Optimal. Leaf size=190 \[ \frac {(a x+1)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (a x+1)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (a x+1)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (a x+1)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {14 (a x+1)^2}{3 a c^4 \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2}}{a c^4}-\frac {7 \sin ^{-1}(a x)}{a c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.44, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6131, 6128, 852, 1635, 789, 669, 641, 216} \[ \frac {(a x+1)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (a x+1)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (a x+1)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (a x+1)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {14 (a x+1)^2}{3 a c^4 \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2}}{a c^4}-\frac {7 \sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 641
Rule 669
Rule 789
Rule 852
Rule 1635
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=\frac {a^4 \int \frac {e^{3 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4 \left (1-a^2 x^2\right )^{3/2}}{(1-a x)^7} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)^7}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {a^4 \int \frac {(1+a x)^6 \left (\frac {7}{a^4}+\frac {9 x}{a^3}+\frac {9 x^2}{a^2}+\frac {9 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^4 \int \frac {(1+a x)^5 \left (\frac {155}{a^4}+\frac {126 x}{a^3}+\frac {63 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{63 c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^4 \int \frac {\left (\frac {945}{a^4}+\frac {315 x}{a^3}\right ) (1+a x)^4}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{315 c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {7 \int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {14 (1+a x)^2}{3 a c^4 \sqrt {1-a^2 x^2}}-\frac {7 \int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {14 (1+a x)^2}{3 a c^4 \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2}}{a c^4}-\frac {7 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {(1+a x)^7}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {34 (1+a x)^6}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {344 (1+a x)^5}{315 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 (1+a x)^4}{3 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {14 (1+a x)^2}{3 a c^4 \sqrt {1-a^2 x^2}}+\frac {7 \sqrt {1-a^2 x^2}}{a c^4}-\frac {7 \sin ^{-1}(a x)}{a c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 77, normalized size = 0.41 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (315 a^5 x^5-6539 a^4 x^4+19780 a^3 x^3-25347 a^2 x^2+15115 a x-3464\right )}{(a x-1)^5}-2205 \sin ^{-1}(a x)}{315 a c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.79, size = 213, normalized size = 1.12 \[ \frac {3464 \, a^{5} x^{5} - 17320 \, a^{4} x^{4} + 34640 \, a^{3} x^{3} - 34640 \, a^{2} x^{2} + 17320 \, a x + 4410 \, {\left (a^{5} x^{5} - 5 \, a^{4} x^{4} + 10 \, a^{3} x^{3} - 10 \, a^{2} x^{2} + 5 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (315 \, a^{5} x^{5} - 6539 \, a^{4} x^{4} + 19780 \, a^{3} x^{3} - 25347 \, a^{2} x^{2} + 15115 \, a x - 3464\right )} \sqrt {-a^{2} x^{2} + 1} - 3464}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.67, size = 288, normalized size = 1.52 \[ -\frac {7 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c^{4} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{4}} - \frac {2 \, {\left (\frac {26136 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {93834 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {188706 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {229194 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {167580 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {75810 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} + \frac {19530 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{14} x^{7}} - \frac {2205 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{8}}{a^{16} x^{8}} - 3149\right )}}{315 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{9} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 300, normalized size = 1.58 \[ -\frac {a \,x^{2}}{c^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {27}{a \,c^{4} \sqrt {-a^{2} x^{2}+1}}+\frac {70 x}{c^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} \sqrt {a^{2}}}+\frac {5002}{315 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8543}{315 a^{2} c^{4} \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {17086 x}{315 c^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {356}{63 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {8}{9 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a x}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.87, size = 579, normalized size = 3.05 \[ \frac {4\,\sqrt {1-a^2\,x^2}}{9\,\sqrt {-a^2}\,\left (5\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+10\,a^2\,c^4\,x^3\,\sqrt {-a^2}-5\,a^3\,c^4\,x^4\,\sqrt {-a^2}+a^4\,c^4\,x^5\,\sqrt {-a^2}-10\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {44\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {7\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}-\frac {8\,a^3\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {1754\,a^4\,\sqrt {1-a^2\,x^2}}{315\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}-\frac {20\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {4964\,\sqrt {1-a^2\,x^2}}{315\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {697\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}+\frac {16\,a^6\,\sqrt {1-a^2\,x^2}}{63\,\left (a^{11}\,c^4\,x^4-4\,a^{10}\,c^4\,x^3+6\,a^9\,c^4\,x^2-4\,a^8\,c^4\,x+a^7\,c^4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{4} \left (\int \frac {x^{4}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a x^{5}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________