Optimal. Leaf size=163 \[ \frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.240775, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6157, 6149, 819, 641, 216} \[ \frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
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Rule 6157
Rule 6149
Rule 819
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (c-\frac{c}{a^2 x^2}\right )^4} \, dx &=\frac{a^8 \int \frac{e^{-\tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac{a^8 \int \frac{x^8 (1-a x)}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^6 \int \frac{x^6 (7-8 a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^4 \int \frac{x^4 (35-48 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{a^2 \int \frac{x^2 (105-192 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{105 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{\int \frac{105-384 a x}{\sqrt{1-a^2 x^2}} \, dx}{105 c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{c^4}\\ &=\frac{a^6 x^7 (1-a x)}{7 c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac{a^4 x^5 (7-8 a x)}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{a^2 x^3 (35-48 a x)}{105 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{x (35-64 a x)}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{128 \sqrt{1-a^2 x^2}}{35 a c^4}+\frac{\sin ^{-1}(a x)}{a c^4}\\ \end{align*}
Mathematica [A] time = 0.0961641, size = 126, normalized size = 0.77 \[ \frac{-105 a^7 x^7-281 a^6 x^6+559 a^5 x^5+965 a^4 x^4-715 a^3 x^3-1065 a^2 x^2+105 (a x-1)^2 (a x+1)^3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)+279 a x+384}{105 a c^4 (a x-1)^2 (a x+1)^3 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 438, normalized size = 2.7 \begin{align*}{\frac{1}{14\,{a}^{5}{c}^{4} \left ( x+{a}^{-1} \right ) ^{4}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{187}{672\,{a}^{4}{c}^{4} \left ( x+{a}^{-1} \right ) ^{3}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{53}{960\,{a}^{4}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-3}}+{\frac{47}{128\,{a}^{3}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-2}}+{\frac{187}{256\,a{c}^{4}}\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}-{\frac{187}{256\,{c}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{1}{160\,{a}^{5}{c}^{4}} \left ( -{a}^{2} \left ( x-{a}^{-1} \right ) ^{2}-2\,a \left ( x-{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}} \left ( x-{a}^{-1} \right ) ^{-4}}+{\frac{35}{32\,{a}^{3}{c}^{4} \left ( x+{a}^{-1} \right ) ^{2}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{443}{256\,a{c}^{4}}\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}+{\frac{443}{256\,{c}^{4}}\arctan \left ({x\sqrt{{a}^{2}}{\frac{1}{\sqrt{-{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) }}}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{112\,{a}^{6}{c}^{4} \left ( x+{a}^{-1} \right ) ^{5}} \left ( -{a}^{2} \left ( x+{a}^{-1} \right ) ^{2}+2\,a \left ( x+{a}^{-1} \right ) \right ) ^{{\frac{3}{2}}}} \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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38672, size = 632, normalized size = 3.88 \begin{align*} \frac{384 \, a^{7} x^{7} + 384 \, a^{6} x^{6} - 1152 \, a^{5} x^{5} - 1152 \, a^{4} x^{4} + 1152 \, a^{3} x^{3} + 1152 \, a^{2} x^{2} - 384 \, a x - 210 \,{\left (a^{7} x^{7} + a^{6} x^{6} - 3 \, a^{5} x^{5} - 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} - a x - 1\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) +{\left (105 \, a^{7} x^{7} + 281 \, a^{6} x^{6} - 559 \, a^{5} x^{5} - 965 \, a^{4} x^{4} + 715 \, a^{3} x^{3} + 1065 \, a^{2} x^{2} - 279 \, a x - 384\right )} \sqrt{-a^{2} x^{2} + 1} - 384}{105 \,{\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\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*} \frac{a^{8} \int \frac{x^{8} \sqrt{- a^{2} x^{2} + 1}}{a^{9} x^{9} + a^{8} x^{8} - 4 a^{7} x^{7} - 4 a^{6} x^{6} + 6 a^{5} x^{5} + 6 a^{4} x^{4} - 4 a^{3} x^{3} - 4 a^{2} x^{2} + a x + 1}\, dx}{c^{4}} \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{\sqrt{-a^{2} x^{2} + 1}}{{\left (a x + 1\right )}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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