Optimal. Leaf size=55 \[ \frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5223, 4410, 4184, 3475} \[ \frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3475
Rule 4184
Rule 4410
Rule 5223
Rubi steps
\begin {align*} \int x \left (a+b \csc ^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b \operatorname {Subst}\left (\int (a+b x) \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b^2 \operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{c^2}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 89, normalized size = 1.62 \[ \frac {a c x \left (a c x+2 b \sqrt {1-\frac {1}{c^2 x^2}}\right )+2 b c x \csc ^{-1}(c x) \left (a c x+b \sqrt {1-\frac {1}{c^2 x^2}}\right )+b^2 c^2 x^2 \csc ^{-1}(c x)^2+2 b^2 \log (c x)}{2 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 2.50, size = 111, normalized size = 2.02 \[ \frac {b^{2} c^{2} x^{2} \operatorname {arccsc}\left (c x\right )^{2} + a^{2} c^{2} x^{2} - 4 \, a b c^{2} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, b^{2} \log \relax (x) + 2 \, {\left (a b c^{2} x^{2} - a b c^{2}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, \sqrt {c^{2} x^{2} - 1} {\left (b^{2} \operatorname {arccsc}\left (c x\right ) + a b\right )}}{2 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.25, size = 427, normalized size = 7.76 \[ \frac {1}{8} \, {\left (\frac {b^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c} + \frac {2 \, a b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {4 \, b^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{2}} + \frac {4 \, a b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {2 \, b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{3}} + \frac {4 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{3}} - \frac {16 \, b^{2} \log \relax (2)}{c^{3}} + \frac {8 \, b^{2} \log \left (2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 2\right )}{c^{3}} - \frac {8 \, b^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{3}} - \frac {8 \, b^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{3}} + \frac {2 \, a^{2}}{c^{3}} - \frac {4 \, b^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {4 \, a b}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {2 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a^{2}}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.26, size = 133, normalized size = 2.42 \[ \frac {a^{2} x^{2}}{2}+\frac {x^{2} b^{2} \mathrm {arccsc}\left (c x \right )^{2}}{2}+\frac {b^{2} \mathrm {arccsc}\left (c x \right ) x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{c}-\frac {b^{2} \ln \left (\frac {1}{c x}\right )}{c^{2}}+a b \,x^{2} \mathrm {arccsc}\left (c x \right )+\frac {a b x}{c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a b}{c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 84, normalized size = 1.53 \[ \frac {1}{2} \, b^{2} x^{2} \operatorname {arccsc}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} + {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} a b + {\left (\frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \operatorname {arccsc}\left (c x\right )}{c} + \frac {\log \relax (x)}{c^{2}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________