Optimal. Leaf size=124 \[ \frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} i b d \csc ^{-1}(c x)^2+\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b d \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.21, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5349, 14,
4815, 6874, 270, 2363, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -d \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b d \csc ^{-1}(c x)^2-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 270
Rule 2221
Rule 2317
Rule 2363
Rule 2438
Rule 3798
Rule 4721
Rule 4815
Rule 5349
Rule 6874
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx &=-\text {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \frac {-\frac {e}{2 x^2}+d \log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \text {Subst}\left (\int \left (-\frac {e}{2 x^2 \sqrt {1-\frac {x^2}{c^2}}}+\frac {d \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \text {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(b e) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \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 )-(b d) \text {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {b e \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} e x^2 \left (a+b \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 )-(b d) \text {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\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 (\frac {1}{x}\right )-d \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 i b d) \text {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=\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 )+(b d) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=\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 \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 108, normalized size = 0.87 \begin {gather*} \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+\text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 4.88, size = 198, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {i b d \mathrm {arccsc}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e \,x^{2}}{2}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e x}{2 c}-\frac {i b e}{2 c^{2}}-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(198\) |
default | \(\frac {a e \,x^{2}}{2}+a d \ln \left (c x \right )+\frac {i b d \mathrm {arccsc}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) e \,x^{2}}{2}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, e x}{2 c}-\frac {i b e}{2 c^{2}}-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\) | \(198\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.92, size = 111, normalized size = 0.90 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________