3.1.31 \(\int \frac {(a+b \text {csch}^{-1}(c x))^3}{x^4} \, dx\) [31]

Optimal. Leaf size=166 \[ -\frac {14}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.13, antiderivative size = 166, 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 = {6421, 5554, 3392, 3377, 2718, 2713} \begin {gather*} \frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {2}{3} b c^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\frac {2}{27} b^3 c^3 \left (\frac {1}{c^2 x^2}+1\right )^{3/2}-\frac {14}{9} b^3 c^3 \sqrt {\frac {1}{c^2 x^2}+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsch[c*x])^3/x^4,x]

[Out]

(-14*b^3*c^3*Sqrt[1 + 1/(c^2*x^2)])/9 + (2*b^3*c^3*(1 + 1/(c^2*x^2))^(3/2))/27 - (2*b^2*(a + b*ArcCsch[c*x]))/
(9*x^3) + (4*b^2*c^2*(a + b*ArcCsch[c*x]))/(3*x) - (2*b*c^3*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/3 +
(b*c*Sqrt[1 + 1/(c^2*x^2)]*(a + b*ArcCsch[c*x])^2)/(3*x^2) - (a + b*ArcCsch[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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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 5554

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

Rule 6421

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b
*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, ArcCsch[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 {csch}^{-1}(c x)\right )^3}{x^4} \, dx &=-\left (c^3 \text {Subst}\left (\int (a+b x)^3 \cosh (x) \sinh ^2(x) \, dx,x,\text {csch}^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-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 \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \sinh ^3(x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-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) \cosh (x) \, dx,x,\text {csch}^{-1}(c x)\right )-\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1+\frac {1}{c^2 x^2}}\right )\\ &=-\frac {2}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^3 c^3\right ) \text {Subst}\left (\int \sinh (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=-\frac {14}{9} b^3 c^3 \sqrt {1+\frac {1}{c^2 x^2}}+\frac {2}{27} b^3 c^3 \left (1+\frac {1}{c^2 x^2}\right )^{3/2}-\frac {2 b^2 \left (a+b \text {csch}^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {b c \sqrt {1+\frac {1}{c^2 x^2}} \left (a+b \text {csch}^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \text {csch}^{-1}(c x)\right )^3}{3 x^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 200, normalized size = 1.20 \begin {gather*} \frac {-9 a^3+2 b^3 c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-20 c^2 x^2\right )+9 a^2 b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-2 c^2 x^2\right )+6 a b^2 \left (-1+6 c^2 x^2\right )+3 b \left (-9 a^2+6 a b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (1-2 c^2 x^2\right )+2 b^2 \left (-1+6 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)-9 b^2 \left (3 a+b c \sqrt {1+\frac {1}{c^2 x^2}} x \left (-1+2 c^2 x^2\right )\right ) \text {csch}^{-1}(c x)^2-9 b^3 \text {csch}^{-1}(c x)^3}{27 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*a^3 + 2*b^3*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 20*c^2*x^2) + 9*a^2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2)
+ 6*a*b^2*(-1 + 6*c^2*x^2) + 3*b*(-9*a^2 + 6*a*b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(1 - 2*c^2*x^2) + 2*b^2*(-1 + 6*c^
2*x^2))*ArcCsch[c*x] - 9*b^2*(3*a + b*c*Sqrt[1 + 1/(c^2*x^2)]*x*(-1 + 2*c^2*x^2))*ArcCsch[c*x]^2 - 9*b^3*ArcCs
ch[c*x]^3)/(27*x^3)

________________________________________________________________________________________

Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsch(c*x))^3/x^4,x)

[Out]

int((a+b*arccsch(c*x))^3/x^4,x)

________________________________________________________________________________________

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*arccsch(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*arccsch(c*x)/x^3) - 1/3*b^3*log(s
qrt(c^2*x^2 + 1) + 1)^3/x^3 - 1/3*a^3/x^3 - integrate((b^3*log(c)^3 - 3*a*b^2*log(c)^2 + (b^3*c^2*x^2 + b^3)*l
og(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)
*x^2)*log(x)^2 + (3*b^3*log(c) - 3*a*b^2 + 3*(b^3*c^2*log(c) - a*b^2*c^2)*x^2 + 3*(b^3*c^2*x^2 + b^3)*log(x) +
sqrt(c^2*x^2 + 1)*(3*b^3*log(c) - 3*a*b^2 + (b^3*c^2*(3*log(c) - 1) - 3*a*b^2*c^2)*x^2 + 3*(b^3*c^2*x^2 + b^3
)*log(x)))*log(sqrt(c^2*x^2 + 1) + 1)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*l
og(c))*x^2)*log(x) - 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2
*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x) + (b^3*log(c)^2 - 2*a*
b^2*log(c) + (b^3*c^2*log(c)^2 - 2*a*b^2*c^2*log(c))*x^2 + (b^3*c^2*x^2 + b^3)*log(x)^2 + 2*(b^3*log(c) - a*b^
2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x))*sqrt(c^2*x^2 + 1))*log(sqrt(c^2*x^2 + 1) + 1) + (b^3*log(c)^3 -
3*a*b^2*log(c)^2 + (b^3*c^2*x^2 + b^3)*log(x)^3 + (b^3*c^2*log(c)^3 - 3*a*b^2*c^2*log(c)^2)*x^2 + 3*(b^3*log(c
) - a*b^2 + (b^3*c^2*log(c) - a*b^2*c^2)*x^2)*log(x)^2 + 3*(b^3*log(c)^2 - 2*a*b^2*log(c) + (b^3*c^2*log(c)^2
- 2*a*b^2*c^2*log(c))*x^2)*log(x))*sqrt(c^2*x^2 + 1))/(c^2*x^6 + x^4 + (c^2*x^6 + x^4)*sqrt(c^2*x^2 + 1)), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (144) = 288\).
time = 0.44, size = 301, normalized size = 1.81 \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*arccsch(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)*sq
rt((c^2*x^2 + 1)/(c^2*x^2)))/x^3

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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*arccsch(c*x))^3/x^4,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\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*asinh(1/(c*x)))^3/x^4,x)

[Out]

int((a + b*asinh(1/(c*x)))^3/x^4, x)

________________________________________________________________________________________