Optimal. Leaf size=170 \[ \frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 4489,
3392, 3377, 2718, 2713} \begin {gather*} \frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2718
Rule 3377
Rule 3392
Rule 4489
Rule 5331
Rubi steps
\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^3}{x^4} \, dx &=-\left (c^3 \text {Subst}\left (\int (a+b x)^3 \cos (x) \sin ^2(x) \, dx,x,\csc ^{-1}(c x)\right )\right )\\ &=-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}+\left (b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (2 b c^3\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\csc ^{-1}(c x)\right )-\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \sin ^3(x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}+\frac {1}{3} \left (4 b^2 c^3\right ) \text {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\csc ^{-1}(c x)\right )+\frac {1}{9} \left (2 b^3 c^3\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )\\ &=\frac {2}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}-\frac {1}{3} \left (4 b^3 c^3\right ) \text {Subst}\left (\int \sin (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {14}{9} b^3 c^3 \sqrt {1-\frac {1}{c^2 x^2}}-\frac {2}{27} b^3 c^3 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}+\frac {2 b^2 \left (a+b \csc ^{-1}(c x)\right )}{9 x^3}+\frac {4 b^2 c^2 \left (a+b \csc ^{-1}(c x)\right )}{3 x}-\frac {2}{3} b c^3 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \csc ^{-1}(c x)\right )^2}{3 x^2}-\frac {\left (a+b \csc ^{-1}(c x)\right )^3}{3 x^3}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 204, normalized size = 1.20 \begin {gather*} \frac {-9 a^3-9 a^2 b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+6 a b^2 \left (1+6 c^2 x^2\right )+2 b^3 c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+20 c^2 x^2\right )+3 b \left (-9 a^2-6 a b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )+2 b^2 \left (1+6 c^2 x^2\right )\right ) \csc ^{-1}(c x)-9 b^2 \left (3 a+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (1+2 c^2 x^2\right )\right ) \csc ^{-1}(c x)^2-9 b^3 \csc ^{-1}(c x)^3}{27 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(298\) vs.
\(2(148)=296\).
time = 0.41, size = 299, normalized size = 1.76
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\mathrm {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\mathrm {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\mathrm {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\mathrm {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
default | \(c^{3} \left (-\frac {a^{3}}{3 c^{3} x^{3}}+b^{3} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{3}}{3 c^{3} x^{3}}-\frac {\mathrm {arccsc}\left (c x \right )^{2} \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3 c^{2} x^{2}}+\frac {4 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{3}+\frac {4 \,\mathrm {arccsc}\left (c x \right )}{3 c x}+\frac {2 \,\mathrm {arccsc}\left (c x \right )}{9 c^{3} x^{3}}+\frac {2 \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{27 c^{2} x^{2}}\right )+3 a \,b^{2} \left (-\frac {\mathrm {arccsc}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {2 \,\mathrm {arccsc}\left (c x \right ) \left (2 c^{2} x^{2}+1\right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}{9 c^{2} x^{2}}+\frac {2}{27 c^{3} x^{3}}+\frac {4}{9 c x}\right )+3 a^{2} b \left (-\frac {\mathrm {arccsc}\left (c x \right )}{3 c^{3} x^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{2} x^{2}+1\right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )\right )\) | \(299\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.50, size = 173, normalized size = 1.02 \begin {gather*} \frac {36 \, a b^{2} c^{2} x^{2} - 9 \, b^{3} \operatorname {arccsc}\left (c x\right )^{3} - 27 \, a b^{2} \operatorname {arccsc}\left (c x\right )^{2} - 9 \, a^{3} + 6 \, a b^{2} + 3 \, {\left (12 \, b^{3} c^{2} x^{2} - 9 \, a^{2} b + 2 \, b^{3}\right )} \operatorname {arccsc}\left (c x\right ) - {\left (2 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} c^{2} x^{2} + 9 \, a^{2} b - 2 \, b^{3} + 9 \, {\left (2 \, b^{3} c^{2} x^{2} + b^{3}\right )} \operatorname {arccsc}\left (c x\right )^{2} + 18 \, {\left (2 \, a b^{2} c^{2} x^{2} + a b^{2}\right )} \operatorname {arccsc}\left (c x\right )\right )} \sqrt {c^{2} x^{2} - 1}}{27 \, x^{3}} \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 \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}}{x^{4}}\, 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. 428 vs.
\(2 (148) = 296\).
time = 0.48, size = 428, normalized size = 2.52 \begin {gather*} \frac {1}{27} \, {\left (9 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right )^{2} + 18 \, a b^{2} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} \arcsin \left (\frac {1}{c x}\right ) - 27 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right )^{2} - \frac {9 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{3}}{x} + 9 \, a^{2} b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 2 \, b^{3} c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 54 \, a b^{2} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} \arcsin \left (\frac {1}{c x}\right ) - \frac {27 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )^{2}}{x} - \frac {9 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )^{3}}{x} - 27 \, a^{2} b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 42 \, b^{3} c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {27 \, a^{2} b c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {6 \, b^{3} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {27 \, a b^{2} c \arcsin \left (\frac {1}{c x}\right )^{2}}{x} + \frac {6 \, a b^{2} c {\left (\frac {1}{c^{2} x^{2}} - 1\right )}}{x} - \frac {27 \, a^{2} b c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, b^{3} c \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {42 \, a b^{2} c}{x} - \frac {9 \, a^{3}}{c x^{3}}\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 \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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