Integrand size = 23, antiderivative size = 76 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{2 \arctan (a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \arctan (a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {5178, 5177} \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {6 (a x+2) e^{2 \arctan (a x)}}{65 a c^2 \sqrt {a^2 c x^2+c}}+\frac {(3 a x+2) e^{2 \arctan (a x)}}{13 a c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 5177
Rule 5178
Rubi steps \begin{align*} \text {integral}& = \frac {e^{2 \arctan (a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{13 c} \\ & = \frac {e^{2 \arctan (a x)} (2+3 a x)}{13 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 e^{2 \arctan (a x)} (2+a x)}{65 a c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.82 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {e^{2 \arctan (a x)} \left (22+21 a x+12 a^2 x^2+6 a^3 x^3\right )}{65 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2}} \]
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Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.74
method | result | size |
gosper | \(\frac {\left (a^{2} x^{2}+1\right ) \left (6 a^{3} x^{3}+12 a^{2} x^{2}+21 a x +22\right ) {\mathrm e}^{2 \arctan \left (a x \right )}}{65 a \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\) | \(56\) |
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Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.95 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {{\left (6 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 21 \, a x + 22\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (2 \, \arctan \left (a x\right )\right )}}{65 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {e^{2 \operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {e^{\left (2 \, \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {e^{2 \arctan (a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {{\mathrm {e}}^{2\,\mathrm {atan}\left (a\,x\right )}\,\left (\frac {22}{65\,a^3\,c^2}+\frac {6\,x^3}{65\,c^2}+\frac {21\,x}{65\,a^2\,c^2}+\frac {12\,x^2}{65\,a\,c^2}\right )}{\frac {\sqrt {c\,a^2\,x^2+c}}{a^2}+x^2\,\sqrt {c\,a^2\,x^2+c}} \]
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