\(\int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx\) [49]

Optimal result

Integrand size = 16, antiderivative size = 102 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\frac {b \csc ^{-1}(c x)}{d e}-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \text {arctanh}\left (\frac {c^2 d+\frac {e}{x}}{c \sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{d \sqrt {c^2 d^2-e^2}} \] Output:

b*arccsc(c*x)/d/e-(a+b*arccsc(c*x))/e/(e*x+d)+b*arctanh((c^2*d+e/x)/c/(c^2 
*d^2-e^2)^(1/2)/(1-1/c^2/x^2)^(1/2))/d/(c^2*d^2-e^2)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=-\frac {a}{e (d+e x)}-\frac {b \csc ^{-1}(c x)}{e (d+e x)}+\frac {b \arcsin \left (\frac {1}{c x}\right )}{d e}+\frac {b \log (d+e x)}{d \sqrt {c^2 d^2-e^2}}-\frac {b \log \left (e+c \left (c d-\sqrt {c^2 d^2-e^2} \sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )}{d \sqrt {c^2 d^2-e^2}} \] Input:

Integrate[(a + b*ArcCsc[c*x])/(d + e*x)^2,x]
 

Output:

-(a/(e*(d + e*x))) - (b*ArcCsc[c*x])/(e*(d + e*x)) + (b*ArcSin[1/(c*x)])/( 
d*e) + (b*Log[d + e*x])/(d*Sqrt[c^2*d^2 - e^2]) - (b*Log[e + c*(c*d - Sqrt 
[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])*x])/(d*Sqrt[c^2*d^2 - e^2])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5750, 1892, 1803, 605, 223, 488, 219}

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.

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

\(\Big \downarrow \) 5750

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

\(\Big \downarrow \) 1892

\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} \left (\frac {d}{x}+e\right ) x^3}dx}{c e}-\frac {a+b \csc ^{-1}(c x)}{e (d+e x)}\)

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 605

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[(a + b*ArcCsc[c*x])/(d + e*x)^2,x]
 

Output:

-((a + b*ArcCsc[c*x])/(e*(d + e*x))) + (b*((c*ArcSin[1/(c*x)])/d + (c*e*Ar 
cTanh[(c*(d + e/(c^2*x)))/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - 1/(c^2*x^2)])])/(d 
*Sqrt[c^2*d^2 - e^2])))/(c*e)
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 605

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[1/d   Int[x^(m - 1)*(a + b*x^2)^p, x], x] - Simp[c/d   Int[x^(m - 1 
)*((a + b*x^2)^p/(c + d*x)), x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 
 0] && LtQ[-1, p, 0]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1892

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^(mn_.))^(q_.)*((a_) + (c_.)*(x_)^(n2_.))^ 
(p_.), x_Symbol] :> Int[x^(m + mn*q)*(e + d/x^mn)^q*(a + c*x^n2)^p, x] /; F 
reeQ[{a, c, d, e, m, mn, p}, x] && EqQ[n2, -2*mn] && IntegerQ[q] && (PosQ[n 
2] ||  !IntegerQ[p])
 

rule 5750

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol 
] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCsc[c*x])/(e*(m + 1))), x] + Simp[b/ 
(c*e*(m + 1))   Int[(d + e*x)^(m + 1)/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] / 
; FreeQ[{a, b, c, d, e, m}, x] && NeQ[m, -1]
 

Maple [A] (verified)

Time = 2.68 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88

method	result	size
parts	\(-\frac {a}{\left (e x +d \right ) e}+\frac {b \left (-\frac {c^{2} \operatorname {arccsc}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 d \,c^{2} x -2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(192\)
derivativedivides	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 d \,c^{2} x -2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(202\)
default	\(\frac {-\frac {a \,c^{2}}{\left (c e x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\left (c e x +c d \right ) e}+\frac {\sqrt {c^{2} x^{2}-1}\, \left (\arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}-\ln \left (\frac {2 \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e -2 d \,c^{2} x -2 e}{c e x +c d}\right )\right )}{e \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x d \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\right )}{c}\)	\(202\)

Input:

int((a+b*arccsc(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-a/(e*x+d)/e+b/c*(-c^2/(c*e*x+c*d)/e*arccsc(c*x)+1/e*(c^2*x^2-1)^(1/2)*(ar 
ctan(1/(c^2*x^2-1)^(1/2))*((c^2*d^2-e^2)/e^2)^(1/2)-ln(2*((c^2*x^2-1)^(1/2 
)*((c^2*d^2-e^2)/e^2)^(1/2)*e-d*c^2*x-e)/(c*e*x+c*d)))/((c^2*x^2-1)/c^2/x^ 
2)^(1/2)/x/d/((c^2*d^2-e^2)/e^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (96) = 192\).

Time = 0.16 (sec) , antiderivative size = 475, normalized size of antiderivative = 4.66 \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\left [-\frac {a c^{2} d^{3} - a d e^{2} - \sqrt {c^{2} d^{2} - e^{2}} {\left (b e^{2} x + b d e\right )} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac {a c^{2} d^{3} - a d e^{2} + 2 \, \sqrt {-c^{2} d^{2} + e^{2}} {\left (b e^{2} x + b d e\right )} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) + {\left (b c^{2} d^{3} - b d e^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \] Input:

integrate((a+b*arccsc(c*x))/(e*x+d)^2,x, algorithm="fricas")
 

Output:

[-(a*c^2*d^3 - a*d*e^2 - sqrt(c^2*d^2 - e^2)*(b*e^2*x + b*d*e)*log((c^3*d^ 
2*x + c*d*e + sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 + sqrt(c^2*d^2 
- e^2)*c*d - e^2)*sqrt(c^2*x^2 - 1))/(e*x + d)) + (b*c^2*d^3 - b*d*e^2)*ar 
ccsc(c*x) + 2*(b*c^2*d^3 - b*d*e^2 + (b*c^2*d^2*e - b*e^3)*x)*arctan(-c*x 
+ sqrt(c^2*x^2 - 1)))/(c^2*d^4*e - d^2*e^3 + (c^2*d^3*e^2 - d*e^4)*x), -(a 
*c^2*d^3 - a*d*e^2 + 2*sqrt(-c^2*d^2 + e^2)*(b*e^2*x + b*d*e)*arctan(-(sqr 
t(-c^2*d^2 + e^2)*sqrt(c^2*x^2 - 1)*e - sqrt(-c^2*d^2 + e^2)*(c*e*x + c*d) 
)/(c^2*d^2 - e^2)) + (b*c^2*d^3 - b*d*e^2)*arccsc(c*x) + 2*(b*c^2*d^3 - b* 
d*e^2 + (b*c^2*d^2*e - b*e^3)*x)*arctan(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^ 
4*e - d^2*e^3 + (c^2*d^3*e^2 - d*e^4)*x)]
 

Sympy [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \] Input:

integrate((a+b*acsc(c*x))/(e*x+d)**2,x)
 

Output:

Integral((a + b*acsc(c*x))/(d + e*x)**2, x)
 

Maxima [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x + d\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccsc(c*x))/(e*x+d)^2,x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*arccsc(c*x))/(e*x+d)^2,x, algorithm="giac")
 

Output:

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve 
cteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{{\left (d+e\,x\right )}^2} \,d x \] Input:

int((a + b*asin(1/(c*x)))/(d + e*x)^2,x)
 

Output:

int((a + b*asin(1/(c*x)))/(d + e*x)^2, x)
 

Reduce [F]

\[ \int \frac {a+b \csc ^{-1}(c x)}{(d+e x)^2} \, dx=\frac {\left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b \,d^{2}+\left (\int \frac {\mathit {acsc} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b d e x +a x}{d \left (e x +d \right )} \] Input:

int((a+b*acsc(c*x))/(e*x+d)^2,x)
 

Output:

(int(acsc(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d**2 + int(acsc(c*x)/(d** 
2 + 2*d*e*x + e**2*x**2),x)*b*d*e*x + a*x)/(d*(d + e*x))