\(\int \frac {a+b \csc ^{-1}(c x)}{d+e x} \, dx\) [48]

Optimal result

Integrand size = 16, antiderivative size = 259 \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x} \, dx=\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {c d e^{i \csc ^{-1}(c x)}}{i e-\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {c d e^{i \csc ^{-1}(c x)}}{i e+\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {c d e^{i \csc ^{-1}(c x)}}{i e-\sqrt {c^2 d^2-e^2}}\right )}{e}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {c d e^{i \csc ^{-1}(c x)}}{i e+\sqrt {c^2 d^2-e^2}}\right )}{e}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e} \] Output:

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

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.59 \[ \int \frac {a+b \csc ^{-1}(c x)}{d+e x} \, dx=\frac {a \log (d+e x)}{e}+\frac {b \left (i \left (\pi -2 \csc ^{-1}(c x)\right )^2+32 i \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right ) \arctan \left (\frac {(c d-e) \cot \left (\frac {1}{4} \left (\pi +2 \csc ^{-1}(c x)\right )\right )}{\sqrt {-c^2 d^2+e^2}}\right )-4 \left (\pi -2 \csc ^{-1}(c x)+4 \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )-4 \left (\pi -2 \csc ^{-1}(c x)-4 \arcsin \left (\frac {\sqrt {1+\frac {e}{c d}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )-8 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+4 \left (\pi -2 \csc ^{-1}(c x)\right ) \log \left (e+\frac {d}{x}\right )+8 \csc ^{-1}(c x) \log \left (e+\frac {d}{x}\right )+8 i \left (\operatorname {PolyLog}\left (2,\frac {i \left (-e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )\right )+4 i \left (\csc ^{-1}(c x)^2+\operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )\right )}{8 e} \] Input:

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

Output:

(a*Log[d + e*x])/e + (b*(I*(Pi - 2*ArcCsc[c*x])^2 + (32*I)*ArcSin[Sqrt[1 + 
 e/(c*d)]/Sqrt[2]]*ArcTan[((c*d - e)*Cot[(Pi + 2*ArcCsc[c*x])/4])/Sqrt[-(c 
^2*d^2) + e^2]] - 4*(Pi - 2*ArcCsc[c*x] + 4*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[ 
2]])*Log[1 + (I*(e - Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))] - 4 
*(Pi - 2*ArcCsc[c*x] - 4*ArcSin[Sqrt[1 + e/(c*d)]/Sqrt[2]])*Log[1 + (I*(e 
+ Sqrt[-(c^2*d^2) + e^2]))/(c*d*E^(I*ArcCsc[c*x]))] - 8*ArcCsc[c*x]*Log[1 
- E^((2*I)*ArcCsc[c*x])] + 4*(Pi - 2*ArcCsc[c*x])*Log[e + d/x] + 8*ArcCsc[ 
c*x]*Log[e + d/x] + (8*I)*(PolyLog[2, (I*(-e + Sqrt[-(c^2*d^2) + e^2]))/(c 
*d*E^(I*ArcCsc[c*x]))] + PolyLog[2, ((-I)*(e + Sqrt[-(c^2*d^2) + e^2]))/(c 
*d*E^(I*ArcCsc[c*x]))]) + (4*I)*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*ArcCs 
c[c*x])])))/(8*e)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5748, 2998}

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.

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

\(\Big \downarrow \) 5748

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

\(\Big \downarrow \) 2998

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2998

Int[Log[v_]*(u_), x_Symbol] :> With[{w = DerivativeDivides[v, u*(1 - v), x] 
}, Simp[w*PolyLog[2, 1 - v], x] /;  !FalseQ[w]]
 

rule 5748

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))/((d_.) + (e_.)*(x_)), x_Symbol] :> S 
imp[(a + b*ArcCsc[c*x])*(Log[1 - I*(e - Sqrt[(-c^2)*d^2 + e^2])*(E^(I*ArcCs 
c[c*x])/(c*d))]/e), x] + (Simp[(a + b*ArcCsc[c*x])*(Log[1 - I*(e + Sqrt[(-c 
^2)*d^2 + e^2])*(E^(I*ArcCsc[c*x])/(c*d))]/e), x] - Simp[(a + b*ArcCsc[c*x] 
)*(Log[1 - E^(2*I*ArcCsc[c*x])]/e), x] + Simp[b/(c*e)   Int[Log[1 - I*(e - 
Sqrt[(-c^2)*d^2 + e^2])*(E^(I*ArcCsc[c*x])/(c*d))]/(x^2*Sqrt[1 - 1/(c^2*x^2 
)]), x], x] + Simp[b/(c*e)   Int[Log[1 - I*(e + Sqrt[(-c^2)*d^2 + e^2])*(E^ 
(I*ArcCsc[c*x])/(c*d))]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x] - Simp[b/(c*e) 
  Int[Log[1 - E^(2*I*ArcCsc[c*x])]/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x]) /; 
FreeQ[{a, b, c, d, e}, x]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 866 vs. \(2 (318 ) = 636\).

Time = 2.17 (sec) , antiderivative size = 867, normalized size of antiderivative = 3.35

method	result	size
derivativedivides	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (-\frac {e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}\right )}{c}\)	\(867\)
default	\(\frac {\frac {a c \ln \left (c e x +c d \right )}{e}+b c \left (-\frac {e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i e \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {i \operatorname {dilog}\left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {-c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-i e +\sqrt {c^{2} d^{2}-e^{2}}}{-i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{2} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}\right )}{c}\)	\(867\)
parts	\(\frac {a \ln \left (e x +d \right )}{e}+\frac {b \left (-\frac {c e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {c e \,\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{3} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {\operatorname {arccsc}\left (c x \right ) \ln \left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{3} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i c \operatorname {dilog}\left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {i \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{3} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right ) c^{3} d^{2}}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {i c e \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e -\sqrt {c^{2} d^{2}-e^{2}}}{i e -\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}+\frac {i c e \operatorname {dilog}\left (\frac {c d \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i e +\sqrt {c^{2} d^{2}-e^{2}}}{i e +\sqrt {c^{2} d^{2}-e^{2}}}\right )}{c^{2} d^{2}-e^{2}}-\frac {c \,\operatorname {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i c \operatorname {dilog}\left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}\right )}{c}\)	\(880\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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