\(\int x \csc ^{-1}(a+b x) \, dx\) [20]

Optimal result

Integrand size = 8, antiderivative size = 79 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{2 b^2}-\frac {a^2 \csc ^{-1}(a+b x)}{2 b^2}+\frac {1}{2} x^2 \csc ^{-1}(a+b x)-\frac {a \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{b^2} \] Output:

1/2*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^2-1/2*a^2*arccsc(b*x+a)/b^2+1/2*x^2*ar 
ccsc(b*x+a)-a*arctanh((1-1/(b*x+a)^2)^(1/2))/b^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.39 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {(a+b x) \sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}+b^2 x^2 \csc ^{-1}(a+b x)-a^2 \arcsin \left (\frac {1}{a+b x}\right )-2 a \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{2 b^2} \] Input:

Integrate[x*ArcCsc[a + b*x],x]
 

Output:

((a + b*x)*Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2] + b^2*x^2*ArcC 
sc[a + b*x] - a^2*ArcSin[(a + b*x)^(-1)] - 2*a*Log[(a + b*x)*(1 + Sqrt[(-1 
 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(2*b^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {5782, 25, 4927, 3042, 4260, 3042, 4254, 24, 4257}

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.

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

\(\Big \downarrow \) 5782

\(\displaystyle -\frac {\int b x (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int -b x (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^2}\)

\(\Big \downarrow \) 4927

\(\displaystyle -\frac {\frac {1}{2} \int b^2 x^2d\csc ^{-1}(a+b x)-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4260

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4254

\(\displaystyle -\frac {\frac {1}{2} \left (-\int 1d\left ((a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-2 a \int \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {\frac {1}{2} \left (-2 a \int \csc \left (\csc ^{-1}(a+b x)\right )d\csc ^{-1}(a+b x)+a^2 \csc ^{-1}(a+b x)-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\)

\(\Big \downarrow \) 4257

\(\displaystyle -\frac {\frac {1}{2} \left (a^2 \csc ^{-1}(a+b x)+2 a \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )-(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{2} b^2 x^2 \csc ^{-1}(a+b x)}{b^2}\)

Input:

Int[x*ArcCsc[a + b*x],x]
 

Output:

-((-1/2*(b^2*x^2*ArcCsc[a + b*x]) + (-((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) 
 + a^2*ArcCsc[a + b*x] + 2*a*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/2)/b^2)
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

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

rule 3042

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

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4257

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

rule 4260

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + 
 (Simp[2*a*b   Int[Csc[c + d*x], x], x] + Simp[b^2   Int[Csc[c + d*x]^2, x] 
, x]) /; FreeQ[{a, b, c, d}, x]
 

rule 4927

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

rule 5782

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.37

method	result	size
derivativedivides	\(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (2 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}}}{b^{2}}\)	\(108\)
default	\(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (2 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-\sqrt {\left (b x +a \right )^{2}-1}\right )}{2 \left (b x +a \right ) \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}}}{b^{2}}\)	\(108\)
parts	\(\frac {x^{2} \operatorname {arccsc}\left (b x +a \right )}{2}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {b^{2}}+2 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b -\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\right )}{2 b^{2} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\)	\(177\)

Input:

int(x*arccsc(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/b^2*(1/2*arccsc(b*x+a)*(b*x+a)^2-arccsc(b*x+a)*a*(b*x+a)-1/2*((b*x+a)^2- 
1)^(1/2)*(2*a*ln(b*x+a+((b*x+a)^2-1)^(1/2))-((b*x+a)^2-1)^(1/2))/(b*x+a)/( 
((b*x+a)^2-1)/(b*x+a)^2)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.29 \[ \int x \csc ^{-1}(a+b x) \, dx=\frac {b^{2} x^{2} \operatorname {arccsc}\left (b x + a\right ) + 2 \, a^{2} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 2 \, a \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{2 \, b^{2}} \] Input:

integrate(x*arccsc(b*x+a),x, algorithm="fricas")
 

Output:

1/2*(b^2*x^2*arccsc(b*x + a) + 2*a^2*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a* 
b*x + a^2 - 1)) + 2*a*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 
sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^2
 

Sympy [F]

\[ \int x \csc ^{-1}(a+b x) \, dx=\int x \operatorname {acsc}{\left (a + b x \right )}\, dx \] Input:

integrate(x*acsc(b*x+a),x)
 

Output:

Integral(x*acsc(a + b*x), x)
 

Maxima [F]

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

integrate(x*arccsc(b*x+a),x, algorithm="maxima")
 

Output:

1/2*x^2*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + integrate(1/2*(b 
^2*x^3 + a*b*x^2)*e^(1/2*log(b*x + a + 1) + 1/2*log(b*x + a - 1))/(b^2*x^2 
 + 2*a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 - 1)*e^(log(b*x + a + 1) + log 
(b*x + a - 1)) - 1), x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.73 \[ \int x \csc ^{-1}(a+b x) \, dx=-\frac {1}{4} \, b {\left (\frac {2 \, {\left (b x + a\right )}^{2} {\left (\frac {2 \, a}{b x + a} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{3}} - \frac {{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 4 \, a \log \left (\frac {1}{2} \, {\left | b x + a \right |} {\left | -2 \, \sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 2 \right |}\right ) - \frac {1}{{\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}}}{b^{3}}\right )} \] Input:

integrate(x*arccsc(b*x+a),x, algorithm="giac")
 

Output:

-1/4*b*(2*(b*x + a)^2*(2*a/(b*x + a) - 1)*arcsin(-1/((b*x + a)*(a/(b*x + a 
) - 1) - a))/b^3 - ((b*x + a)*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 4*a*log(1/2 
*abs(b*x + a)*abs(-2*sqrt(-1/(b*x + a)^2 + 1) + 2)) - 1/((b*x + a)*(sqrt(- 
1/(b*x + a)^2 + 1) - 1)))/b^3)
 

Mupad [F(-1)]

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

int(x*asin(1/(a + b*x)),x)
 

Output:

int(x*asin(1/(a + b*x)), x)
 

Reduce [F]

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

int(x*acsc(b*x+a),x)
 

Output:

int(acsc(a + b*x)*x,x)