Optimal. Leaf size=165 \[ \frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}-\frac {a \left (3+4 a^2\right ) e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4}+\frac {a \left (3+4 a^2\right ) \cosh ^{-1}(a+b x)}{8 b^4} \]
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Rubi [A]
time = 0.12, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6015, 2320, 12,
1642} \begin {gather*} -\frac {\left (4 a^2+3\right ) a e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (4 a^2+3\right ) a \cosh ^{-1}(a+b x)}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}+\frac {\left (6 a^2+1\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4} \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^3 \, dx &=\frac {\text {Subst}\left (\int e^x \left (-\frac {a}{b}+\frac {\cosh (x)}{b}\right )^3 \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 )^3}{16 b^3 x^4} \, 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 )^3}{x^4} \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{x^4}+\frac {6 a}{x^3}-\frac {2 \left (1+6 a^2\right )}{x^2}+\frac {2 a \left (3+4 a^2\right )}{x}-2 a \left (3+4 a^2\right ) x+2 \left (1+6 a^2\right ) x^2-6 a x^3+x^4\right ) \, dx,x,e^{\cosh ^{-1}(a+b x)}\right )}{16 b^4}\\ &=\frac {e^{-3 \cosh ^{-1}(a+b x)}}{48 b^4}-\frac {3 a e^{-2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{-\cosh ^{-1}(a+b x)}}{8 b^4}-\frac {a \left (3+4 a^2\right ) e^{2 \cosh ^{-1}(a+b x)}}{16 b^4}+\frac {\left (1+6 a^2\right ) e^{3 \cosh ^{-1}(a+b x)}}{24 b^4}-\frac {3 a e^{4 \cosh ^{-1}(a+b x)}}{32 b^4}+\frac {e^{5 \cosh ^{-1}(a+b x)}}{80 b^4}+\frac {a \left (3+4 a^2\right ) \cosh ^{-1}(a+b x)}{8 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 138, normalized size = 0.84 \begin {gather*} \frac {30 a b^4 x^4+24 b^5 x^5+\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-16-83 a^2-6 a^4+a \left (29+6 a^2\right ) b x-2 \left (4+3 a^2\right ) b^2 x^2+6 a b^3 x^3+24 b^4 x^4\right )+15 a \left (3+4 a^2\right ) \log \left (a+b x+\sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{120 b^4} \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 = 0.08, size = 376, normalized size = 2.28
method | result | size |
default | \(-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (-24 \,\mathrm {csgn}\left (b \right ) b^{4} x^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 \,\mathrm {csgn}\left (b \right ) a \,b^{3} x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 \,\mathrm {csgn}\left (b \right ) a^{2} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a^{3} b x +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a^{4}+8 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) b^{2} x^{2}-29 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a b x +83 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right ) a^{2}-60 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )+b x +a \right ) \mathrm {csgn}\left (b \right )\right ) a^{3}+16 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )-45 \ln \left (\left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \mathrm {csgn}\left (b \right )+b x +a \right ) \mathrm {csgn}\left (b \right )\right ) a \right ) \mathrm {csgn}\left (b \right )}{120 b^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}+\frac {b \,x^{5}}{5}+\frac {a \,x^{4}}{4}\) | \(376\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 495 vs.
\(2 (247) = 494\).
time = 0.30, size = 495, normalized size = 3.00 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} x^{2}}{5 \, b^{2}} - \frac {{\left (a^{2} - 1\right )} 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 )}{5 \, b^{4}} - \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} a x}{20 \, b^{3}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} a x}{5 \, b^{3}} + \frac {{\left (a^{2} - 1\right )}^{2} 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 )}{5 \, b^{4}} + \frac {7 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} a^{2}}{12 \, b^{4}} + \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (a^{2} - 1\right )} a^{2}}{5 \, b^{4}} + \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} 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 )}{40 \, b^{6}} - \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{\frac {3}{2}} {\left (a^{2} - 1\right )}}{15 \, b^{4}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a x}{40 \, b^{5}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} {\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 )}{40 \, b^{6}} - \frac {7 \, {\left (5 \, a^{2} b^{2} - {\left (a^{2} - 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} a^{2}}{40 \, b^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 133, normalized size = 0.81 \begin {gather*} \frac {24 \, b^{5} x^{5} + 30 \, a b^{4} x^{4} + {\left (24 \, b^{4} x^{4} + 6 \, a b^{3} x^{3} - 2 \, {\left (3 \, a^{2} + 4\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (6 \, a^{3} + 29 \, a\right )} b x - 83 \, a^{2} - 16\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 15 \, {\left (4 \, a^{3} + 3 \, a\right )} \log \left (-b x + \sqrt {b x + a + 1} \sqrt {b x + a - 1} - a\right )}{120 \, b^{4}} \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^{3} \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. 564 vs.
\(2 (247) = 494\).
time = 0.43, size = 564, normalized size = 3.42 \begin {gather*} \frac {24 \, b^{2} x^{5} + 30 \, a b x^{4} + 5 \, {\left ({\left (b x + a + 1\right )} {\left (2 \, {\left (b x + a + 1\right )} {\left (\frac {3 \, {\left (b x + a + 1\right )}}{b^{3}} - \frac {12 \, a b^{12} + 13 \, b^{12}}{b^{15}}\right )} + \frac {36 \, a^{2} b^{12} + 84 \, a b^{12} + 43 \, b^{12}}{b^{15}}\right )} - \frac {3 \, {\left (8 \, a^{3} b^{12} + 36 \, a^{2} b^{12} + 36 \, a b^{12} + 13 \, b^{12}\right )}}{b^{15}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 5 \, {\left ({\left ({\left (b x + a + 1\right )} {\left (2 \, {\left (b x + a + 1\right )} {\left (\frac {3 \, {\left (b x + a + 1\right )}}{b^{3}} - \frac {12 \, a b^{12} + 13 \, b^{12}}{b^{15}}\right )} + \frac {36 \, a^{2} b^{12} + 84 \, a b^{12} + 43 \, b^{12}}{b^{15}}\right )} - \frac {3 \, {\left (8 \, a^{3} b^{12} + 36 \, a^{2} b^{12} + 36 \, a b^{12} + 13 \, b^{12}\right )}}{b^{15}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - \frac {6 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{3}}\right )} a + {\left ({\left ({\left (2 \, {\left (b x + a + 1\right )} {\left (3 \, {\left (b x + a + 1\right )} {\left (\frac {4 \, {\left (b x + a + 1\right )}}{b^{4}} - \frac {20 \, a b^{20} + 21 \, b^{20}}{b^{24}}\right )} + \frac {120 \, a^{2} b^{20} + 260 \, a b^{20} + 133 \, b^{20}}{b^{24}}\right )} - \frac {5 \, {\left (48 \, a^{3} b^{20} + 168 \, a^{2} b^{20} + 172 \, a b^{20} + 59 \, b^{20}\right )}}{b^{24}}\right )} {\left (b x + a + 1\right )} + \frac {15 \, {\left (8 \, a^{4} b^{20} + 48 \, a^{3} b^{20} + 72 \, a^{2} b^{20} + 52 \, a b^{20} + 13 \, b^{20}\right )}}{b^{24}}\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + \frac {30 \, {\left (8 \, a^{4} + 16 \, a^{3} + 24 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{4}}\right )} b - \frac {30 \, {\left (8 \, a^{3} + 12 \, a^{2} + 12 \, a + 3\right )} \log \left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}{b^{3}}}{120 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 69.81, size = 1408, normalized size = 8.53 \begin {gather*} \frac {a\,x^4}{4}-\frac {\frac {\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )\,\left (2\,a^3+\frac {3\,a}{2}\right )}{b^4\,\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{19}\,\left (2\,a^3+\frac {3\,a}{2}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{19}}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^3\,\left (\frac {58\,a^3}{3}+\frac {29\,a}{2}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^3}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{17}\,\left (\frac {58\,a^3}{3}+\frac {29\,a}{2}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{17}}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^5\,\left (\frac {3584\,a^5}{5}-\frac {4552\,a^3}{3}+654\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^5}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{15}\,\left (\frac {3584\,a^5}{5}-\frac {4552\,a^3}{3}+654\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{15}}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^7\,\left (11776\,a^5-16024\,a^3+4622\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^7}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{13}\,\left (11776\,a^5-16024\,a^3+4622\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{13}}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^9\,\left (39936\,a^5-48012\,a^3+11095\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^9}-\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{11}\,\left (39936\,a^5-48012\,a^3+11095\,a\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{11}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4\,\sqrt {a-1}\,\sqrt {a+1}\,\left (64\,a^4-128\,a^2+64\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{16}\,\sqrt {a-1}\,\sqrt {a+1}\,\left (64\,a^4-128\,a^2+64\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{16}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6\,\sqrt {a-1}\,\sqrt {a+1}\,\left (3712\,a^4-\frac {12544\,a^2}{3}+\frac {1408}{3}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{14}\,\sqrt {a-1}\,\sqrt {a+1}\,\left (3712\,a^4-\frac {12544\,a^2}{3}+\frac {1408}{3}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{14}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8\,\sqrt {a-1}\,\sqrt {a+1}\,\left (25536\,a^4-\frac {56960\,a^2}{3}+\frac {4928}{3}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{12}\,\sqrt {a-1}\,\sqrt {a+1}\,\left (25536\,a^4-\frac {56960\,a^2}{3}+\frac {4928}{3}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{12}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}\,\sqrt {a-1}\,\sqrt {a+1}\,\left (\frac {231168\,a^4}{5}-\frac {160256\,a^2}{5}+\frac {11008}{5}\right )}{b^4\,{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}}{\frac {45\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^4}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^4}-\frac {10\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^2}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^2}-\frac {120\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^6}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^6}+\frac {210\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^8}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^8}-\frac {252\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{10}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{10}}+\frac {210\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{12}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{12}}-\frac {120\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{14}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{14}}+\frac {45\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{16}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{16}}-\frac {10\,{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{18}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{18}}+\frac {{\left (\sqrt {a-1}-\sqrt {a+b\,x-1}\right )}^{20}}{{\left (\sqrt {a+1}-\sqrt {a+b\,x+1}\right )}^{20}}+1}+\frac {b\,x^5}{5}+\frac {a\,\mathrm {atanh}\left (\frac {\sqrt {a-1}-\sqrt {a+b\,x-1}}{\sqrt {a+1}-\sqrt {a+b\,x+1}}\right )\,\left (4\,a^2+3\right )}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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