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

Optimal result

Integrand size = 10, antiderivative size = 203 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=-\frac {\left (2+17 a^2\right ) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^4}-\frac {x^2 \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{12 b^2}+\frac {a (a+b x) \sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{3 b^4}-\frac {a^4 \text {sech}^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \text {sech}^{-1}(a+b x)+\frac {a \left (1+2 a^2\right ) \arctan \left (\frac {\sqrt {\frac {1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{2 b^4} \] Output:

-1/12*(17*a^2+2)*((-b*x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)/b^4-1/12*x^2*((-b* 
x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)/b^2+1/3*a*(b*x+a)*((-b*x-a+1)/(b*x+a+1)) 
^(1/2)*(b*x+a+1)/b^4-1/4*a^4*arcsech(b*x+a)/b^4+1/4*x^4*arcsech(b*x+a)+1/2 
*a*(2*a^2+1)*arctan(((-b*x-a+1)/(b*x+a+1))^(1/2)*(b*x+a+1)/(b*x+a))/b^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.11 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=-\frac {\sqrt {-\frac {-1+a+b x}{1+a+b x}} \left (2+2 a+13 a^2+13 a^3+\left (2-4 a+9 a^2\right ) b x+(1-3 a) b^2 x^2+b^3 x^3\right )-3 b^4 x^4 \text {sech}^{-1}(a+b x)-3 a^4 \log (a+b x)+3 a^4 \log \left (1+\sqrt {-\frac {-1+a+b x}{1+a+b x}}+a \sqrt {-\frac {-1+a+b x}{1+a+b x}}+b x \sqrt {-\frac {-1+a+b x}{1+a+b x}}\right )+6 i a \left (1+2 a^2\right ) \log \left (-2 i (a+b x)+2 \sqrt {-\frac {-1+a+b x}{1+a+b x}} (1+a+b x)\right )}{12 b^4} \] Input:

Integrate[x^3*ArcSech[a + b*x],x]
 

Output:

-1/12*(Sqrt[-((-1 + a + b*x)/(1 + a + b*x))]*(2 + 2*a + 13*a^2 + 13*a^3 + 
(2 - 4*a + 9*a^2)*b*x + (1 - 3*a)*b^2*x^2 + b^3*x^3) - 3*b^4*x^4*ArcSech[a 
 + b*x] - 3*a^4*Log[a + b*x] + 3*a^4*Log[1 + Sqrt[-((-1 + a + b*x)/(1 + a 
+ b*x))] + a*Sqrt[-((-1 + a + b*x)/(1 + a + b*x))] + b*x*Sqrt[-((-1 + a + 
b*x)/(1 + a + b*x))]] + (6*I)*a*(1 + 2*a^2)*Log[(-2*I)*(a + b*x) + 2*Sqrt[ 
-((-1 + a + b*x)/(1 + a + b*x))]*(1 + a + b*x)])/b^4
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6875, 25, 5991, 3042, 4269, 3042, 4536, 2009}

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.

\(\displaystyle \int x^3 \text {sech}^{-1}(a+b x) \, dx\)

\(\Big \downarrow \) 6875

\(\displaystyle -\frac {\int b^3 x^3 (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)d\text {sech}^{-1}(a+b x)}{b^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int -b^3 x^3 (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1) \text {sech}^{-1}(a+b x)d\text {sech}^{-1}(a+b x)}{b^4}\)

\(\Big \downarrow \) 5991

\(\displaystyle -\frac {\frac {1}{4} \int b^4 x^4d\text {sech}^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \text {sech}^{-1}(a+b x)}{b^4}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4269

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \int -b x \left (3 a^3+8 (a+b x)^2 a-\left (9 a^2+2\right ) (a+b x)\right )d\text {sech}^{-1}(a+b x)+\frac {1}{3} b^2 x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )-\frac {1}{4} b^4 x^4 \text {sech}^{-1}(a+b x)}{b^4}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {-\frac {1}{4} b^4 x^4 \text {sech}^{-1}(a+b x)+\frac {1}{4} \left (\frac {1}{3} b^2 x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)+\frac {1}{3} \int \left (a-\csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right ) \left (3 a^3+8 \csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )^2 a+\left (-9 a^2-2\right ) \csc \left (i \text {sech}^{-1}(a+b x)+\frac {\pi }{2}\right )\right )d\text {sech}^{-1}(a+b x)\right )}{b^4}\)

\(\Big \downarrow \) 4536

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (6 a^4-12 \left (2 a^2+1\right ) (a+b x) a+2 \left (17 a^2+2\right ) (a+b x)^2\right )d\text {sech}^{-1}(a+b x)-4 a (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )+\frac {1}{3} b^2 x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )-\frac {1}{4} b^4 x^4 \text {sech}^{-1}(a+b x)}{b^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (6 a^4 \text {sech}^{-1}(a+b x)-12 \left (2 a^2+1\right ) a \arctan \left (\frac {\sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )+2 \left (17 a^2+2\right ) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )-4 a (a+b x) \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )+\frac {1}{3} b^2 x^2 \sqrt {\frac {-a-b x+1}{a+b x+1}} (a+b x+1)\right )-\frac {1}{4} b^4 x^4 \text {sech}^{-1}(a+b x)}{b^4}\)

Input:

Int[x^3*ArcSech[a + b*x],x]
 

Output:

-((-1/4*(b^4*x^4*ArcSech[a + b*x]) + ((b^2*x^2*Sqrt[(1 - a - b*x)/(1 + a + 
 b*x)]*(1 + a + b*x))/3 + (-4*a*(a + b*x)*Sqrt[(1 - a - b*x)/(1 + a + b*x) 
]*(1 + a + b*x) + (2*(2 + 17*a^2)*Sqrt[(1 - a - b*x)/(1 + a + b*x)]*(1 + a 
 + b*x) + 6*a^4*ArcSech[a + b*x] - 12*a*(1 + 2*a^2)*ArcTan[(Sqrt[(1 - a - 
b*x)/(1 + a + b*x)]*(1 + a + b*x))/(a + b*x)])/2)/3)/4)/b^4)
 

Defintions of rubi rules used

rule 25

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 4269

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C 
ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[1/(n - 1) 
   Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) + 3* 
a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] / 
; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]
 

rule 4536

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-b)*C*Csc[e + 
 f*x]*(Cot[e + f*x]/(2*f)), x] + Simp[1/2   Int[Simp[2*A*a + (2*B*a + b*(2* 
A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a 
, b, e, f, A, B, C}, x]
 

rule 5991

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

rule 6875

Int[((a_.) + ArcSech[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[-(d^(m + 1))^(-1)   Subst[Int[(a + b*x)^p*Sech[x]*T 
anh[x]*(d*e - c*f + f*Sech[x])^m, x], x, ArcSech[c + d*x]], x] /; FreeQ[{a, 
 b, c, d, e, f}, x] && IGtQ[p, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.02 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arcsech}\left (b x +a \right ) a^{4}}{4}-\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+12 a^{3} \arcsin \left (b x +a \right )+18 a^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+6 a \arcsin \left (b x +a \right )+2 \sqrt {1-\left (b x +a \right )^{2}}\right )}{12 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{4}}\)	\(250\)
default	\(\frac {\frac {\operatorname {arcsech}\left (b x +a \right ) a^{4}}{4}-\operatorname {arcsech}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arcsech}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arcsech}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arcsech}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 a^{4} \operatorname {arctanh}\left (\frac {1}{\sqrt {1-\left (b x +a \right )^{2}}}\right )+12 a^{3} \arcsin \left (b x +a \right )+18 a^{2} \sqrt {1-\left (b x +a \right )^{2}}-6 a \left (b x +a \right ) \sqrt {1-\left (b x +a \right )^{2}}+\sqrt {1-\left (b x +a \right )^{2}}\, \left (b x +a \right )^{2}+6 a \arcsin \left (b x +a \right )+2 \sqrt {1-\left (b x +a \right )^{2}}\right )}{12 \sqrt {1-\left (b x +a \right )^{2}}}}{b^{4}}\)	\(250\)
parts	\(\frac {x^{4} \operatorname {arcsech}\left (b x +a \right )}{4}-\frac {\sqrt {-\frac {b x +a -1}{b x +a}}\, \left (b x +a \right ) \sqrt {\frac {b x +a +1}{b x +a}}\, \left (3 \,\operatorname {csgn}\left (b \right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}}\right ) a^{4}+\operatorname {csgn}\left (b \right ) b^{2} x^{2} \sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}-4 \sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}\, \operatorname {csgn}\left (b \right ) a b x +13 \sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}\, \operatorname {csgn}\left (b \right ) a^{2}+12 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}}\right ) a^{3}+2 \sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}\, \operatorname {csgn}\left (b \right )+6 \arctan \left (\frac {\operatorname {csgn}\left (b \right ) \left (b x +a \right )}{\sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}}\right ) a \right ) \operatorname {csgn}\left (b \right )}{12 b^{4} \sqrt {-b^{2} x^{2}-2 b x a -a^{2}+1}}\)	\(296\)

Input:

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

Output:

1/b^4*(1/4*arcsech(b*x+a)*a^4-arcsech(b*x+a)*a^3*(b*x+a)+3/2*arcsech(b*x+a 
)*a^2*(b*x+a)^2-arcsech(b*x+a)*a*(b*x+a)^3+1/4*arcsech(b*x+a)*(b*x+a)^4-1/ 
12*(-(b*x+a-1)/(b*x+a))^(1/2)*(b*x+a)*((b*x+a+1)/(b*x+a))^(1/2)*(3*a^4*arc 
tanh(1/(1-(b*x+a)^2)^(1/2))+12*a^3*arcsin(b*x+a)+18*a^2*(1-(b*x+a)^2)^(1/2 
)-6*a*(b*x+a)*(1-(b*x+a)^2)^(1/2)+(1-(b*x+a)^2)^(1/2)*(b*x+a)^2+6*a*arcsin 
(b*x+a)+2*(1-(b*x+a)^2)^(1/2))/(1-(b*x+a)^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.70 \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\frac {6 \, b^{4} x^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) - 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) + 3 \, a^{4} \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) + 12 \, {\left (2 \, a^{3} + a\right )} \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 2 \, {\left (b^{3} x^{3} - 3 \, a b^{2} x^{2} + 13 \, a^{3} + {\left (9 \, a^{2} + 2\right )} b x + 2 \, a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{24 \, b^{4}} \] Input:

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

Output:

1/24*(6*b^4*x^4*log(((b*x + a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^ 
2 + 2*a*b*x + a^2)) + 1)/(b*x + a)) - 3*a^4*log(((b*x + a)*sqrt(-(b^2*x^2 
+ 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) + 1)/x) + 3*a^4*log(((b*x 
+ a)*sqrt(-(b^2*x^2 + 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)) - 1)/x 
) + 12*(2*a^3 + a)*arctan((b^2*x^2 + 2*a*b*x + a^2)*sqrt(-(b^2*x^2 + 2*a*b 
*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2))/(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 
2*(b^3*x^3 - 3*a*b^2*x^2 + 13*a^3 + (9*a^2 + 2)*b*x + 2*a)*sqrt(-(b^2*x^2 
+ 2*a*b*x + a^2 - 1)/(b^2*x^2 + 2*a*b*x + a^2)))/b^4
 

Sympy [F]

\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int x^{3} \operatorname {asech}{\left (a + b x \right )}\, dx \] Input:

integrate(x**3*asech(b*x+a),x)
 

Output:

Integral(x**3*asech(a + b*x), x)
 

Maxima [F]

\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arsech}\left (b x + a\right ) \,d x } \] Input:

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

Output:

1/8*(2*b^4*x^4*log(sqrt(b*x + a + 1)*sqrt(-b*x - a + 1)*b*x + sqrt(b*x + a 
 + 1)*sqrt(-b*x - a + 1)*a + b*x + a) - 2*b^4*x^4*log(b*x + a) - b^2*x^2 + 
 6*a*b*x - (a^4 + 4*a^3 + 6*a^2 + 4*a + 1)*log(b*x + a + 1) - 2*(b^4*x^4 - 
 a^4)*log(b*x + a) - (a^4 - 4*a^3 + 6*a^2 - 4*a + 1)*log(-b*x - a + 1))/b^ 
4 + integrate(1/4*(b^2*x^5 + a*b*x^4)/(b^2*x^2 + 2*a*b*x + a^2 + (b^2*x^2 
+ 2*a*b*x + a^2 - 1)*e^(1/2*log(b*x + a + 1) + 1/2*log(-b*x - a + 1)) - 1) 
, x)
 

Giac [F]

\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arsech}\left (b x + a\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int x^3\,\mathrm {acosh}\left (\frac {1}{a+b\,x}\right ) \,d x \] Input:

int(x^3*acosh(1/(a + b*x)),x)
 

Output:

int(x^3*acosh(1/(a + b*x)), x)
 

Reduce [F]

\[ \int x^3 \text {sech}^{-1}(a+b x) \, dx=\int \mathit {asech} \left (b x +a \right ) x^{3}d x \] Input:

int(x^3*asech(b*x+a),x)
 

Output:

int(asech(a + b*x)*x**3,x)