Integrand size = 18, antiderivative size = 69 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=-\frac {3}{8} \sqrt {-1+x} \sqrt {1+x}+\frac {1}{24} (7-2 x) (-1+x)^{3/2} \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x^2 \sqrt {1+x}+\frac {3 \text {arccosh}(x)}{8} \]
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Time = 0.01 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {102, 152, 52, 54} \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {3 \text {arccosh}(x)}{8}+\frac {1}{4} (x-1)^{3/2} \sqrt {x+1} x^2+\frac {1}{24} (7-2 x) (x-1)^{3/2} \sqrt {x+1}-\frac {3}{8} \sqrt {x-1} \sqrt {x+1} \]
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Rule 52
Rule 54
Rule 102
Rule 152
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} (-1+x)^{3/2} x^2 \sqrt {1+x}+\frac {1}{4} \int \frac {(2-x) \sqrt {-1+x} x}{\sqrt {1+x}} \, dx \\ & = \frac {1}{24} (7-2 x) (-1+x)^{3/2} \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x^2 \sqrt {1+x}-\frac {3}{8} \int \frac {\sqrt {-1+x}}{\sqrt {1+x}} \, dx \\ & = -\frac {3}{8} \sqrt {-1+x} \sqrt {1+x}+\frac {1}{24} (7-2 x) (-1+x)^{3/2} \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x^2 \sqrt {1+x}+\frac {3}{8} \int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx \\ & = -\frac {3}{8} \sqrt {-1+x} \sqrt {1+x}+\frac {1}{24} (7-2 x) (-1+x)^{3/2} \sqrt {1+x}+\frac {1}{4} (-1+x)^{3/2} x^2 \sqrt {1+x}+\frac {3}{8} \cosh ^{-1}(x) \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {\sqrt {\frac {-1+x}{1+x}} \left (\sqrt {-1+x} \left (-16-7 x+x^2-2 x^3+6 x^4\right )-18 \sqrt {1+x} \log \left (\sqrt {-1+x}-\sqrt {1+x}\right )\right )}{24 \sqrt {-1+x}} \]
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Time = 1.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87
method | result | size |
risch | \(\frac {\left (6 x^{3}-8 x^{2}+9 x -16\right ) \sqrt {x -1}\, \sqrt {x +1}}{24}+\frac {3 \ln \left (x +\sqrt {x^{2}-1}\right ) \sqrt {\left (x -1\right ) \left (x +1\right )}}{8 \sqrt {x -1}\, \sqrt {x +1}}\) | \(60\) |
default | \(\frac {\sqrt {x -1}\, \sqrt {x +1}\, \left (6 x^{3} \sqrt {x^{2}-1}-8 x^{2} \sqrt {x^{2}-1}+9 x \sqrt {x^{2}-1}+9 \ln \left (x +\sqrt {x^{2}-1}\right )-16 \sqrt {x^{2}-1}\right )}{24 \sqrt {x^{2}-1}}\) | \(76\) |
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {1}{24} \, {\left (6 \, x^{3} - 8 \, x^{2} + 9 \, x - 16\right )} \sqrt {x + 1} \sqrt {x - 1} - \frac {3}{8} \, \log \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) \]
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\[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\int \frac {x^{3} \sqrt {x - 1}}{\sqrt {x + 1}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {1}{4} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} x - \frac {1}{3} \, {\left (x^{2} - 1\right )}^{\frac {3}{2}} + \frac {5}{8} \, \sqrt {x^{2} - 1} x - \sqrt {x^{2} - 1} + \frac {3}{8} \, \log \left (2 \, x + 2 \, \sqrt {x^{2} - 1}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {1}{24} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {x - 1} - \frac {3}{4} \, \log \left (\sqrt {x + 1} - \sqrt {x - 1}\right ) \]
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Time = 32.74 (sec) , antiderivative size = 473, normalized size of antiderivative = 6.86 \[ \int \frac {\sqrt {-1+x} x^3}{\sqrt {1+x}} \, dx=\frac {3\,\mathrm {atanh}\left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )}{2}+\frac {\frac {23\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{2\,{\left (\sqrt {x+1}-1\right )}^3}-\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^4\,64{}\mathrm {i}}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {333\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^5}{2\,{\left (\sqrt {x+1}-1\right )}^5}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^6\,256{}\mathrm {i}}{3\,{\left (\sqrt {x+1}-1\right )}^6}+\frac {671\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^7}{2\,{\left (\sqrt {x+1}-1\right )}^7}-\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^8\,128{}\mathrm {i}}{3\,{\left (\sqrt {x+1}-1\right )}^8}+\frac {671\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^9}{2\,{\left (\sqrt {x+1}-1\right )}^9}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{10}\,256{}\mathrm {i}}{3\,{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {333\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{11}}{2\,{\left (\sqrt {x+1}-1\right )}^{11}}-\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{12}\,64{}\mathrm {i}}{{\left (\sqrt {x+1}-1\right )}^{12}}+\frac {23\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{13}}{2\,{\left (\sqrt {x+1}-1\right )}^{13}}-\frac {3\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{15}}{2\,{\left (\sqrt {x+1}-1\right )}^{15}}-\frac {3\,\left (\sqrt {x-1}-\mathrm {i}\right )}{2\,\left (\sqrt {x+1}-1\right )}}{1+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {70\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {56\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{10}}{{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {28\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{12}}{{\left (\sqrt {x+1}-1\right )}^{12}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^{14}}{{\left (\sqrt {x+1}-1\right )}^{14}}+\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^{16}}{{\left (\sqrt {x+1}-1\right )}^{16}}-\frac {8\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}} \]
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