Optimal. Leaf size=86 \[ -\frac {2 i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {2 i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5253, 5217, 3758, 4183, 2279, 2391} \[ -\frac {2 i \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {2 i \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2279
Rule 2391
Rule 3758
Rule 4183
Rule 5217
Rule 5253
Rubi steps
\begin {align*} \int \csc ^{-1}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \csc ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=-\frac {\operatorname {Subst}\left (\int x^2 \cot (x) \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}-\frac {2 \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {2 \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}-\frac {2 \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}(a+b x)^2}{b}+\frac {4 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}-\frac {2 i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b}+\frac {2 i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 99, normalized size = 1.15 \[ \frac {-2 i \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )+2 i \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )+\csc ^{-1}(a+b x) \left ((a+b x) \csc ^{-1}(a+b x)-2 \log \left (1-e^{i \csc ^{-1}(a+b x)}\right )+2 \log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )}{b} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {arccsc}\left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arccsc}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 167, normalized size = 1.94 \[ x \mathrm {arccsc}\left (b x +a \right )^{2}+\frac {\mathrm {arccsc}\left (b x +a \right )^{2} a}{b}-\frac {2 \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}+\frac {2 \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}-\frac {2 i \dilog \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b}+\frac {2 i \dilog \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{2} - \frac {1}{4} \, x \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \int \frac {2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} b x \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} + {\left (a^{2} - 1\right )} b x + {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acsc}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________