3.47 \(\int \frac {(a+b \text {sech}^{-1}(c x))^3}{x^2} \, dx\)

Optimal. Leaf size=102 \[ -\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\frac {6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x} \]

[Out]

-6*b^2*(a+b*arcsech(c*x))/x-(a+b*arcsech(c*x))^3/x+6*b^3*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/x+3*b*(c*x+1)*(a+b*a
rcsech(c*x))^2*((-c*x+1)/(c*x+1))^(1/2)/x

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Rubi [A]  time = 0.10, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6285, 3296, 2637} \[ -\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\frac {6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^3/x^2,x]

[Out]

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 6285

Int[((a_.) + ArcSech[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> -Dist[(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Sech[x]^(m + 1)*Tanh[x], x], x, ArcSech[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] &
& (GtQ[n, 0] || LtQ[m, -1])

Rubi steps

\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x^2} \, dx &=-\left (c \operatorname {Subst}\left (\int (a+b x)^3 \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+(3 b c) \operatorname {Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}-\left (6 b^2 c\right ) \operatorname {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}+\left (6 b^3 c\right ) \operatorname {Subst}\left (\int \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {6 b^3 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{x}-\frac {6 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {3 b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{x}\\ \end {align*}

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Mathematica [A]  time = 0.36, size = 165, normalized size = 1.62 \[ -\frac {a^3+3 b \text {sech}^{-1}(c x) \left (a^2-2 a b \sqrt {\frac {1-c x}{c x+1}} (c x+1)+2 b^2\right )-3 a^2 b \sqrt {\frac {1-c x}{c x+1}} (c x+1)-3 b^2 \text {sech}^{-1}(c x)^2 \left (b \sqrt {\frac {1-c x}{c x+1}} (c x+1)-a\right )+6 a b^2-6 b^3 \sqrt {\frac {1-c x}{c x+1}} (c x+1)+b^3 \text {sech}^{-1}(c x)^3}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.54, size = 228, normalized size = 2.24 \[ -\frac {b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} - 3 \, {\left (a^{2} b + 2 \, b^{3}\right )} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + a^{3} + 6 \, a b^{2} - 3 \, {\left (b^{3} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} - 3 \, {\left (2 \, a b^{2} c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - a^{2} b - 2 \, b^{3}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arcsech(c*x) + a)^3/x^2, x)

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maple [B]  time = 0.17, size = 227, normalized size = 2.23 \[ c \left (-\frac {a^{3}}{c x}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{c x}+3 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {6 \,\mathrm {arcsech}\left (c x \right )}{c x}+6 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{c x}+2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}-\frac {2}{c x}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{c x}+\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))^3/x^2,x)

[Out]

c*(-a^3/c/x+b^3*(-1/c/x*arcsech(c*x)^3+3*arcsech(c*x)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-6/c/x*arcsech
(c*x)+6*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2))+3*a*b^2*(-1/c/x*arcsech(c*x)^2+2*arcsech(c*x)*(-(c*x-1)/c/x)
^(1/2)*((c*x+1)/c/x)^(1/2)-2/c/x)+3*a^2*b*(-1/c/x*arcsech(c*x)+(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)))

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maxima [A]  time = 0.33, size = 144, normalized size = 1.41 \[ -\frac {b^{3} \operatorname {arsech}\left (c x\right )^{3}}{x} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} a^{2} b + 6 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right ) - \frac {1}{x}\right )} a b^{2} + 3 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} \operatorname {arsech}\left (c x\right )^{2} + 2 \, c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {2 \, \operatorname {arsech}\left (c x\right )}{x}\right )} b^{3} - \frac {3 \, a b^{2} \operatorname {arsech}\left (c x\right )^{2}}{x} - \frac {a^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*arcsech(c*x)^3/x + 3*(c*sqrt(1/(c^2*x^2) - 1) - arcsech(c*x)/x)*a^2*b + 6*(c*sqrt(1/(c^2*x^2) - 1)*arcsec
h(c*x) - 1/x)*a*b^2 + 3*(c*sqrt(1/(c^2*x^2) - 1)*arcsech(c*x)^2 + 2*c*sqrt(1/(c^2*x^2) - 1) - 2*arcsech(c*x)/x
)*b^3 - 3*a*b^2*arcsech(c*x)^2/x - a^3/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))^3/x^2,x)

[Out]

int((a + b*acosh(1/(c*x)))^3/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**3/x**2,x)

[Out]

Integral((a + b*asech(c*x))**3/x**2, x)

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