Optimal. Leaf size=124 \[ -\frac {256 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{693 a c^3 \sqrt {c-a^2 c x^2}}-\frac {32 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{693 a c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (1-10 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{99 a c \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6136, 6135} \[ -\frac {256 (1-2 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{693 a c^3 \sqrt {c-a^2 c x^2}}-\frac {32 (1-6 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{693 a c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {2 (1-10 a x) e^{\frac {1}{2} \tanh ^{-1}(a x)}}{99 a c \left (c-a^2 c x^2\right )^{5/2}} \]
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 )^{7/2}} \, dx &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{99 a c \left (c-a^2 c x^2\right )^{5/2}}+\frac {80 \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{99 c}\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{99 a c \left (c-a^2 c x^2\right )^{5/2}}-\frac {32 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{693 a c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {128 \int \frac {e^{\frac {1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{231 c^2}\\ &=-\frac {2 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-10 a x)}{99 a c \left (c-a^2 c x^2\right )^{5/2}}-\frac {32 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{693 a c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {256 e^{\frac {1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{693 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 96, normalized size = 0.77 \[ \frac {2 \sqrt {1-a^2 x^2} \left (256 a^5 x^5-128 a^4 x^4-608 a^3 x^3+272 a^2 x^2+422 a x-151\right )}{693 a c^3 (1-a x)^{11/4} (a x+1)^{9/4} \sqrt {c-a^2 c x^2}} \]
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 \[ \int \frac {\sqrt {\frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 87, normalized size = 0.70 \[ -\frac {2 \left (a x -1\right ) \left (a x +1\right ) \left (256 x^{5} a^{5}-128 x^{4} a^{4}-608 x^{3} a^{3}+272 a^{2} x^{2}+422 a x -151\right ) \sqrt {\frac {a x +1}{\sqrt {-a^{2} x^{2}+1}}}}{693 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}} \]
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 = 1.42, size = 139, normalized size = 1.12 \[ -\frac {\sqrt {\frac {a\,x+1}{\sqrt {1-a^2\,x^2}}}\,\left (\frac {302}{693\,a^5\,c^3}-\frac {512\,x^5}{693\,c^3}-\frac {844\,x}{693\,a^4\,c^3}+\frac {256\,x^4}{693\,a\,c^3}+\frac {1216\,x^3}{693\,a^2\,c^3}-\frac {544\,x^2}{693\,a^3\,c^3}\right )}{\frac {\sqrt {c-a^2\,c\,x^2}}{a^4}+x^4\,\sqrt {c-a^2\,c\,x^2}-\frac {2\,x^2\,\sqrt {c-a^2\,c\,x^2}}{a^2}} \]
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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