Integrand size = 20, antiderivative size = 311 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {11776 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}} \]
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Time = 0.17 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6311, 6316, 96, 91, 79, 37} \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\frac {11776 (c-a c x)^{7/2}}{63 a^5 x^4 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {2 x \left (a-\frac {1}{x}\right )^5 (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}-\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 x \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}+\frac {5120 (c-a c x)^{7/2}}{63 a^4 x^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}}-\frac {512 (c-a c x)^{7/2}}{63 a^3 x^2 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {\frac {1}{a x}+1}} \]
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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^{-3 \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 )^5}{x^{11/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{\left (1-\frac {1}{a x}\right )^{7/2}} \\ & = \frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (20 \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^{9/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{9 a \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {\left (320 \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^{7/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{63 a^2 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (256 \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^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{21 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}-\frac {512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (512 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {-\frac {5}{a}+\frac {3 x}{2 a^2}}{x^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {\left (5888 \left (\frac {1}{x}\right )^{7/2} (c-a c x)^{7/2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2}} \\ & = -\frac {40 \left (a-\frac {1}{x}\right )^4 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}}+\frac {11776 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^4}+\frac {5120 (c-a c x)^{7/2}}{63 a^4 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^3}-\frac {512 (c-a c x)^{7/2}}{63 a^3 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x^2}+\frac {128 \left (a-\frac {1}{x}\right )^3 (c-a c x)^{7/2}}{63 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}} x}+\frac {2 \left (a-\frac {1}{x}\right )^5 x (c-a c x)^{7/2}}{9 a^5 \left (1-\frac {1}{a x}\right )^{7/2} \sqrt {1+\frac {1}{a x}}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.24 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 c^3 \sqrt {c-a c x} \left (5797+2867 a x-638 a^2 x^2+214 a^3 x^3-55 a^4 x^4+7 a^5 x^5\right )}{63 a^2 \sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Time = 0.45 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.26
method | result | size |
gosper | \(\frac {2 \left (a x +1\right ) \left (7 a^{5} x^{5}-55 a^{4} x^{4}+214 a^{3} x^{3}-638 a^{2} x^{2}+2867 a x +5797\right ) \left (-a c x +c \right )^{\frac {7}{2}} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{63 a \left (a x -1\right )^{5}}\) | \(80\) |
default | \(-\frac {2 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}\, c^{3} \left (7 a^{5} x^{5}-55 a^{4} x^{4}+214 a^{3} x^{3}-638 a^{2} x^{2}+2867 a x +5797\right )}{63 \left (a x -1\right )^{2} a}\) | \(84\) |
risch | \(\frac {2 \left (7 a^{4} x^{4}-62 a^{3} x^{3}+276 a^{2} x^{2}-914 a x +3781\right ) \left (a x +1\right ) c^{4} \sqrt {\frac {a x -1}{a x +1}}}{63 a \sqrt {-c \left (a x -1\right )}}+\frac {64 c^{4} \sqrt {\frac {a x -1}{a x +1}}}{a \sqrt {-c \left (a x -1\right )}}\) | \(103\) |
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Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.30 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (7 \, a^{5} c^{3} x^{5} - 55 \, a^{4} c^{3} x^{4} + 214 \, a^{3} c^{3} x^{3} - 638 \, a^{2} c^{3} x^{2} + 2867 \, a c^{3} x + 5797 \, c^{3}\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}}{63 \, {\left (a^{2} x - a\right )}} \]
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Timed out. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.44 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (7 \, a^{6} \sqrt {-c} c^{3} x^{6} - 48 \, a^{5} \sqrt {-c} c^{3} x^{5} + 159 \, a^{4} \sqrt {-c} c^{3} x^{4} - 424 \, a^{3} \sqrt {-c} c^{3} x^{3} + 2229 \, a^{2} \sqrt {-c} c^{3} x^{2} + 8664 \, a \sqrt {-c} c^{3} x + 5797 \, \sqrt {-c} c^{3}\right )} {\left (a x - 1\right )}^{2}}{63 \, {\left (a^{3} x^{2} - 2 \, a^{2} x + a\right )} {\left (a x + 1\right )}^{\frac {3}{2}}} \]
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Exception generated. \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.85 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.33 \[ \int e^{-3 \coth ^{-1}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}\,\left (7\,a^4\,x^4-48\,a^3\,x^3+166\,a^2\,x^2-472\,a\,x+2395\right )}{63\,a}-\frac {16384\,c^3\,\sqrt {c-a\,c\,x}\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{63\,a\,\left (a\,x-1\right )} \]
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