Integrand size = 24, antiderivative size = 153 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \]
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Time = 0.14 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6302, 6276, 685, 655, 201, 223, 209} \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a}-\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {(a x+1) \left (c-a^2 c x^2\right )^{7/2}}{8 a}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a} \]
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Rule 201
Rule 209
Rule 223
Rule 655
Rule 685
Rule 6276
Rule 6302
Rubi steps \begin{align*} \text {integral}& = -\int e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx \\ & = -\left (c \int (1+a x)^2 \left (c-a^2 c x^2\right )^{5/2} \, dx\right ) \\ & = \frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{8} (9 c) \int (1+a x) \left (c-a^2 c x^2\right )^{5/2} \, dx \\ & = \frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{8} (9 c) \int \left (c-a^2 c x^2\right )^{5/2} \, dx \\ & = -\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{16} \left (15 c^2\right ) \int \left (c-a^2 c x^2\right )^{3/2} \, dx \\ & = -\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{64} \left (45 c^3\right ) \int \sqrt {c-a^2 c x^2} \, dx \\ & = -\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{128} \left (45 c^4\right ) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx \\ & = -\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {1}{128} \left (45 c^4\right ) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right ) \\ & = -\frac {45}{128} c^3 x \sqrt {c-a^2 c x^2}-\frac {15}{64} c^2 x \left (c-a^2 c x^2\right )^{3/2}-\frac {3}{16} c x \left (c-a^2 c x^2\right )^{5/2}+\frac {9 \left (c-a^2 c x^2\right )^{7/2}}{56 a}+\frac {(1+a x) \left (c-a^2 c x^2\right )^{7/2}}{8 a}-\frac {45 c^{7/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{128 a} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {c^3 \sqrt {c-a^2 c x^2} \left (\sqrt {1+a x} \left (256-837 a x-187 a^2 x^2+978 a^3 x^3+558 a^4 x^4-600 a^5 x^5-424 a^6 x^6+144 a^7 x^7+112 a^8 x^8\right )+630 \sqrt {1-a x} \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{896 a \sqrt {1-a x} \sqrt {1-a^2 x^2}} \]
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Time = 0.59 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.80
method | result | size |
risch | \(\frac {\left (112 a^{7} x^{7}+256 a^{6} x^{6}-168 a^{5} x^{5}-768 a^{4} x^{4}-210 a^{3} x^{3}+768 a^{2} x^{2}+581 a x -256\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{896 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {45 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right ) c^{4}}{128 \sqrt {a^{2} c}}\) | \(122\) |
default | \(\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{8}+\frac {7 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{8}+\frac {\frac {2 \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}{7}-2 a c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{12 a^{2} c}+\frac {5 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \left (-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 a^{2} c \left (x -\frac {1}{a}\right )-2 a c \right ) \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )}{a}\) | \(375\) |
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Time = 0.27 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.87 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\left [\frac {315 \, \sqrt {-c} c^{3} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - 2 \, {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{1792 \, a}, \frac {315 \, c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) - {\left (112 \, a^{7} c^{3} x^{7} + 256 \, a^{6} c^{3} x^{6} - 168 \, a^{5} c^{3} x^{5} - 768 \, a^{4} c^{3} x^{4} - 210 \, a^{3} c^{3} x^{3} + 768 \, a^{2} c^{3} x^{2} + 581 \, a c^{3} x - 256 \, c^{3}\right )} \sqrt {-a^{2} c x^{2} + c}}{896 \, a}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 614 vs. \(2 (139) = 278\).
Time = 2.97 (sec) , antiderivative size = 614, normalized size of antiderivative = 4.01 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\begin {cases} \frac {- 2 c^{3} \left (\begin {cases} \left (\frac {a^{2} x^{2}}{3} - \frac {1}{3}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{2} \sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) + 4 c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{4} x^{4}}{5} - \frac {a^{2} x^{2}}{15} - \frac {2}{15}\right ) & \text {for}\: c \neq 0 \\\frac {a^{4} \sqrt {c} x^{4}}{4} & \text {otherwise} \end {cases}\right ) - 2 c^{3} \left (\begin {cases} \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{6} x^{6}}{7} - \frac {a^{4} x^{4}}{35} - \frac {4 a^{2} x^{2}}{105} - \frac {8}{105}\right ) & \text {for}\: c \neq 0 \\\frac {a^{6} \sqrt {c} x^{6}}{6} & \text {otherwise} \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {5 c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{128} + \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{7} x^{7}}{8} - \frac {a^{5} x^{5}}{48} - \frac {5 a^{3} x^{3}}{192} - \frac {5 a x}{128}\right ) & \text {for}\: c \neq 0 \\\frac {a^{7} \sqrt {c} x^{7}}{7} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{16} + \sqrt {- a^{2} c x^{2} + c} \left (\frac {a^{5} x^{5}}{6} - \frac {a^{3} x^{3}}{24} - \frac {a x}{16}\right ) & \text {for}\: c \neq 0 \\\frac {a^{5} \sqrt {c} x^{5}}{5} & \text {otherwise} \end {cases}\right ) + c^{3} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{8} + \left (\frac {a^{3} x^{3}}{4} - \frac {a x}{8}\right ) \sqrt {- a^{2} c x^{2} + c} & \text {for}\: c \neq 0 \\\frac {a^{3} \sqrt {c} x^{3}}{3} & \text {otherwise} \end {cases}\right ) - c^{3} \left (\begin {cases} \frac {a x \sqrt {- a^{2} c x^{2} + c}}{2} + \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a c x + 2 \sqrt {- c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- c}} & \text {for}\: c \neq 0 \\\frac {a x \log {\left (a x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} & \text {for}\: c \neq 0 \\a \sqrt {c} x & \text {otherwise} \end {cases}\right )}{a} & \text {for}\: a \neq 0 \\- c^{\frac {7}{2}} x & \text {otherwise} \end {cases} \]
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Time = 0.29 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.13 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {1}{8} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} x - \frac {3}{16} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c x - \frac {15}{64} \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2} x - \frac {5}{8} \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3} x + \frac {35}{128} \, \sqrt {-a^{2} c x^{2} + c} c^{3} x - \frac {5 \, c^{5} \arcsin \left (a x - 2\right )}{8 \, a \left (-c\right )^{\frac {3}{2}}} + \frac {35 \, c^{\frac {7}{2}} \arcsin \left (a x\right )}{128 \, a} + \frac {2 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}{7 \, a} + \frac {5 \, \sqrt {a^{2} c x^{2} - 4 \, a c x + 3 \, c} c^{3}}{4 \, a} \]
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Time = 0.32 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.92 \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {45 \, c^{4} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{128 \, \sqrt {-c} {\left | a \right |}} + \frac {1}{896} \, \sqrt {-a^{2} c x^{2} + c} {\left (\frac {256 \, c^{3}}{a} - {\left (581 \, c^{3} + 2 \, {\left (384 \, a c^{3} - {\left (105 \, a^{2} c^{3} + 4 \, {\left (96 \, a^{3} c^{3} + {\left (21 \, a^{4} c^{3} - 2 \, {\left (7 \, a^{6} c^{3} x + 16 \, a^{5} c^{3}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
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Timed out. \[ \int e^{2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^{7/2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \]
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