} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{\frac {15}{2}} x^{3} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{\frac {15}{2}} x^{3} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {4 b c^{\frac {15}{2}} x^{2} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{\frac {13}{2}} x^{4} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{\frac {11}{2}} x^{7} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{\frac {11}{2}} x^{7} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {4 b c^{\frac {11}{2}} x^{6} \sqrt {- c}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{7} x^{3} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{7} x^{3} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {b c^{7} x^{3} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {2 b c^{7} x^{3} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{5} x^{7} \sqrt {- c} \log {\left (- \sqrt {c} + x \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{5} x^{7} \sqrt {- c} \log {\left (x - \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} + \frac {b c^{5} x^{7} \sqrt {- c} \log {\left (x + \sqrt {- c} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \sqrt {- c}} - \frac {2 b c^{5} x^{7} \sqrt {- c} \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{- 6 c^{\frac {17}{2}} x^{3} \sqrt {- c} + 6 c^{\frac {13}{2}} x^{7} \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**4,x)

[Out]

Piecewise((-a/(3*x**3), Eq(c, 0)), (-(a - oo*b)/(3*x**3), Eq(c, -x**2)), (-(a + oo*b)/(3*x**3), Eq(c, x**2)),
(2*a*c**(17/2)*sqrt(-c)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*a*c**(13/2)*x**4*sqrt(-c)
/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 2*b*c**(17/2)*sqrt(-c)*atanh(c/x**2)/(-6*c**(17/2)
*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + b*c**(15/2)*x**3*log(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) +
6*c**(13/2)*x**7*sqrt(-c)) - b*c**(15/2)*x**3*log(x + sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**
7*sqrt(-c)) + 4*b*c**(15/2)*x**2*sqrt(-c)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*b*c**(1
3/2)*x**4*sqrt(-c)*atanh(c/x**2)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - b*c**(11/2)*x**7*l
og(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + b*c**(11/2)*x**7*log(x + sqrt(-c))
/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 4*b*c**(11/2)*x**6*sqrt(-c)/(-6*c**(17/2)*x**3*sqr
t(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + 2*b*c**7*x**3*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*
c**(13/2)*x**7*sqrt(-c)) - b*c**7*x**3*sqrt(-c)*log(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x*
*7*sqrt(-c)) - b*c**7*x**3*sqrt(-c)*log(x + sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c))
+ 2*b*c**7*x**3*sqrt(-c)*atanh(c/x**2)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*b*c**5*x*
*7*sqrt(-c)*log(-sqrt(c) + x)/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + b*c**5*x**7*sqrt(-c)*
log(x - sqrt(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) + b*c**5*x**7*sqrt(-c)*log(x + sqrt
(-c))/(-6*c**(17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)) - 2*b*c**5*x**7*sqrt(-c)*atanh(c/x**2)/(-6*c**(
17/2)*x**3*sqrt(-c) + 6*c**(13/2)*x**7*sqrt(-c)), True))

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Giac [A]
time = 0.45, size = 72, normalized size = 1.11 \begin {gather*} -\frac {b \arctan \left (\frac {x}{\sqrt {-c}}\right )}{3 \, \sqrt {-c} c} - \frac {b \arctan \left (\frac {x}{\sqrt {c}}\right )}{3 \, c^{\frac {3}{2}}} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{6 \, x^{3}} - \frac {2 \, b x^{2} + a c}{3 \, c x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c/x^2))/x^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.99, size = 69, normalized size = 1.06 \begin {gather*} \frac {b\,\ln \left (x^2-c\right )}{6\,x^3}-\frac {2\,b}{3\,c\,x}-\frac {b\,\mathrm {atan}\left (\frac {x}{\sqrt {c}}\right )}{3\,c^{3/2}}-\frac {b\,\ln \left (x^2+c\right )}{6\,x^3}-\frac {a}{3\,x^3}-\frac {b\,\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{3\,c^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))/x^4,x)

[Out]

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

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