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

Optimal. Leaf size=257 \[ \frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (\sqrt {e^2-c^2 d^2}+e\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e+\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.41, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5225, 2518} \[ -\frac {i b \text {PolyLog}\left (2,\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {PolyLog}\left (2,\frac {i \left (\sqrt {e^2-c^2 d^2}+e\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i \left (\sqrt {e^2-c^2 d^2}+e\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )}{e} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])/(d + e*x),x]

[Out]

((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*ArcCsc[c*x]))/(c*d)])/e - (I*b*PolyL
og[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

Rule 2518

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

Rule 5225

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

Rubi steps

\begin {align*} \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 {i \left (e-\sqrt {-c^2 d^2+e^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 (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e}+\frac {b \int \frac {\log \left (1-\frac {i \left (e-\sqrt {-c^2 d^2+e^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 {-c^2 d^2+e^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 {-c^2 d^2+e^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 (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\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 \text {Li}_2\left (\frac {i \left (e-\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}-\frac {i b \text {Li}_2\left (\frac {i \left (e+\sqrt {-c^2 d^2+e^2}\right ) e^{i \csc ^{-1}(c x)}}{c d}\right )}{e}+\frac {i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )}{2 e}\\ \end {align*}

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Mathematica [A]  time = 0.69, size = 411, normalized size = 1.60 \[ \frac {a \log (d+e x)}{e}+\frac {b \left (8 i \left (\text {Li}_2\left (\frac {i \left (\sqrt {e^2-c^2 d^2}-e\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )+\text {Li}_2\left (-\frac {i \left (e+\sqrt {e^2-c^2 d^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right )\right )-4 \log \left (1+\frac {i \left (e-\sqrt {e^2-c^2 d^2}\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right ) \left (4 \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right )-2 \csc ^{-1}(c x)+\pi \right )-4 \log \left (1+\frac {i \left (\sqrt {e^2-c^2 d^2}+e\right ) e^{-i \csc ^{-1}(c x)}}{c d}\right ) \left (-4 \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right )-2 \csc ^{-1}(c x)+\pi \right )+32 i \sin ^{-1}\left (\frac {\sqrt {\frac {e}{c d}+1}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {(c d-e) \cot \left (\frac {1}{4} \left (2 \csc ^{-1}(c x)+\pi \right )\right )}{\sqrt {e^2-c^2 d^2}}\right )+4 \left (\pi -2 \csc ^{-1}(c x)\right ) \log \left (\frac {d}{x}+e\right )+8 \csc ^{-1}(c x) \log \left (\frac {d}{x}+e\right )+4 i \left (\csc ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\right )+i \left (\pi -2 \csc ^{-1}(c x)\right )^2-8 \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )\right )}{8 e} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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)]/Sqr
t[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[Sq
rt[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]*Lo
g[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)*(PolyL
og[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)*ArcCsc[c*x])])))/(8*e)

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arccsc}\left (c x\right ) + a}{e x + d}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(x)]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 1.11, size = 869, normalized size = 3.38 \[ \frac {a \ln \left (c e x +d c \right )}{e}-\frac {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {-d c \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 {b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (\frac {d c \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^{2} b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {-d c \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 )}{e \left (c^{2} d^{2}-e^{2}\right )}+\frac {c^{2} b \,d^{2} \mathrm {arccsc}\left (c x \right ) \ln \left (\frac {d c \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 )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i c^{2} b \,d^{2} \dilog \left (\frac {-d c \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 )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i c^{2} b \,d^{2} \dilog \left (\frac {d c \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 )}{e \left (c^{2} d^{2}-e^{2}\right )}-\frac {i b \dilog \left (\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}+\frac {i b e \dilog \left (\frac {-d c \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 b e \dilog \left (\frac {d c \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 b \dilog \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e}-\frac {b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{e x + d}\,{d x} + \frac {a \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{d+e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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