3.1.62 \(\int \frac {a+b \tanh ^{-1}(c x^2)}{x^4} \, dx\) [62]

Optimal. Leaf size=63 \[ -\frac {2 b c}{3 x}-\frac {1}{3} b c^{3/2} \text {ArcTan}\left (\sqrt {c} x\right )+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{3 x^3} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 63, 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, 331, 304, 209, 212} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^2\right )}{3 x^3}-\frac {1}{3} b c^{3/2} \text {ArcTan}\left (\sqrt {c} x\right )+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {2 b c}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])/x^4,x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 91, normalized size = 1.44 \begin {gather*} -\frac {a}{3 x^3}-\frac {2 b c}{3 x}-\frac {1}{3} b c^{3/2} \text {ArcTan}\left (\sqrt {c} x\right )-\frac {b \tanh ^{-1}\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 ) \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*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 + Sqrt[c]*x])/6

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Maple [A]
time = 0.06, size = 51, normalized size = 0.81

method	result	size
default	\(-\frac {a}{3 x^{3}}-\frac {b \arctanh \left (c \,x^{2}\right )}{3 x^{3}}+\frac {b \,c^{\frac {3}{2}} \arctanh \left (x \sqrt {c}\right )}{3}-\frac {b \,c^{\frac {3}{2}} \arctan \left (x \sqrt {c}\right )}{3}-\frac {2 b c}{3 x}\)	\(51\)
risch	\(-\frac {b \ln \left (c \,x^{2}+1\right )}{6 x^{3}}-\frac {-c \sqrt {-c}\, b \ln \left (c^{4} \sqrt {-c}-x \,c^{5}\right ) x^{3}+c \sqrt {-c}\, b \ln \left (-c^{4} \sqrt {-c}-x \,c^{5}\right ) x^{3}-c^{\frac {3}{2}} b \ln \left (-c^{\frac {11}{2}}-x \,c^{6}\right ) x^{3}+c^{\frac {3}{2}} b \ln \left (c^{\frac {11}{2}}-x \,c^{6}\right ) x^{3}+4 b c \,x^{2}-b \ln \left (-c \,x^{2}+1\right )+2 a}{6 x^{3}}\)	\(143\)

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

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Maxima [A]
time = 0.46, size = 65, normalized size = 1.03 \begin {gather*} -\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}} \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*((2*sqrt(c)*arctan(sqrt(c)*x) + sqrt(c)*log((c*x - sqrt(c))/(c*x + sqrt(c))) + 4/x)*c + 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. 95 vs. \(2 (47) = 94\).
time = 0.35, size = 181, normalized size = 2.87 \begin {gather*} \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 ] \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*c^(3/2)*x^3*arctan(sqrt(c)*x) - b*c^(3/2)*x^3*log((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, -1/6*(2*b*sqrt(-c)*c*x^3*arctan(sqrt(-c)*x) - b*sqrt(-c)*c*x^3
*log((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]

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1904 vs. \(2 (60) = 120\).
time = 6.34, size = 1904, normalized size = 30.22 \begin {gather*} \text {Too large to display} \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 - oo*b)/(3*x**3), Eq(c, -1/x**2)), (-(a + oo*b)/(3*x**3), Eq(c, x**(-2))), (-a/(3*x**3), Eq(c,
0)), (-a*c*x**4*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) + a*c*x**4*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)
+ 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*sqr
t(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*s
qrt(-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)*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*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*sqrt(1/c)/c) - b*c*x**4*
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*x**4*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*s
qrt(1/c)/c) - b*c*x**3*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*x**3*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*x**3*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*x**3*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*x**3*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*x**3*ata
nh(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*x**2*sq
rt(-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) - 2*b*x**2*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*sqrt(-1/c)*ata
nh(c*x**2)/(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*sqrt(1/
c)*atanh(c*x**2)/(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)), True
))

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Giac [A]
time = 0.42, size = 93, normalized size = 1.48 \begin {gather*} -\frac {b c^{3} \arctan \left (x \sqrt {{\left | c \right |}}\right )}{3 \, {\left | c \right |}^{\frac {3}{2}}} + \frac {1}{6} \, b c \sqrt {{\left | c \right |}} \log \left ({\left | x + \frac {1}{\sqrt {{\left | c \right |}}} \right |}\right ) - \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}} \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*c^3*arctan(x*sqrt(abs(c)))/abs(c)^(3/2) + 1/6*b*c*sqrt(abs(c))*log(abs(x + 1/sqrt(abs(c)))) - 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

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Mupad [B]
time = 0.99, size = 71, normalized size = 1.13 \begin {gather*} \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} \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(1 - c*x^2))/(6*x^3) - (b*c^(3/2)*atan(c^(1/2)*x))/3 - (b*c^(3/2)*atan(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)

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