Integrand size = 24, antiderivative size = 239 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac {42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {5040 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt {c-a^2 c x^2}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6320, 6319} \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {5040 (n-a x) e^{n \coth ^{-1}(a x)}}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt {c-a^2 c x^2}}-\frac {840 (n-3 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {42 (n-5 a x) e^{n \coth ^{-1}(a x)}}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {(n-7 a x) e^{n \coth ^{-1}(a x)}}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}} \]
[In]
[Out]
Rule 6319
Rule 6320
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}+\frac {42 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{c \left (49-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac {42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}+\frac {840 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{c^2 \left (25-n^2\right ) \left (49-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac {42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}+\frac {5040 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right )} \\ & = -\frac {e^{n \coth ^{-1}(a x)} (n-7 a x)}{a c \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{7/2}}-\frac {42 e^{n \coth ^{-1}(a x)} (n-5 a x)}{a c^2 \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{5/2}}-\frac {840 e^{n \coth ^{-1}(a x)} (n-3 a x)}{a c^3 \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \left (c-a^2 c x^2\right )^{3/2}}-\frac {5040 e^{n \coth ^{-1}(a x)} (n-a x)}{a c^4 \left (1-n^2\right ) \left (9-n^2\right ) \left (25-n^2\right ) \left (49-n^2\right ) \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 1.95 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.09 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {a e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right ) x^2 \left (-\frac {35 n}{-1+n^2}+\frac {35 a x}{-1+n^2}-\frac {63 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (3 \coth ^{-1}(a x)\right )}{-9+n^2}+\frac {35 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (5 \coth ^{-1}(a x)\right )}{-25+n^2}-\frac {7 a \sqrt {1-\frac {1}{a^2 x^2}} x \cosh \left (7 \coth ^{-1}(a x)\right )}{-49+n^2}+\frac {21 a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (3 \coth ^{-1}(a x)\right )}{-9+n^2}-\frac {7 a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (5 \coth ^{-1}(a x)\right )}{-25+n^2}+\frac {a n \sqrt {1-\frac {1}{a^2 x^2}} x \sinh \left (7 \coth ^{-1}(a x)\right )}{-49+n^2}\right )}{64 c^3 \left (c-a^2 c x^2\right )^{3/2}} \]
[In]
[Out]
Time = 0.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.91
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (a x +1\right ) \left (5040 a^{7} x^{7}-5040 n \,a^{6} x^{6}+2520 a^{5} n^{2} x^{5}-840 a^{4} n^{3} x^{4}-17640 a^{5} x^{5}+210 a^{3} n^{4} x^{3}+15960 n \,a^{4} x^{4}-42 a^{2} n^{5} x^{2}-7140 a^{3} n^{2} x^{3}+7 a \,n^{6} x +2100 a^{2} n^{3} x^{2}-n^{7}+22050 a^{3} x^{3}-455 a \,n^{4} x -17178 n \,x^{2} a^{2}+77 n^{5}+6433 n^{2} x a -1519 n^{3}-11025 a x +6483 n \right ) {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )}}{a \left (n^{8}-84 n^{6}+1974 n^{4}-12916 n^{2}+11025\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\) | \(218\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.90 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {{\left (5040 \, a^{7} x^{7} - 5040 \, a^{6} n x^{6} - n^{7} + 2520 \, {\left (a^{5} n^{2} - 7 \, a^{5}\right )} x^{5} + 77 \, n^{5} - 840 \, {\left (a^{4} n^{3} - 19 \, a^{4} n\right )} x^{4} + 210 \, {\left (a^{3} n^{4} - 34 \, a^{3} n^{2} + 105 \, a^{3}\right )} x^{3} - 1519 \, n^{3} - 42 \, {\left (a^{2} n^{5} - 50 \, a^{2} n^{3} + 409 \, a^{2} n\right )} x^{2} + 7 \, {\left (a n^{6} - 65 \, a n^{4} + 919 \, a n^{2} - 1575 \, a\right )} x + 6483 \, n\right )} \sqrt {-a^{2} c x^{2} + c} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a c^{5} n^{8} - 84 \, a c^{5} n^{6} + 1974 \, a c^{5} n^{4} + {\left (a^{9} c^{5} n^{8} - 84 \, a^{9} c^{5} n^{6} + 1974 \, a^{9} c^{5} n^{4} - 12916 \, a^{9} c^{5} n^{2} + 11025 \, a^{9} c^{5}\right )} x^{8} - 12916 \, a c^{5} n^{2} - 4 \, {\left (a^{7} c^{5} n^{8} - 84 \, a^{7} c^{5} n^{6} + 1974 \, a^{7} c^{5} n^{4} - 12916 \, a^{7} c^{5} n^{2} + 11025 \, a^{7} c^{5}\right )} x^{6} + 11025 \, a c^{5} + 6 \, {\left (a^{5} c^{5} n^{8} - 84 \, a^{5} c^{5} n^{6} + 1974 \, a^{5} c^{5} n^{4} - 12916 \, a^{5} c^{5} n^{2} + 11025 \, a^{5} c^{5}\right )} x^{4} - 4 \, {\left (a^{3} c^{5} n^{8} - 84 \, a^{3} c^{5} n^{6} + 1974 \, a^{3} c^{5} n^{4} - 12916 \, a^{3} c^{5} n^{2} + 11025 \, a^{3} c^{5}\right )} x^{2}} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}} \,d x } \]
[In]
[Out]
Time = 4.67 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.85 \[ \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {5040\,x^7}{c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {-n^7+77\,n^5-1519\,n^3+6483\,n}{a^7\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {5040\,n\,x^6}{a\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^5\,\left (2520\,n^2-17640\right )}{a^2\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {x^3\,\left (210\,n^4-7140\,n^2+22050\right )}{a^4\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}+\frac {7\,x\,\left (n^6-65\,n^4+919\,n^2-1575\right )}{a^6\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {840\,n\,x^4\,\left (n^2-19\right )}{a^3\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}-\frac {42\,n\,x^2\,\left (n^4-50\,n^2+409\right )}{a^5\,c^4\,\left (n^8-84\,n^6+1974\,n^4-12916\,n^2+11025\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}\right )} \]
[In]
[Out]