3.2.62 \(\int \frac {a+b \tanh ^{-1}(\frac {c}{x^2})}{x^3} \, dx\) [162]

Optimal. Leaf size=37 \[ -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{2 x^2}-\frac {b \log \left (1-\frac {c^2}{x^4}\right )}{4 c} \]

[Out]

1/2*(-a-b*arctanh(c/x^2))/x^2-1/4*b*ln(1-c^2/x^4)/c

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Rubi [A]
time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6037, 266} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{2 x^2}-\frac {b \log \left (1-\frac {c^2}{x^4}\right )}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c/x^2])/x^3,x]

[Out]

-1/2*(a + b*ArcTanh[c/x^2])/x^2 - (b*Log[1 - c^2/x^4])/(4*c)

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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^3} \, dx &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{2 x^2}-(b c) \int \frac {1}{\left (1-\frac {c^2}{x^4}\right ) x^5} \, dx\\ &=-\frac {a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{2 x^2}-\frac {b \log \left (1-\frac {c^2}{x^4}\right )}{4 c}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 42, normalized size = 1.14 \begin {gather*} -\frac {a}{2 x^2}-\frac {b \tanh ^{-1}\left (\frac {c}{x^2}\right )}{2 x^2}-\frac {b \log \left (1-\frac {c^2}{x^4}\right )}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*a/x^2 - (b*ArcTanh[c/x^2])/(2*x^2) - (b*Log[1 - c^2/x^4])/(4*c)

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Maple [A]
time = 0.10, size = 39, normalized size = 1.05

method	result	size
derivativedivides	\(-\frac {\frac {c a}{x^{2}}+\frac {b c \arctanh \left (\frac {c}{x^{2}}\right )}{x^{2}}+\frac {b \ln \left (1-\frac {c^{2}}{x^{4}}\right )}{2}}{2 c}\)	\(39\)
default	\(-\frac {\frac {c a}{x^{2}}+\frac {b c \arctanh \left (\frac {c}{x^{2}}\right )}{x^{2}}+\frac {b \ln \left (1-\frac {c^{2}}{x^{4}}\right )}{2}}{2 c}\)	\(39\)
risch	\(-\frac {b \ln \left (x^{2}+c \right )}{4 x^{2}}-\frac {-i \pi b c \,\mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )+2 i \pi b c \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (i \left (-x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )-2 i \pi b c -i \pi b c \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{3}-i \pi b c \,\mathrm {csgn}\left (\frac {i}{x^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (-x^{2}+c \right )}{x^{2}}\right )^{2}+i \pi b c \,\mathrm {csgn}\left (i \left (x^{2}+c \right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{2}-i \pi b c \mathrm {csgn}\left (\frac {i \left (x^{2}+c \right )}{x^{2}}\right )^{3}-8 b \ln \left (x \right ) x^{2}+2 b \ln \left (x^{4}-c^{2}\right ) x^{2}-2 \ln \left (-x^{2}+c \right ) b c +4 a c}{8 c \,x^{2}}\)	\(323\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/c*(c/x^2*a+b*c/x^2*arctanh(c/x^2)+1/2*b*ln(1-c^2/x^4))

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Maxima [A]
time = 0.25, size = 37, normalized size = 1.00 \begin {gather*} -\frac {b {\left (\frac {2 \, c \operatorname {artanh}\left (\frac {c}{x^{2}}\right )}{x^{2}} + \log \left (-\frac {c^{2}}{x^{4}} + 1\right )\right )}}{4 \, c} - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.36, size = 55, normalized size = 1.49 \begin {gather*} -\frac {b x^{2} \log \left (x^{4} - c^{2}\right ) - 4 \, b x^{2} \log \left (x\right ) + b c \log \left (\frac {x^{2} + c}{x^{2} - c}\right ) + 2 \, a c}{4 \, c x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (32) = 64\).
time = 4.29, size = 76, normalized size = 2.05 \begin {gather*} \begin {cases} - \frac {a}{2 x^{2}} - \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 x^{2}} + \frac {b \log {\left (x \right )}}{c} - \frac {b \log {\left (x - \sqrt {- c} \right )}}{2 c} - \frac {b \log {\left (x + \sqrt {- c} \right )}}{2 c} + \frac {b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}}{2 c} & \text {for}\: c \neq 0 \\- \frac {a}{2 x^{2}} & \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**3,x)

[Out]

Piecewise((-a/(2*x**2) - b*atanh(c/x**2)/(2*x**2) + b*log(x)/c - b*log(x - sqrt(-c))/(2*c) - b*log(x + sqrt(-c
))/(2*c) + b*atanh(c/x**2)/(2*c), Ne(c, 0)), (-a/(2*x**2), True))

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Giac [A]
time = 0.41, size = 52, normalized size = 1.41 \begin {gather*} -\frac {b \log \left (x^{4} - c^{2}\right )}{4 \, c} + \frac {b \log \left (x\right )}{c} - \frac {b \log \left (\frac {x^{2} + c}{x^{2} - c}\right )}{4 \, x^{2}} - \frac {a}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.84, size = 56, normalized size = 1.51 \begin {gather*} \frac {b\,\ln \left (x\right )}{c}-\frac {b\,\ln \left (x^4-c^2\right )}{4\,c}-\frac {a}{2\,x^2}-\frac {b\,\ln \left (x^2+c\right )}{4\,x^2}+\frac {b\,\ln \left (x^2-c\right )}{4\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c/x^2))/x^3,x)

[Out]

(b*log(x))/c - (b*log(x^4 - c^2))/(4*c) - a/(2*x^2) - (b*log(c + x^2))/(4*x^2) + (b*log(x^2 - c))/(4*x^2)

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