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 = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, 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 6135
Rule 6136
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*}
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Mathematica [A] time = 0.06, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 103, normalized size = 0.62 \[ \frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (2048 a^{7} x^{7}-1024 x^{6} a^{6}-6912 x^{5} a^{5}+3200 x^{4} a^{4}+8240 x^{3} a^{3}-3384 a^{2} x^{2}-3838 a x +1241\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{6435 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.49, size = 183, normalized size = 1.11 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {2482}{6435\,a^7\,c^4}+\frac {4096\,x^7}{6435\,c^4}-\frac {7676\,x}{6435\,a^6\,c^4}-\frac {2048\,x^6}{6435\,a\,c^4}-\frac {1536\,x^5}{715\,a^2\,c^4}+\frac {1280\,x^4}{1287\,a^3\,c^4}+\frac {3296\,x^3}{1287\,a^4\,c^4}-\frac {752\,x^2}{715\,a^5\,c^4}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^6}-x^6\,\sqrt {c-a^2\,c\,x^2}+\frac {3\,x^4\,\sqrt {c-a^2\,c\,x^2}}{a^2}-\frac {3\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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