Optimal. Leaf size=47 \[ \frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{(a+b x)^2}}\right )}{b} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4916, 5359,
379, 272, 65, 212} \begin {gather*} \frac {c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{(a+b x)^2}}\right )}{b}+\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 272
Rule 379
Rule 4916
Rule 5359
Rubi steps
\begin {align*} \int \sin ^{-1}\left (\frac {c}{a+b x}\right ) \, dx &=\int \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right ) \, dx\\ &=\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\int \frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right ) \sqrt {1-\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}} \, dx\\ &=\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^2}} x} \, dx,x,\frac {a}{c}+\frac {b x}{c}\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}\right )}{2 b}\\ &=\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-\frac {c^2}{(a+b x)^2}}\right )}{b}\\ &=\frac {(a+b x) \csc ^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \tanh ^{-1}\left (\sqrt {1-\frac {c^2}{(a+b x)^2}}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(345\) vs. \(2(47)=94\).
time = 0.64, size = 345, normalized size = 7.34 \begin {gather*} x \text {ArcSin}\left (\frac {c}{a+b x}\right )+\frac {(a+b x) \sqrt {\frac {a^2-c^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a \left (b+\sqrt {b^2}\right ) \text {ArcTan}\left (\frac {a+\sqrt {b^2} x-\sqrt {a^2-c^2+2 a b x+b^2 x^2}}{c}\right )+2 a \left (-b+\sqrt {b^2}\right ) \text {ArcTan}\left (\frac {a-\sqrt {b^2} x+\sqrt {a^2-c^2+2 a b x+b^2 x^2}}{c}\right )-c \left (\sqrt {b^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )+\left (-b+\sqrt {b^2}\right ) \log \left (a-\sqrt {b^2} x+\sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )+b \log \left (a b^2+\left (b^2\right )^{3/2} x-b^2 \sqrt {a^2-c^2+2 a b x+b^2 x^2}\right )\right )\right )}{2 b^2 \sqrt {a^2-c^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 47, normalized size = 1.00
method | result | size |
derivativedivides | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arcsin \left (\frac {c}{b x +a}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(47\) |
default | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arcsin \left (\frac {c}{b x +a}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1-\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(47\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (45) = 90\).
time = 1.03, size = 141, normalized size = 3.00 \begin {gather*} \frac {b x \arcsin \left (\frac {c}{b x + a}\right ) - 2 \, a \arctan \left (-\frac {b x - {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + a}{c}\right ) - c \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right )}{b} \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 \operatorname {asin}{\left (\frac {c}{a + b x} \right )}\, 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. 95 vs.
\(2 (45) = 90\).
time = 0.43, size = 95, normalized size = 2.02 \begin {gather*} \frac {b {\left (\frac {c^{2} {\left (\log \left (\sqrt {-\frac {c^{2}}{{\left (b x + a\right )}^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {c^{2}}{{\left (b x + a\right )}^{2}} + 1} + 1\right )\right )}}{b^{2}} + \frac {2 \, {\left (b x + a\right )} c \arcsin \left (-\frac {c}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{2}}\right )}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.67, size = 42, normalized size = 0.89 \begin {gather*} \frac {c\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {c^2}{{\left (a+b\,x\right )}^2}}}\right )}{b}+\frac {\mathrm {asin}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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