Optimal. Leaf size=31 \[ a x+b x \csc ^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c} \]
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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5323, 272, 65,
214} \begin {gather*} a x+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 5323
Rubi steps
\begin {align*} \int \left (a+b \csc ^{-1}(c x)\right ) \, dx &=a x+b \int \csc ^{-1}(c x) \, dx\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \csc ^{-1}(c x)-\frac {b \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=a x+b x \csc ^{-1}(c x)+(b c) \text {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 58, normalized size = 1.87 \begin {gather*} a x+b x \csc ^{-1}(c x)+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x \tanh ^{-1}\left (\frac {c x}{\sqrt {-1+c^2 x^2}}\right )}{\sqrt {-1+c^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 37, normalized size = 1.19
method | result | size |
default | \(a x +b x \,\mathrm {arccsc}\left (c x \right )+\frac {b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c}\) | \(37\) |
derivativedivides | \(\frac {a c x +\mathrm {arccsc}\left (c x \right ) b c x +\ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) b}{c}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 53, normalized size = 1.71 \begin {gather*} a x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (29) = 58\).
time = 0.35, size = 64, normalized size = 2.06 \begin {gather*} \frac {a c x - 2 \, b c \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c x - b c\right )} \operatorname {arccsc}\left (c x\right ) - b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.63, size = 32, normalized size = 1.03 \begin {gather*} a x + b \left (x \operatorname {acsc}{\left (c x \right )} + \frac {\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}}{c}\right ) \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. 62 vs.
\(2 (29) = 58\).
time = 0.42, size = 62, normalized size = 2.00 \begin {gather*} \frac {1}{2} \, b c {\left (\frac {2 \, x \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}}\right )} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 33, normalized size = 1.06 \begin {gather*} a\,x+b\,x\,\mathrm {asin}\left (\frac {1}{c\,x}\right )+\frac {b\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{c^2\,x^2}}}\right )}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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