Optimal. Leaf size=270 \[ -\frac{2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{12 (1-a x)^4}{7 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{10 (1-a x)^3}{3 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{(1-a x)^2}{3 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.42703, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {6159, 6129, 98, 150, 143, 41, 216} \[ -\frac{2 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{82 (a x+1) (1-a x)^4}{105 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^3 (37 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{12 (1-a x)^4}{7 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{10 (1-a x)^3}{3 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{(1-a x)^2}{3 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6159
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \, dx &=\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{e^{-2 \tanh ^{-1}(a x)} x^7}{(1-a x)^{7/2} (1+a x)^{7/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x^7}{(1-a x)^{5/2} (1+a x)^{9/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x^5 (6+4 a x)}{(1-a x)^{3/2} (1+a x)^{9/2}} \, dx}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x^4 \left (-50 a-14 a^2 x\right )}{\sqrt{1-a x} (1+a x)^{9/2}} \, dx}{3 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x^3 \left (-144 a^2-62 a^3 x\right )}{\sqrt{1-a x} (1+a x)^{7/2}} \, dx}{21 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}-\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x^2 \left (-246 a^3-228 a^4 x\right )}{\sqrt{1-a x} (1+a x)^{5/2}} \, dx}{105 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}-\frac{2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}-\frac{\left ((1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{x \left (-36 a^4-666 a^5 x\right )}{\sqrt{1-a x} (1+a x)^{3/2}} \, dx}{315 a^{10} \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}-\frac{2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}-\frac{2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}-\frac{\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}-\frac{2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}-\frac{2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}-\frac{\left (2 (1-a x)^{7/2} (1+a x)^{7/2}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{a^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{(1-a x)^2}{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}-\frac{10 (1-a x)^3}{3 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{12 (1-a x)^4}{7 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}-\frac{82 (1-a x)^4 (1+a x)}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}-\frac{2 (1-a x)^4 (1+a x)^2}{35 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^5}-\frac{2 (1-a x)^4 (1+a x)^3 (72+37 a x)}{35 a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}-\frac{2 (1-a x)^{7/2} (1+a x)^{7/2} \sin ^{-1}(a x)}{a^8 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ \end{align*}
Mathematica [A] time = 0.120931, size = 131, normalized size = 0.49 \[ \frac{-105 a^6 x^6-562 a^5 x^5-74 a^4 x^4+1226 a^3 x^3+636 a^2 x^2+210 (a x-1) (a x+1)^3 \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-654 a x-432}{105 a^2 x (a x-1) \sqrt{c-\frac{c}{a^2 x^2}} (a c x+c)^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.15, size = 572, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.97124, size = 1045, normalized size = 3.87 \begin{align*} \left [\frac{105 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{c} \log \left (2 \, a^{2} c x^{2} + 2 \, a^{2} \sqrt{c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) -{\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}, -\frac{210 \,{\left (a^{6} x^{6} + 2 \, a^{5} x^{5} - a^{4} x^{4} - 4 \, a^{3} x^{3} - a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{a^{2} \sqrt{-c} x^{2} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) +{\left (105 \, a^{7} x^{7} + 562 \, a^{6} x^{6} + 74 \, a^{5} x^{5} - 1226 \, a^{4} x^{4} - 636 \, a^{3} x^{3} + 654 \, a^{2} x^{2} + 432 \, a x\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{105 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a x}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx - \int - \frac{1}{a c^{3} x \sqrt{c - \frac{c}{a^{2} x^{2}}} + c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a x} - \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{2} x^{2}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{3} x^{3}} + \frac{3 c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{4} x^{4}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{5} x^{5}} - \frac{c^{3} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{a^{6} x^{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{a^{2} x^{2} - 1}{{\left (a x + 1\right )}^{2}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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