3.15 \(\int x^3 (a+b \text{csch}^{-1}(c x))^2 \, dx\)
Optimal. Leaf size=105 \[ \frac{b x^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{6 c}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 x^2}{12 c^2}-\frac{b^2 \log (x)}{3 c^4} \]
[Out]
(b^2*x^2)/(12*c^2) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x]))/(3*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3*(
a + b*ArcCsch[c*x]))/(6*c) + (x^4*(a + b*ArcCsch[c*x])^2)/4 - (b^2*Log[x])/(3*c^4)
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Rubi [A] time = 0.118674, antiderivative size = 105, normalized size of antiderivative = 1.,
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 =
{6286, 5452, 4185, 4184, 3475} \[ \frac{b x^3 \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{6 c}-\frac{b x \sqrt{\frac{1}{c^2 x^2}+1} \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 x^2}{12 c^2}-\frac{b^2 \log (x)}{3 c^4} \]
Antiderivative was successfully verified.
[In]
Int[x^3*(a + b*ArcCsch[c*x])^2,x]
[Out]
(b^2*x^2)/(12*c^2) - (b*Sqrt[1 + 1/(c^2*x^2)]*x*(a + b*ArcCsch[c*x]))/(3*c^3) + (b*Sqrt[1 + 1/(c^2*x^2)]*x^3*(
a + b*ArcCsch[c*x]))/(6*c) + (x^4*(a + b*ArcCsch[c*x])^2)/4 - (b^2*Log[x])/(3*c^4)
Rule 6286
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])
Rule 5452
Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Si
mp[((c + d*x)^m*Csch[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csch[a + b*x]^n, x], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Rule 4185
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int x^3 \left (a+b \text{csch}^{-1}(c x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \text{csch}^4(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{c^4}\\ &=\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{b \operatorname{Subst}\left (\int (a+b x) \text{csch}^4(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{2 c^4}\\ &=\frac{b^2 x^2}{12 c^2}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{6 c}+\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b \operatorname{Subst}\left (\int (a+b x) \text{csch}^2(x) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^4}\\ &=\frac{b^2 x^2}{12 c^2}-\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{6 c}+\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2+\frac{b^2 \operatorname{Subst}\left (\int \coth (x) \, dx,x,\text{csch}^{-1}(c x)\right )}{3 c^4}\\ &=\frac{b^2 x^2}{12 c^2}-\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x \left (a+b \text{csch}^{-1}(c x)\right )}{3 c^3}+\frac{b \sqrt{1+\frac{1}{c^2 x^2}} x^3 \left (a+b \text{csch}^{-1}(c x)\right )}{6 c}+\frac{1}{4} x^4 \left (a+b \text{csch}^{-1}(c x)\right )^2-\frac{b^2 \log (x)}{3 c^4}\\ \end{align*}
Mathematica [A] time = 0.205312, size = 122, normalized size = 1.16 \[ \frac{c x \left (3 a^2 c^3 x^3+2 a b \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 x^2-2\right )+b^2 c x\right )+2 b c x \text{csch}^{-1}(c x) \left (3 a c^3 x^3+b \sqrt{\frac{1}{c^2 x^2}+1} \left (c^2 x^2-2\right )\right )+3 b^2 c^4 x^4 \text{csch}^{-1}(c x)^2-4 b^2 \log (x)}{12 c^4} \]
Antiderivative was successfully verified.
[In]
Integrate[x^3*(a + b*ArcCsch[c*x])^2,x]
[Out]
(c*x*(b^2*c*x + 3*a^2*c^3*x^3 + 2*a*b*Sqrt[1 + 1/(c^2*x^2)]*(-2 + c^2*x^2)) + 2*b*c*x*(3*a*c^3*x^3 + b*Sqrt[1
+ 1/(c^2*x^2)]*(-2 + c^2*x^2))*ArcCsch[c*x] + 3*b^2*c^4*x^4*ArcCsch[c*x]^2 - 4*b^2*Log[x])/(12*c^4)
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Maple [F] time = 0.187, size = 0, normalized size = 0. \begin{align*} \int{x}^{3} \left ( a+b{\rm arccsch} \left (cx\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^3*(a+b*arccsch(c*x))^2,x)
[Out]
int(x^3*(a+b*arccsch(c*x))^2,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*arccsch(c*x))^2,x, algorithm="maxima")
[Out]
1/4*a^2*x^4 + 1/6*(3*x^4*arccsch(c*x) + (c^2*x^3*(1/(c^2*x^2) + 1)^(3/2) - 3*x*sqrt(1/(c^2*x^2) + 1))/c^3)*a*b
+ 1/288*(72*x^4*log(sqrt(c^2*x^2 + 1) + 1)^2 + 1152*c^2*integrate(1/2*x^5*log(x)/(sqrt(c^2*x^2 + 1)*c^2*x^2 +
c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) - 1152*c^2*integrate(1/2*x^5*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2
*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x)*log(c) + 576*c^2*integrate(1/2*sqrt(c^2*x^2 + 1)*x^5*
log(x)^2/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x) - 1152*c^2*integrate(1/2*sqrt(c^2*x
^2 + 1)*x^5*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x
) + 576*c^2*integrate(1/2*x^5*log(x)^2/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x) - 115
2*c^2*integrate(1/2*x^5*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2
+ 1) + 1), x) + 1152*integrate(1/2*x^3*log(x)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x
)*log(c) - 1152*integrate(1/2*x^3*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x
^2 + 1) + 1), x)*log(c) - 24*(6*c^2*x^2 - 3*(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(3/2) - 12*sqrt(c^2*x^2 + 1) + 6
)*log(c)^2/c^4 - 48*(3*c^2*x^2 - 2*(c^2*x^2 + 1)^(3/2) + 6*sqrt(c^2*x^2 + 1) - 3*log(c^2*x^2 + 1) + 3)*log(c)^
2/c^4 + 144*(c^2*x^2 - 2*sqrt(c^2*x^2 + 1) + 1)*log(c)^2/c^4 + 144*(2*sqrt(c^2*x^2 + 1) - log(c^2*x^2 + 1))*lo
g(c)^2/c^4 - 48*(6*c^2*x^2 - 3*(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(3/2) - 12*sqrt(c^2*x^2 + 1) + 6)*log(c)*log(
x)/c^4 + 288*(c^2*x^2 - 2*sqrt(c^2*x^2 + 1) + 1)*log(c)*log(x)/c^4 + 48*(6*c^2*x^2 - 3*(c^2*x^2 + 1)^2 + 4*(c^
2*x^2 + 1)^(3/2) - 12*sqrt(c^2*x^2 + 1) + 6)*log(c)*log(sqrt(c^2*x^2 + 1) + 1)/c^4 - 288*(c^2*x^2 - 2*sqrt(c^2
*x^2 + 1) + 1)*log(c)*log(sqrt(c^2*x^2 + 1) + 1)/c^4 + 4*(18*c^2*x^2 - 9*(c^2*x^2 + 1)^2 + 16*(c^2*x^2 + 1)^(3
/2) - 96*sqrt(c^2*x^2 + 1) + 66*log(sqrt(c^2*x^2 + 1) + 1) - 30*log(sqrt(c^2*x^2 + 1) - 1) + 18)*log(c)/c^4 +
4*(6*c^2*x^2 + 9*(c^2*x^2 + 1)^2 - 28*(c^2*x^2 + 1)^(3/2) + 132*sqrt(c^2*x^2 + 1) - 132*log(sqrt(c^2*x^2 + 1)
+ 1) + 6)*log(c)/c^4 - 144*(c^2*x^2 - 4*sqrt(c^2*x^2 + 1) + 3*log(sqrt(c^2*x^2 + 1) + 1) - log(sqrt(c^2*x^2 +
1) - 1) + 1)*log(c)/c^4 + 144*(c^2*x^2 - 6*sqrt(c^2*x^2 + 1) + 6*log(sqrt(c^2*x^2 + 1) + 1) + 1)*log(c)/c^4 +
12*(6*c^2*x^2 - 3*(c^2*x^2 + 1)^2 + 4*(c^2*x^2 + 1)^(3/2) - 12*sqrt(c^2*x^2 + 1) + 6)*log(sqrt(c^2*x^2 + 1) +
1)/c^4 + (6*c^2*x^2 + 9*(c^2*x^2 + 1)^2 - 28*(c^2*x^2 + 1)^(3/2) + 132*sqrt(c^2*x^2 + 1) - 132*log(sqrt(c^2*x^
2 + 1) + 1) + 6)/c^4 + 576*integrate(1/2*sqrt(c^2*x^2 + 1)*x^3*log(x)^2/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 +
sqrt(c^2*x^2 + 1) + 1), x) - 1152*integrate(1/2*sqrt(c^2*x^2 + 1)*x^3*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt
(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x) + 576*integrate(1/2*x^3*log(x)^2/(sqrt(c^2*x^2 +
1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x) - 1152*integrate(1/2*x^3*log(x)*log(sqrt(c^2*x^2 + 1) + 1)/(
sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sqrt(c^2*x^2 + 1) + 1), x))*b^2
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Fricas [B] time = 2.35634, size = 603, normalized size = 5.74 \begin{align*} \frac{3 \, b^{2} c^{4} x^{4} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, a^{2} c^{4} x^{4} + 6 \, a b c^{4} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 6 \, a b c^{4} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + b^{2} c^{2} x^{2} - 4 \, b^{2} \log \left (x\right ) + 2 \,{\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{4} +{\left (b^{2} c^{3} x^{3} - 2 \, 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 ) + 2 \,{\left (a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*arccsch(c*x))^2,x, algorithm="fricas")
[Out]
1/12*(3*b^2*c^4*x^4*log((c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) + 1)/(c*x))^2 + 3*a^2*c^4*x^4 + 6*a*b*c^4*log(c*x*s
qrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x + 1) - 6*a*b*c^4*log(c*x*sqrt((c^2*x^2 + 1)/(c^2*x^2)) - c*x - 1) + b^2*c^2
*x^2 - 4*b^2*log(x) + 2*(3*a*b*c^4*x^4 - 3*a*b*c^4 + (b^2*c^3*x^3 - 2*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*(a*b*c^3*x^3 - 2*a*b*c*x)*sqrt((c^2*x^2 + 1)/(c^2*x^2))
)/c^4
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (a + b \operatorname{acsch}{\left (c x \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**3*(a+b*acsch(c*x))**2,x)
[Out]
Integral(x**3*(a + b*acsch(c*x))**2, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}^{2} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^3*(a+b*arccsch(c*x))^2,x, algorithm="giac")
[Out]
integrate((b*arccsch(c*x) + a)^2*x^3, x)