Optimal. Leaf size=107 \[ \frac {b^2 x^2}{12 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{3 c^4} \]
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Rubi [A]
time = 0.08, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5331, 4495,
4270, 4269, 3556} \begin {gather*} \frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{6 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{3 c^4}+\frac {b^2 x^2}{12 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3556
Rule 4269
Rule 4270
Rule 4495
Rule 5331
Rubi steps
\begin {align*} \int x^3 \left (a+b \csc ^{-1}(c x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^2 \cot (x) \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{c^4}\\ &=\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b \text {Subst}\left (\int (a+b x) \csc ^4(x) \, dx,x,\csc ^{-1}(c x)\right )}{2 c^4}\\ &=\frac {b^2 x^2}{12 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b \text {Subst}\left (\int (a+b x) \csc ^2(x) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^4}\\ &=\frac {b^2 x^2}{12 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b^2 \text {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(c x)\right )}{3 c^4}\\ &=\frac {b^2 x^2}{12 c^2}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \left (a+b \csc ^{-1}(c x)\right )}{3 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3 \left (a+b \csc ^{-1}(c x)\right )}{6 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )^2+\frac {b^2 \log (x)}{3 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 124, normalized size = 1.16 \begin {gather*} \frac {c x \left (b^2 c x+3 a^2 c^3 x^3+2 a b \sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )\right )+2 b c x \left (3 a c^3 x^3+b \sqrt {1-\frac {1}{c^2 x^2}} \left (2+c^2 x^2\right )\right ) \csc ^{-1}(c x)+3 b^2 c^4 x^4 \csc ^{-1}(c x)^2+4 b^2 \log (x)}{12 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 181, normalized size = 1.69
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} c^{4} x^{4}}{4}+\frac {b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {b^{2} c^{2} x^{2}}{12}+\frac {b^{2} \mathrm {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {b^{2} \ln \left (\frac {1}{c x}\right )}{3}+2 a b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(181\) |
default | \(\frac {\frac {a^{2} c^{4} x^{4}}{4}+\frac {b^{2} \mathrm {arccsc}\left (c x \right )^{2} c^{4} x^{4}}{4}+\frac {b^{2} \mathrm {arccsc}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}{6}+\frac {b^{2} c^{2} x^{2}}{12}+\frac {b^{2} \mathrm {arccsc}\left (c x \right ) c x \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}-\frac {b^{2} \ln \left (\frac {1}{c x}\right )}{3}+2 a b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\) | \(181\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (93) = 186\).
time = 0.51, size = 197, normalized size = 1.84 \begin {gather*} \frac {1}{4} \, b^{2} x^{4} \operatorname {arccsc}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} a b + \frac {{\left (2 \, c^{4} x^{4} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 2 \, c^{2} x^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + {\left (c^{2} x^{2} + 2 \, \log \left (x^{2}\right )\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 4 \, \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )} b^{2}}{12 \, \sqrt {c x + 1} \sqrt {c x - 1} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 146, normalized size = 1.36 \begin {gather*} \frac {3 \, b^{2} c^{4} x^{4} \operatorname {arccsc}\left (c x\right )^{2} + 3 \, a^{2} c^{4} x^{4} - 12 \, a b c^{4} \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + b^{2} c^{2} x^{2} + 4 \, b^{2} \log \left (x\right ) + 6 \, {\left (a b c^{4} x^{4} - a b c^{4}\right )} \operatorname {arccsc}\left (c x\right ) + 2 \, {\left (a b c^{2} x^{2} + 2 \, a b + {\left (b^{2} c^{2} x^{2} + 2 \, b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 811 vs.
\(2 (93) = 186\).
time = 0.53, size = 811, normalized size = 7.58 \begin {gather*} \frac {1}{192} \, {\left (\frac {3 \, b^{2} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )^{2}}{c} + \frac {6 \, a b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a^{2} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {4 \, b^{2} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3} \arcsin \left (\frac {1}{c x}\right )}{c^{2}} + \frac {4 \, a b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {12 \, 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^{3}} + \frac {24 \, a b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {12 \, a^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {4 \, b^{2} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {36 \, b^{2} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )} \arcsin \left (\frac {1}{c x}\right )}{c^{4}} + \frac {36 \, a b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {18 \, b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{5}} + \frac {36 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{5}} - \frac {128 \, b^{2} \log \left (2\right )}{c^{5}} + \frac {64 \, b^{2} \log \left (2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 2\right )}{c^{5}} - \frac {64 \, b^{2} \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{5}} - \frac {64 \, b^{2} \log \left (\frac {1}{{\left | c \right |} {\left | x \right |}}\right )}{c^{5}} + \frac {18 \, a^{2}}{c^{5}} + \frac {8 \, b^{2}}{c^{5}} - \frac {36 \, b^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} - \frac {36 \, a b}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {24 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a^{2}}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {4 \, b^{2}}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {4 \, b^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} - \frac {4 \, a b}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b^{2} \arcsin \left (\frac {1}{c x}\right )^{2}}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {6 \, a b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a^{2}}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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