3.194 \(\int \frac {a+b \tanh ^{-1}(c \sqrt {x})}{x^4} \, dx\)

Optimal. Leaf size=73 \[ -\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b c^5}{3 \sqrt {x}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c}{15 x^{5/2}} \]

[Out]

-1/15*b*c/x^(5/2)-1/9*b*c^3/x^(3/2)+1/3*b*c^6*arctanh(c*x^(1/2))+1/3*(-a-b*arctanh(c*x^(1/2)))/x^3-1/3*b*c^5/x
^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6097, 51, 63, 206} \[ -\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b c}{15 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*Sqrt[x]])/x^4,x]

[Out]

-(b*c)/(15*x^(5/2)) - (b*c^3)/(9*x^(3/2)) - (b*c^5)/(3*Sqrt[x]) + (b*c^6*ArcTanh[c*Sqrt[x]])/3 - (a + b*ArcTan
h[c*Sqrt[x]])/(3*x^3)

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

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

Rule 6097

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcTa
nh[c*x^n]))/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 - c^2*x^(2*n)), x], x
] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{x^4} \, dx &=-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} (b c) \int \frac {1}{x^{7/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^3\right ) \int \frac {1}{x^{5/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^5\right ) \int \frac {1}{x^{3/2} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{6} \left (b c^7\right ) \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}+\frac {1}{3} \left (b c^7\right ) \operatorname {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )\\ &=-\frac {b c}{15 x^{5/2}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c^5}{3 \sqrt {x}}+\frac {1}{3} b c^6 \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {a+b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 99, normalized size = 1.36 \[ -\frac {a}{3 x^3}-\frac {1}{6} b c^6 \log \left (1-c \sqrt {x}\right )+\frac {1}{6} b c^6 \log \left (c \sqrt {x}+1\right )-\frac {b c^5}{3 \sqrt {x}}-\frac {b c^3}{9 x^{3/2}}-\frac {b c}{15 x^{5/2}}-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*a/x^3 - (b*c)/(15*x^(5/2)) - (b*c^3)/(9*x^(3/2)) - (b*c^5)/(3*Sqrt[x]) - (b*ArcTanh[c*Sqrt[x]])/(3*x^3) -
(b*c^6*Log[1 - c*Sqrt[x]])/6 + (b*c^6*Log[1 + c*Sqrt[x]])/6

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fricas [A]  time = 0.77, size = 74, normalized size = 1.01 \[ \frac {15 \, {\left (b c^{6} x^{3} - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right ) - 2 \, {\left (15 \, b c^{5} x^{2} + 5 \, b c^{3} x + 3 \, b c\right )} \sqrt {x} - 30 \, a}{90 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^4,x, algorithm="fricas")

[Out]

1/90*(15*(b*c^6*x^3 - b)*log(-(c^2*x + 2*c*sqrt(x) + 1)/(c^2*x - 1)) - 2*(15*b*c^5*x^2 + 5*b*c^3*x + 3*b*c)*sq
rt(x) - 30*a)/x^3

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giac [B]  time = 0.22, size = 534, normalized size = 7.32 \[ \frac {2}{45} \, c {\left (\frac {15 \, {\left (\frac {3 \, {\left (c \sqrt {x} + 1\right )}^{5} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {10 \, {\left (c \sqrt {x} + 1\right )}^{3} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {3 \, {\left (c \sqrt {x} + 1\right )} b c^{5}}{c \sqrt {x} - 1}\right )} \log \left (-\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1}\right )}{\frac {{\left (c \sqrt {x} + 1\right )}^{6}}{{\left (c \sqrt {x} - 1\right )}^{6}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {20 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1} + \frac {\frac {90 \, {\left (c \sqrt {x} + 1\right )}^{5} a c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {300 \, {\left (c \sqrt {x} + 1\right )}^{3} a c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {90 \, {\left (c \sqrt {x} + 1\right )} a c^{5}}{c \sqrt {x} - 1} + \frac {45 \, {\left (c \sqrt {x} + 1\right )}^{5} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {135 \, {\left (c \sqrt {x} + 1\right )}^{4} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {230 \, {\left (c \sqrt {x} + 1\right )}^{3} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {210 \, {\left (c \sqrt {x} + 1\right )}^{2} b c^{5}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {93 \, {\left (c \sqrt {x} + 1\right )} b c^{5}}{c \sqrt {x} - 1} + 23 \, b c^{5}}{\frac {{\left (c \sqrt {x} + 1\right )}^{6}}{{\left (c \sqrt {x} - 1\right )}^{6}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}^{5}}{{\left (c \sqrt {x} - 1\right )}^{5}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{4}}{{\left (c \sqrt {x} - 1\right )}^{4}} + \frac {20 \, {\left (c \sqrt {x} + 1\right )}^{3}}{{\left (c \sqrt {x} - 1\right )}^{3}} + \frac {15 \, {\left (c \sqrt {x} + 1\right )}^{2}}{{\left (c \sqrt {x} - 1\right )}^{2}} + \frac {6 \, {\left (c \sqrt {x} + 1\right )}}{c \sqrt {x} - 1} + 1}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^4,x, algorithm="giac")

[Out]

2/45*c*(15*(3*(c*sqrt(x) + 1)^5*b*c^5/(c*sqrt(x) - 1)^5 + 10*(c*sqrt(x) + 1)^3*b*c^5/(c*sqrt(x) - 1)^3 + 3*(c*
sqrt(x) + 1)*b*c^5/(c*sqrt(x) - 1))*log(-(c*sqrt(x) + 1)/(c*sqrt(x) - 1))/((c*sqrt(x) + 1)^6/(c*sqrt(x) - 1)^6
+ 6*(c*sqrt(x) + 1)^5/(c*sqrt(x) - 1)^5 + 15*(c*sqrt(x) + 1)^4/(c*sqrt(x) - 1)^4 + 20*(c*sqrt(x) + 1)^3/(c*sq
rt(x) - 1)^3 + 15*(c*sqrt(x) + 1)^2/(c*sqrt(x) - 1)^2 + 6*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1) + (90*(c*sqrt(x
) + 1)^5*a*c^5/(c*sqrt(x) - 1)^5 + 300*(c*sqrt(x) + 1)^3*a*c^5/(c*sqrt(x) - 1)^3 + 90*(c*sqrt(x) + 1)*a*c^5/(c
*sqrt(x) - 1) + 45*(c*sqrt(x) + 1)^5*b*c^5/(c*sqrt(x) - 1)^5 + 135*(c*sqrt(x) + 1)^4*b*c^5/(c*sqrt(x) - 1)^4 +
230*(c*sqrt(x) + 1)^3*b*c^5/(c*sqrt(x) - 1)^3 + 210*(c*sqrt(x) + 1)^2*b*c^5/(c*sqrt(x) - 1)^2 + 93*(c*sqrt(x)
+ 1)*b*c^5/(c*sqrt(x) - 1) + 23*b*c^5)/((c*sqrt(x) + 1)^6/(c*sqrt(x) - 1)^6 + 6*(c*sqrt(x) + 1)^5/(c*sqrt(x)
- 1)^5 + 15*(c*sqrt(x) + 1)^4/(c*sqrt(x) - 1)^4 + 20*(c*sqrt(x) + 1)^3/(c*sqrt(x) - 1)^3 + 15*(c*sqrt(x) + 1)^
2/(c*sqrt(x) - 1)^2 + 6*(c*sqrt(x) + 1)/(c*sqrt(x) - 1) + 1))

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maple [A]  time = 0.04, size = 73, normalized size = 1.00 \[ -\frac {a}{3 x^{3}}-\frac {b \arctanh \left (c \sqrt {x}\right )}{3 x^{3}}-\frac {b c}{15 x^{\frac {5}{2}}}-\frac {b \,c^{3}}{9 x^{\frac {3}{2}}}-\frac {b \,c^{5}}{3 \sqrt {x}}-\frac {c^{6} b \ln \left (c \sqrt {x}-1\right )}{6}+\frac {c^{6} b \ln \left (1+c \sqrt {x}\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(1/2)))/x^4,x)

[Out]

-1/3*a/x^3-1/3*b/x^3*arctanh(c*x^(1/2))-1/15*b*c/x^(5/2)-1/9*b*c^3/x^(3/2)-1/3*b*c^5/x^(1/2)-1/6*c^6*b*ln(c*x^
(1/2)-1)+1/6*c^6*b*ln(1+c*x^(1/2))

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maxima [A]  time = 0.32, size = 72, normalized size = 0.99 \[ \frac {1}{90} \, {\left ({\left (15 \, c^{5} \log \left (c \sqrt {x} + 1\right ) - 15 \, c^{5} \log \left (c \sqrt {x} - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{2} + 5 \, c^{2} x + 3\right )}}{x^{\frac {5}{2}}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c \sqrt {x}\right )}{x^{3}}\right )} b - \frac {a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(1/2)))/x^4,x, algorithm="maxima")

[Out]

1/90*((15*c^5*log(c*sqrt(x) + 1) - 15*c^5*log(c*sqrt(x) - 1) - 2*(15*c^4*x^2 + 5*c^2*x + 3)/x^(5/2))*c - 30*ar
ctanh(c*sqrt(x))/x^3)*b - 1/3*a/x^3

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mupad [B]  time = 1.39, size = 69, normalized size = 0.95 \[ \frac {b\,c^6\,\mathrm {atanh}\left (c\,\sqrt {x}\right )}{3}-\frac {b\,\left (15\,\ln \left (c\,\sqrt {x}+1\right )-15\,\ln \left (1-c\,\sqrt {x}\right )+6\,c\,\sqrt {x}+10\,c^3\,x^{3/2}+30\,c^5\,x^{5/2}\right )}{90\,x^3}-\frac {a}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x^(1/2)))/x^4,x)

[Out]

(b*c^6*atanh(c*x^(1/2)))/3 - (b*(15*log(c*x^(1/2) + 1) - 15*log(1 - c*x^(1/2)) + 6*c*x^(1/2) + 10*c^3*x^(3/2)
+ 30*c^5*x^(5/2)))/(90*x^3) - a/(3*x^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**(1/2)))/x**4,x)

[Out]

Timed out

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