Optimal. Leaf size=66 \[ \frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{c^2 x^2}\right )}{c m (1+m)} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5329, 346, 371}
\begin {gather*} \frac {(d x)^{m+1} \left (a+b \csc ^{-1}(c x)\right )}{d (m+1)}+\frac {b (d x)^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{c^2 x^2}\right )}{c m (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 346
Rule 371
Rule 5329
Rubi steps
\begin {align*} \int (d x)^m \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {(b d) \int \frac {(d x)^{-1+m}}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{c (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}-\frac {\left (b \left (\frac {1}{x}\right )^m (d x)^m\right ) \text {Subst}\left (\int \frac {x^{-1-m}}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c (1+m)}\\ &=\frac {(d x)^{1+m} \left (a+b \csc ^{-1}(c x)\right )}{d (1+m)}+\frac {b (d x)^m \, _2F_1\left (\frac {1}{2},-\frac {m}{2};1-\frac {m}{2};\frac {1}{c^2 x^2}\right )}{c m (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 83, normalized size = 1.26 \begin {gather*} \frac {(d x)^m \left ((1+m) x \left (a+b \csc ^{-1}(c x)\right )+\frac {b \sqrt {1-c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};c^2 x^2\right )}{c \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{(1+m)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.83, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )\, dx\]
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 [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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d\,x\right )}^m\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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