Optimal. Leaf size=67 \[ \frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}-\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A]
time = 0.05, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6015, 2320, 12,
1642} \begin {gather*} -\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2}+\frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1642
Rule 2320
Rule 6015
Rubi steps
\begin {align*} \int e^{\cosh ^{-1}(a+b x)} x \, dx &=\frac {\text {Subst}\left (\int e^x \left (-\frac {a}{b}+\frac {\cosh (x)}{b}\right ) \sinh (x) \, dx,x,\cosh ^{-1}(a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+2 a x-x^2\right )}{4 b x^2} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{b}\\ &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right ) \left (-1+2 a x-x^2\right )}{x^2} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^2}+\frac {2 a}{x}-2 a x+x^2\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{4 b^2}\\ &=\frac {e^{-\cosh ^{-1}(a+b x)}}{4 b^2}-\frac {a e^{2 \cosh ^{-1}(a+b x)}}{4 b^2}+\frac {e^{3 \cosh ^{-1}(a+b x)}}{12 b^2}+\frac {a \cosh ^{-1}(a+b x)}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 93, normalized size = 1.39 \begin {gather*} \frac {1}{6} \left (3 a x^2+2 b x^3+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-2-a^2+a b x+2 b^2 x^2\right )}{b^2}+\frac {3 a \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{b^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.07, size = 194, normalized size = 2.90
method | result | size |
default | \(-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) b^{2} x^{2}-\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a b x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a^{2}+2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )-3 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )+b x +a \right ) \mathrm {csgn}\left (b \right )\right ) a \right ) \mathrm {csgn}\left (b \right )}{6 b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+\frac {b \,x^{3}}{3}+\frac {a \,x^{2}}{2}\) | \(194\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 177, normalized size = 2.64 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} + \frac {a^{3} \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{2 \, b} - \frac {{\left (a^{2} - 1\right )} a \log \left (2 \, b^{2} x + 2 \, a b + 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} b\right )}{2 \, b^{2}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{2 \, b^{2}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}}}{3 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 88, normalized size = 1.31 \begin {gather*} \frac {2 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} + {\left (2 \, b^{2} x^{2} + a b x - a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 3 \, a \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{6 \, b^{2}} \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 \left (a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}\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. 280 vs.
\(2 (101) = 202\).
time = 0.41, size = 280, normalized size = 4.18 \begin {gather*} \frac {2 \, b^{2} x^{3} + {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left ({\left (b x + a + 1\right )} {\left (\frac {2 \, {\left (b x + a + 1\right )}}{b^{2}} - \frac {6 \, a b^{6} + 7 \, b^{6}}{b^{8}}\right )} + \frac {3 \, {\left (2 \, a^{2} b^{6} + 6 \, a b^{6} + 3 \, b^{6}\right )}}{b^{8}}\right )} + \frac {6 \, {\left (2 \, a^{2} + 2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{2}}\right )} b + \frac {3 \, {\left ({\left (b x + a + 1\right )}^{2} - 2 \, {\left (b x + a + 1\right )} a - 2 \, b x - 2 \, a - 2\right )} a}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )} a}{b} + \frac {3 \, {\left (\sqrt {b x + a + 1} \sqrt {b x + a - 1} {\left (b x - a - 2\right )} - 2 \, {\left (2 \, a + 1\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )\right )}}{b}}{6 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.72, size = 852, normalized size = 12.72 \begin {gather*} \frac {a\,x^2}{2}-\frac {\frac {2\,a\,\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}{b^2\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\left (42\,a-\frac {160\,a^3}{3}\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^9\,\left (42\,a-\frac {160\,a^3}{3}\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^9}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\left (212\,a-288\,a^3\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^7\,\left (212\,a-288\,a^3\right )}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^7}+\frac {2\,a\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{11}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{11}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2\,\left (8\,a^2-8\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}\,\left (8\,a^2-8\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\left (160\,a^2-32\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8\,\left (160\,a^2-32\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6\,\left (\frac {1040\,a^2}{3}-\frac {272}{3}\right )\,\sqrt {a-1}\,\sqrt {a+1}}{b^2\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}}{\frac {15\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {6\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-\frac {20\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}+\frac {15\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}-\frac {6\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{12}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{12}}+1}+\frac {b\,x^3}{3}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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