Optimal. Leaf size=165 \[ -\frac{2048 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^4 \sqrt{c-a^2 c x^2}}-\frac{256 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{112 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-14 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{195 a c \left (c-a^2 c x^2\right )^{7/2}} \]
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Rubi [A] time = 0.197033, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6136, 6135} \[ -\frac{2048 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^4 \sqrt{c-a^2 c x^2}}-\frac{256 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{112 (1-10 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{2 (1-14 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{195 a c \left (c-a^2 c x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6136
Rule 6135
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}+\frac{56 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx}{65 c}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}+\frac{896 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{1287 c^2}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{256 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}+\frac{1024 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{2145 c^3}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-14 a x)}{195 a c \left (c-a^2 c x^2\right )^{7/2}}-\frac{112 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{6435 a c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac{256 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{6435 a c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2048 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{6435 a c^4 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0527619, size = 112, normalized size = 0.68 \[ -\frac{2 \sqrt{1-a^2 x^2} \left (2048 a^7 x^7-1024 a^6 x^6-6912 a^5 x^5+3200 a^4 x^4+8240 a^3 x^3-3384 a^2 x^2-3838 a x+1241\right )}{6435 a c^4 (1-a x)^{15/4} (a x+1)^{13/4} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.031, size = 103, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) \left ( 2048\,{a}^{7}{x}^{7}-1024\,{x}^{6}{a}^{6}-6912\,{x}^{5}{a}^{5}+3200\,{x}^{4}{a}^{4}+8240\,{x}^{3}{a}^{3}-3384\,{a}^{2}{x}^{2}-3838\,ax+1241 \right ) }{6435\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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