3.1.15 \(\int x^3 (a+b \text {csch}^{-1}(c x))^2 \, dx\) [15]

3.1.15.1 Optimal result

Integrand size = 14, antiderivative size = 105 \[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\frac {b^2 x^2}{12 c^2}-\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x \left (a+b \text {csch}^{-1}(c x)\right )}{3 c^3}+\frac {b \sqrt {1+\frac {1}{c^2 x^2}} x^3 \left (a+b \text {csch}^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2-\frac {b^2 \log (x)}{3 c^4} \]

output

1/12*b^2*x^2/c^2+1/4*x^4*(a+b*arccsch(c*x))^2-1/3*b^2*ln(x)/c^4-1/3*b*x*(a 
+b*arccsch(c*x))*(1+1/c^2/x^2)^(1/2)/c^3+1/6*b*x^3*(a+b*arccsch(c*x))*(1+1 
/c^2/x^2)^(1/2)/c
 

3.1.15.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.16 \[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\frac {c x \left (b^2 c x+3 a^2 c^3 x^3+2 a b \sqrt {1+\frac {1}{c^2 x^2}} \left (-2+c^2 x^2\right )\right )+2 b c x \left (3 a c^3 x^3+b \sqrt {1+\frac {1}{c^2 x^2}} \left (-2+c^2 x^2\right )\right ) \text {csch}^{-1}(c x)+3 b^2 c^4 x^4 \text {csch}^{-1}(c x)^2-4 b^2 \log (x)}{12 c^4} \]

input

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

output

(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)
 

3.1.15.3 Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {6840, 5975, 3042, 4673, 25, 3042, 25, 4672, 26, 3042, 26, 3956}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx\)

\(\Big \downarrow \) 6840

\(\displaystyle -\frac {\int c^5 \sqrt {1+\frac {1}{c^2 x^2}} x^5 \left (a+b \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)}{c^4}\)

\(\Big \downarrow \) 5975

\(\displaystyle -\frac {\frac {1}{2} b \int c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \int \left (a+b \text {csch}^{-1}(c x)\right ) \csc \left (i \text {csch}^{-1}(c x)\right )^4d\text {csch}^{-1}(c x)}{c^4}\)

\(\Big \downarrow \) 4673

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {2}{3} \int -c^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\frac {1}{2} b \left (-\frac {2}{3} \int c^2 x^2 \left (a+b \text {csch}^{-1}(c x)\right )d\text {csch}^{-1}(c x)-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \left (-\frac {2}{3} \int -\left (\left (a+b \text {csch}^{-1}(c x)\right ) \csc \left (i \text {csch}^{-1}(c x)\right )^2\right )d\text {csch}^{-1}(c x)-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \int \left (a+b \text {csch}^{-1}(c x)\right ) \csc \left (i \text {csch}^{-1}(c x)\right )^2d\text {csch}^{-1}(c x)-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (c x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-i b \int -i c \sqrt {1+\frac {1}{c^2 x^2}} xd\text {csch}^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {2}{3} \left (c x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-b \int c \sqrt {1+\frac {1}{c^2 x^2}} xd\text {csch}^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (c x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-b \int -i \tan \left (i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2+\frac {1}{2} b \left (\frac {2}{3} \left (c x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )+i b \int \tan \left (i \text {csch}^{-1}(c x)+\frac {\pi }{2}\right )d\text {csch}^{-1}(c x)\right )-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )}{c^4}\)

\(\Big \downarrow \) 3956

\(\displaystyle -\frac {\frac {1}{2} b \left (\frac {2}{3} \left (c x \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-b \log \left (\frac {1}{c x}\right )\right )-\frac {1}{3} c^3 x^3 \sqrt {\frac {1}{c^2 x^2}+1} \left (a+b \text {csch}^{-1}(c x)\right )-\frac {1}{6} b c^2 x^2\right )-\frac {1}{4} c^4 x^4 \left (a+b \text {csch}^{-1}(c x)\right )^2}{c^4}\)

input

Int[x^3*(a + b*ArcCsch[c*x])^2,x]
 

output

-((-1/4*(c^4*x^4*(a + b*ArcCsch[c*x])^2) + (b*(-1/6*(b*c^2*x^2) - (c^3*Sqr 
t[1 + 1/(c^2*x^2)]*x^3*(a + b*ArcCsch[c*x]))/3 + (2*(c*Sqrt[1 + 1/(c^2*x^2 
)]*x*(a + b*ArcCsch[c*x]) - b*Log[1/(c*x)]))/3))/2)/c^4)
 

3.1.15.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[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 4673

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] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S 
imp[b^2*((n - 2)/(n - 1))   Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) 
 /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
 

rule 5975

Int[Coth[(a_.) + (b_.)*(x_)]^(p_.)*Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csch[a + b*x]^n/(b*n)) 
, x] + Simp[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 6840

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ 
-(c^(m + 1))^(-1)   Subst[Int[(a + b*x)^n*Csch[x]^(m + 1)*Coth[x], x], x, A 
rcCsch[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G 
tQ[n, 0] || LtQ[m, -1])
 

3.1.15.4 Maple [F]

\[\int x^{3} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{2}d x\]

input

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

output

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

3.1.15.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (91) = 182\).

Time = 0.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.59 \[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\frac {3 \, b^{2} c^{4} x^{4} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + 3 \, a^{2} c^{4} x^{4} + 6 \, a b c^{4} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - 6 \, a b c^{4} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + b^{2} c^{2} x^{2} - 4 \, b^{2} \log \left (x\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{4} + {\left (b^{2} c^{3} x^{3} - 2 \, b^{2} c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, {\left (a b c^{3} x^{3} - 2 \, a b c x\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{12 \, c^{4}} \]

input

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

output

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*sqrt((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
 

3.1.15.6 Sympy [F]

\[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\int x^{3} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{2}\, dx \]

input

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

output

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

3.1.15.7 Maxima [F]

\[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

input

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

output

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*lo 
g(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) - 1 
152*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) + 57 
6*c^2*integrate(1/2*x^5*log(x)^2/(sqrt(c^2*x^2 + 1)*c^2*x^2 + c^2*x^2 + sq 
rt(c^2*x^2 + 1) + 1), x) - 1152*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))*log(c)^2/c^4 - 48*(6*c^2*x^2 - 3*(c^2*x^2 + 1)^2 ...
 

3.1.15.8 Giac [F]

\[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{2} x^{3} \,d x } \]

input

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

output

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

3.1.15.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \left (a+b \text {csch}^{-1}(c x)\right )^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]

input

int(x^3*(a + b*asinh(1/(c*x)))^2,x)
 

output

int(x^3*(a + b*asinh(1/(c*x)))^2, x)