Integrand size = 20, antiderivative size = 161 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {4096 c^4 \sqrt {1-\frac {1}{a^2 x^2}} x}{315 \sqrt {c-a c x}}+\frac {1024}{315} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x \sqrt {c-a c x}+\frac {128}{105} c^2 \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{3/2}+\frac {32}{63} c \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{5/2}+\frac {2}{9} \sqrt {1-\frac {1}{a^2 x^2}} x (c-a c x)^{7/2} \]
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Time = 0.15 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.58, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {5504 \sqrt {\frac {1}{a x}+1} (c-a c x)^{7/2}}{315 a^4 x^3 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 x \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {32 \sqrt {\frac {1}{a x}+1} \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {256 \sqrt {\frac {1}{a x}+1} (c-a c x)^{7/2}}{45 a^3 x^2 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {128 \sqrt {\frac {1}{a x}+1} (c-a c x)^{7/2}}{105 a^2 x \left (1-\frac {1}{a x}\right )^{7/2}} \]
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Rule 37
Rule 79
Rule 91
Rule 96
Rule 6311
Rule 6316
Rubi steps \begin{align*} \text {integral}& = \frac {(c-a c x)^{7/2} \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{7/2} x^{7/2} \, dx}{\left (1-\frac {1}{a x}\right )^{7/2} x^{7/2}} \\ & = -\frac {\left (\left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^{11/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {\left (16 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^3}{x^{9/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{9 a \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {2 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (64 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^2}{x^{7/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{21 a^2 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {128 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (128 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {-\frac {7}{a}+\frac {5 x}{2 a^2}}{x^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 a^2 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {256 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{45 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {128 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}-\frac {\left (2752 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {1}{x^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{315 a^4 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {32 \left (a-\frac {1}{x}\right )^3 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2}}+\frac {5504 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{315 a^4 \left (1-\frac {1}{a x}\right )^{7/2} x^3}-\frac {256 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{45 a^3 \left (1-\frac {1}{a x}\right )^{7/2} x^2}+\frac {128 \sqrt {1+\frac {1}{a x}} (c-a c x)^{7/2}}{105 a^2 \left (1-\frac {1}{a x}\right )^{7/2} x}+\frac {2 \left (a-\frac {1}{x}\right )^4 \sqrt {1+\frac {1}{a x}} x (c-a c x)^{7/2}}{9 a^4 \left (1-\frac {1}{a x}\right )^{7/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.48 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x} \left (2867-1276 a x+642 a^2 x^2-220 a^3 x^3+35 a^4 x^4\right )}{315 a \sqrt {1-\frac {1}{a x}}} \]
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Time = 0.43 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.43
method | result | size |
risch | \(\frac {2 c^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right ) \left (a x +1\right )}{315 \sqrt {-c \left (a x -1\right )}\, a}\) | \(69\) |
gosper | \(\frac {2 \left (a x +1\right ) \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right ) \left (-a c x +c \right )^{\frac {7}{2}} \sqrt {\frac {a x -1}{a x +1}}}{315 a \left (a x -1\right )^{4}}\) | \(72\) |
default | \(-\frac {2 \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{3} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right )}{315 \left (a x -1\right ) a}\) | \(76\) |
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Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} c^{3} x^{5} - 185 \, a^{4} c^{3} x^{4} + 422 \, a^{3} c^{3} x^{3} - 634 \, a^{2} c^{3} x^{2} + 1591 \, a c^{3} x + 2867 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{315 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.70 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{5} \sqrt {-c} c^{3} x^{5} - 185 \, a^{4} \sqrt {-c} c^{3} x^{4} + 422 \, a^{3} \sqrt {-c} c^{3} x^{3} - 634 \, a^{2} \sqrt {-c} c^{3} x^{2} + 1591 \, a \sqrt {-c} c^{3} x + 2867 \, \sqrt {-c} c^{3}\right )} {\left (a x - 1\right )}}{315 \, {\left (a^{2} x - a\right )} \sqrt {a x + 1}} \]
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Exception generated. \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.35 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63 \[ \int e^{-\coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (35\,a^4\,x^4-150\,a^3\,x^3+272\,a^2\,x^2-362\,a\,x+1229\right )}{315\,a}-\frac {8192\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{315\,a\,\left (a\,x-1\right )} \]
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