3.1.49 \(\int \frac {(a+b \text {sech}^{-1}(c x))^3}{x^4} \, dx\) [49]

Optimal. Leaf size=213 \[ \frac {14 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3} \]

[Out]

2/27*b^3*((-c*x+1)/(c*x+1))^(3/2)*(c*x+1)^3/x^3-2/9*b^2*(a+b*arcsech(c*x))/x^3-4/3*b^2*c^2*(a+b*arcsech(c*x))/
x-1/3*(a+b*arcsech(c*x))^3/x^3+14/9*b^3*c^2*(c*x+1)*((-c*x+1)/(c*x+1))^(1/2)/x+1/3*b*(c*x+1)*(a+b*arcsech(c*x)
)^2*((-c*x+1)/(c*x+1))^(1/2)/x^3+2/3*b*c^2*(c*x+1)*(a+b*arcsech(c*x))^2*((-c*x+1)/(c*x+1))^(1/2)/x

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Rubi [A]
time = 0.12, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 5555, 3392, 3377, 2717, 2713} \begin {gather*} -\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}+\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{c x+1}\right )^{3/2} (c x+1)^3}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSech[c*x])^3/x^4,x]

[Out]

(14*b^3*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x))/(9*x) + (2*b^3*((1 - c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3)/(27*x
^3) - (2*b^2*(a + b*ArcSech[c*x]))/(9*x^3) - (4*b^2*c^2*(a + b*ArcSech[c*x]))/(3*x) + (b*Sqrt[(1 - c*x)/(1 + c
*x)]*(1 + c*x)*(a + b*ArcSech[c*x])^2)/(3*x^3) + (2*b*c^2*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcSech[c
*x])^2)/(3*x) - (a + b*ArcSech[c*x])^3/(3*x^3)

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2717

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

Rule 3377

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 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 5555

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c +
d*x)^m*(Cosh[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Cosh[a + b*x]^
(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 6420

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^4} \, dx &=-\left (c^3 \text {Subst}\left (\int (a+b x)^3 \cosh ^2(x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \cosh ^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \cosh ^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^2 c^3\right ) \text {Subst}\left (\int (a+b x) \sinh (x) \, dx,x,\text {sech}^{-1}(c x)\right )+\frac {1}{9} \left (2 i b^3 c^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\frac {i \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )\\ &=\frac {2 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (4 b^3 c^3\right ) \text {Subst}\left (\int \cosh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {14 b^3 c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{9 x}+\frac {2 b^3 \left (\frac {1-c x}{1+c x}\right )^{3/2} (1+c x)^3}{27 x^3}-\frac {2 b^2 \left (a+b \text {sech}^{-1}(c x)\right )}{9 x^3}-\frac {4 b^2 c^2 \left (a+b \text {sech}^{-1}(c x)\right )}{3 x}+\frac {b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x^3}+\frac {2 b c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )^2}{3 x}-\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 x^3}\\ \end {align*}

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Mathematica [A]
time = 0.24, size = 256, normalized size = 1.20 \begin {gather*} \frac {-9 a^3-6 a b^2 \left (1+6 c^2 x^2\right )+9 a^2 b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )+2 b^3 \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+20 c^2 x^2+20 c^3 x^3\right )-3 b \left (9 a^2+2 b^2 \left (1+6 c^2 x^2\right )-6 a b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )\right ) \text {sech}^{-1}(c x)+9 b^2 \left (-3 a+b \sqrt {\frac {1-c x}{1+c x}} \left (1+c x+2 c^2 x^2+2 c^3 x^3\right )\right ) \text {sech}^{-1}(c x)^2-9 b^3 \text {sech}^{-1}(c x)^3}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSech[c*x])^3/x^4,x]

[Out]

(-9*a^3 - 6*a*b^2*(1 + 6*c^2*x^2) + 9*a^2*b*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3) + 2*b^
3*Sqrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 20*c^2*x^2 + 20*c^3*x^3) - 3*b*(9*a^2 + 2*b^2*(1 + 6*c^2*x^2) - 6*a*b*S
qrt[(1 - c*x)/(1 + c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x] + 9*b^2*(-3*a + b*Sqrt[(1 - c*x)/(1 +
c*x)]*(1 + c*x + 2*c^2*x^2 + 2*c^3*x^3))*ArcSech[c*x]^2 - 9*b^3*ArcSech[c*x]^3)/(27*x^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(386\) vs. \(2(191)=382\).
time = 0.26, size = 387, normalized size = 1.82

method	result	size
derivativedivides	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3}+\frac {\mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3 c^{2} x^{2}}-\frac {4 \,\mathrm {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\mathrm {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\)	\(387\)
default	\(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{3}}{3 c^{3} x^{3}}+\frac {2 \mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3}+\frac {\mathrm {arcsech}\left (c x \right )^{2} \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{3 c^{2} x^{2}}-\frac {4 \,\mathrm {arcsech}\left (c x \right )}{3 c x}+\frac {40 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27}-\frac {2 \,\mathrm {arcsech}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arcsech}\left (c x \right )^{2}}{3 c^{3} x^{3}}+\frac {4 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9}+\frac {2 \,\mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}}{9 c^{2} x^{2}}-\frac {4}{9 c x}-\frac {2}{27 c^{3} x^{3}}\right )+3 a^{2} b \left (-\frac {\mathrm {arcsech}\left (c x \right )}{3 c^{3} x^{3}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{2} x^{2}+1\right )}{9 c^{2} x^{2}}\right )\right )\)	\(387\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsech(c*x))^3/x^4,x,method=_RETURNVERBOSE)

[Out]

c^3*(-1/3*a^3/c^3/x^3+b^3*(-1/3*arcsech(c*x)^3/c^3/x^3+2/3*arcsech(c*x)^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(
1/2)+1/3*arcsech(c*x)^2/c^2/x^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-4/3/c/x*arcsech(c*x)+40/27*(-(c*x-1)/
c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-2/9*arcsech(c*x)/c^3/x^3+2/27*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)/c^2/x^2)
+3*a*b^2*(-1/3*arcsech(c*x)^2/c^3/x^3+4/9*arcsech(c*x)*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)+2/9*arcsech(c*
x)/c^2/x^2*(-(c*x-1)/c/x)^(1/2)*((c*x+1)/c/x)^(1/2)-4/9/c/x-2/27/c^3/x^3)+3*a^2*b*(-1/3*arcsech(c*x)/c^3/x^3+1
/9*(-(c*x-1)/c/x)^(1/2)/c^2/x^2*((c*x+1)/c/x)^(1/2)*(2*c^2*x^2+1)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^4,x, algorithm="maxima")

[Out]

1/3*a^2*b*((c^4*(1/(c^2*x^2) - 1)^(3/2) + 3*c^4*sqrt(1/(c^2*x^2) - 1))/c - 3*arcsech(c*x)/x^3) - 1/3*a^3/x^3 +
integrate(b^3*log(sqrt(1/(c*x) + 1)*sqrt(1/(c*x) - 1) + 1/(c*x))^3/x^4 + 3*a*b^2*log(sqrt(1/(c*x) + 1)*sqrt(1
/(c*x) - 1) + 1/(c*x))^2/x^4, x)

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Fricas [A]
time = 0.35, size = 305, normalized size = 1.43 \begin {gather*} -\frac {36 \, a b^{2} c^{2} x^{2} + 9 \, b^{3} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{3} + 9 \, a^{3} + 6 \, a b^{2} + 9 \, {\left (3 \, a b^{2} - {\left (2 \, b^{3} c^{3} x^{3} + b^{3} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{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 (12 \, b^{3} c^{2} x^{2} + 9 \, a^{2} b + 2 \, b^{3} - 6 \, {\left (2 \, a b^{2} c^{3} x^{3} + a b^{2} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, {\left (9 \, a^{2} b + 20 \, b^{3}\right )} c^{3} x^{3} + {\left (9 \, a^{2} b + 2 \, b^{3}\right )} c x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{27 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^4,x, algorithm="fricas")

[Out]

-1/27*(36*a*b^2*c^2*x^2 + 9*b^3*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x))^3 + 9*a^3 + 6*a*b^2 + 9*(3
*a*b^2 - (2*b^3*c^3*x^3 + b^3*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1
)/(c*x))^2 + 3*(12*b^3*c^2*x^2 + 9*a^2*b + 2*b^3 - 6*(2*a*b^2*c^3*x^3 + a*b^2*c*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^
2)))*log((c*x*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)) + 1)/(c*x)) - (2*(9*a^2*b + 20*b^3)*c^3*x^3 + (9*a^2*b + 2*b^3)*c
*x)*sqrt(-(c^2*x^2 - 1)/(c^2*x^2)))/x^3

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asech(c*x))**3/x**4,x)

[Out]

Integral((a + b*asech(c*x))**3/x**4, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsech(c*x))^3/x^4,x, algorithm="giac")

[Out]

integrate((b*arcsech(c*x) + a)^3/x^4, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(1/(c*x)))^3/x^4,x)

[Out]

int((a + b*acosh(1/(c*x)))^3/x^4, x)

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