3.1.86 \(\int \frac {(d+e x^2) (a+b \csc ^{-1}(c x))}{x} \, dx\) [86]

3.1.86.1 Optimal result

Integrand size = 19, antiderivative size = 124 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b d \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]

output

1/2*I*b*d*arccsc(c*x)^2+1/2*e*x^2*(a+b*arccsc(c*x))-b*d*arccsc(c*x)*ln(1-( 
I/c/x+(1-1/c^2/x^2)^(1/2))^2)+b*d*arccsc(c*x)*ln(1/x)-d*(a+b*arccsc(c*x))* 
ln(1/x)+1/2*I*b*d*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)+1/2*b*e*x*(1-1/ 
c^2/x^2)^(1/2)/c
 

3.1.86.2 Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.87 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+\frac {1}{2} b e x^2 \csc ^{-1}(c x)-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a d \log (x)+\frac {1}{2} i b d \left (\csc ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right ) \]

input

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

output

(a*e*x^2)/2 + (b*e*x*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(2*c) + (b*e*x^2*ArcC 
sc[c*x])/2 - b*d*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a*d*Log[x] + 
 (I/2)*b*d*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCsc[c*x])])
 

3.1.86.3 Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5764, 5230, 27, 7293, 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.

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

\(\Big \downarrow \) 5764

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

\(\Big \downarrow \) 5230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -d \log \left (\frac {1}{x}\right ) \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )+\frac {1}{2} e x^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )-\frac {b \left (-i c d \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )-i c d \arcsin \left (\frac {1}{c x}\right )^2+2 c d \arcsin \left (\frac {1}{c x}\right ) \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )-2 c d \log \left (\frac {1}{x}\right ) \arcsin \left (\frac {1}{c x}\right )-e x \sqrt {1-\frac {1}{c^2 x^2}}\right )}{2 c}\)

input

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

output

(e*x^2*(a + b*ArcSin[1/(c*x)]))/2 - d*(a + b*ArcSin[1/(c*x)])*Log[x^(-1)] 
- (b*(-(e*Sqrt[1 - 1/(c^2*x^2)]*x) - I*c*d*ArcSin[1/(c*x)]^2 + 2*c*d*ArcSi 
n[1/(c*x)]*Log[1 - E^((2*I)*ArcSin[1/(c*x)])] - 2*c*d*ArcSin[1/(c*x)]*Log[ 
x^(-1)] - I*c*d*PolyLog[2, E^((2*I)*ArcSin[1/(c*x)])]))/(2*c)
 

3.1.86.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 5230

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp 
[(a + b*ArcSin[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - 
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 
0] && IntegerQ[p] && (GtQ[p, 0] || (IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))
 

rule 5764

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^( 
m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] 
&& IntegerQ[m] && IntegerQ[p]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.1.86.4 Maple [A] (verified)

Time = 2.04 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.53

method	result	size
parts	\(\frac {a e \,x^{2}}{2}+a d \ln \left (x \right )+b \left (\frac {i \operatorname {arccsc}\left (c x \right )^{2} d}{2}+\frac {e \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-i\right )}{2 c^{2}}-d \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-d \,\operatorname {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i d \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i d \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )\)	\(190\)
derivativedivides	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {b \left (\frac {i c^{2} d \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {e \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-i\right )}{2}-\ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \,\operatorname {arccsc}\left (c x \right )-\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \,\operatorname {arccsc}\left (c x \right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d +i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \right )}{c^{2}}\)	\(207\)
default	\(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {b \left (\frac {i c^{2} d \operatorname {arccsc}\left (c x \right )^{2}}{2}+\frac {e \left (c^{2} x^{2} \operatorname {arccsc}\left (c x \right )+x c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}-i\right )}{2}-\ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \,\operatorname {arccsc}\left (c x \right )-\ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \,\operatorname {arccsc}\left (c x \right )+i \operatorname {polylog}\left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d +i \operatorname {polylog}\left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) c^{2} d \right )}{c^{2}}\)	\(207\)

input

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

output

1/2*a*e*x^2+a*d*ln(x)+b*(1/2*I*arccsc(c*x)^2*d+1/2*e*(c^2*x^2*arccsc(c*x)+ 
x*c*((c^2*x^2-1)/c^2/x^2)^(1/2)-I)/c^2-d*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x 
^2)^(1/2))-d*arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+I*d*polylog(2,-I/ 
c/x-(1-1/c^2/x^2)^(1/2))+I*d*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2)))
 

3.1.86.5 Fricas [F]

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

input

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

output

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccsc(c*x))/x, x)
 

3.1.86.6 Sympy [F]

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

input

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

output

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

3.1.86.7 Maxima [F]

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

input

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

output

1/2*a*e*x^2 + a*d*log(x) + 1/4*(2*I*b*c^2*d*log(-c*x + 1)*log(x) + 2*I*b*c 
^2*d*log(x)^2 + 2*I*b*c^2*d*dilog(c*x) + 2*I*b*c^2*d*dilog(-c*x) + 2*(b*c^ 
2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + I*b*c^2*log(c))*e*x^2 - I*(b*e 
*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2) + 8*b*d*integrate(1/2*log(x)/(c^2*x 
^3 - x), x))*c^2 + 4*c^2*integrate(1/2*(b*e*x^2 + 2*b*d*log(x))*sqrt(c*x + 
 1)*sqrt(c*x - 1)/(c^2*x^3 - x), x) + I*b*e*log(c*x - 1) + (-I*b*c^2*e*x^2 
 - 2*I*b*c^2*d*log(x))*log(c^2*x^2) + (2*I*b*c^2*d*log(x) + I*b*e)*log(c*x 
 + 1) - 2*(-I*b*c^2*e*x^2 - 2*(b*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1 
)) + I*b*c^2*log(c))*d)*log(x))/c^2
 

3.1.86.8 Giac [F(-2)]

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

input

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

output

Exception raised: RuntimeError >> an error occurred running a Giac command 
:INPUT:sage2OUTPUT:Limit: Max order reached or unable to make series expan 
sion Error: Bad Argument Value
 

3.1.86.9 Mupad [B] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx=\frac {a\,e\,x^2}{2}-a\,d\,\ln \left (\frac {1}{x}\right )-b\,d\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right )+\frac {b\,e\,x\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+c\,x\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{2\,c}+\frac {b\,d\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2} \]

input

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

output

(b*d*polylog(2, exp(asin(1/(c*x))*2i))*1i)/2 - a*d*log(1/x) + (b*d*asin(1/ 
(c*x))^2*1i)/2 + (a*e*x^2)/2 - b*d*log(1 - exp(asin(1/(c*x))*2i))*asin(1/( 
c*x)) + (b*e*x*((1 - 1/(c^2*x^2))^(1/2) + c*x*asin(1/(c*x))))/(2*c)