Optimal. Leaf size=283 \[ \frac{142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{(a x+1)^2}{7 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.502057, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6167, 6159, 6129, 98, 150, 143, 41, 216} \[ \frac{142 (a x+1)^2 (1-a x)^4}{35 a^6 x^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^3 (107 a x+72) (1-a x)^4}{35 a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{782 (a x+1)^2 (1-a x)^3}{105 a^5 x^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{124 (a x+1)^2 (1-a x)^2}{105 a^4 x^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^2 (1-a x)}{5 a^3 x^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{(a x+1)^2}{7 a^2 x \left (c-\frac{c}{a^2 x^2}\right )^{7/2}}-\frac{2 (a x+1)^{7/2} (1-a x)^{7/2} \sin ^{-1}(a x)}{a^8 x^7 \left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6159
Rule 6129
Rule 98
Rule 150
Rule 143
Rule 41
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{2 \coth ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2}} \, dx &=-\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)^{9/2} (1+a x)^{5/2}} \, dx}{\left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{(1+a x)^2}{7 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+8 a x)}{(1-a x)^{7/2} (1+a x)^{5/2}} \, dx}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{(1+a x)^2}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 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 (-70 a-54 a^2 x\right )}{(1-a x)^{5/2} (1+a x)^{5/2}} \, dx}{35 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{(1+a x)^2}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 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 (496 a^2+286 a^3 x\right )}{(1-a x)^{3/2} (1+a x)^{5/2}} \, dx}{105 a^6 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{(1+a x)^2}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}+\frac{782 (1-a x)^3 (1+a x)^2}{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 (-2346 a^3-1068 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}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}+\frac{782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}+\frac{142 (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 (-2556 a^4-1926 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}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}+\frac{782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}+\frac{142 (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+107 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}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}+\frac{782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}+\frac{142 (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+107 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}{7 a^2 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x}+\frac{2 (1-a x) (1+a x)^2}{5 a^3 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^2}-\frac{124 (1-a x)^2 (1+a x)^2}{105 a^4 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^3}+\frac{782 (1-a x)^3 (1+a x)^2}{105 a^5 \left (c-\frac{c}{a^2 x^2}\right )^{7/2} x^4}+\frac{142 (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+107 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 = 0.12093, size = 133, normalized size = 0.47 \[ \frac{105 a^6 x^6-562 a^5 x^5+74 a^4 x^4+1226 a^3 x^3-636 a^2 x^2+210 (a x-1)^3 (a x+1) \sqrt{a^2 x^2-1} \log \left (\sqrt{a^2 x^2-1}+a x\right )-654 a x+432}{105 a^2 c^3 x (a x-1)^3 (a x+1) \sqrt{c-\frac{c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.212, size = 572, normalized size = 2. \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 x + 1}{{\left (a x - 1\right )}{\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 = 2.60046, size = 1045, normalized size = 3.69 \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 + 1}{\left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{\frac{7}{2}} \left (a x - 1\right )}\, 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 x + 1}{{\left (a x - 1\right )}{\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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