Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(a \sec (e+f x)+a)^2}{(c+d x)^2},x\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx &=\int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 29.12, size = 0, normalized size = 0.00 \[ \int \frac {(a+a \sec (e+f x))^2}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} \sec \left (f x + e\right )^{2} + 2 \, a^{2} \sec \left (f x + e\right ) + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 5.14, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sec \left (f x +e \right )\right )^{2}}{\left (d x +c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{2} d f x + a^{2} c f - 2 \, a^{2} d \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (a^{2} d f x + a^{2} c f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) - 4 \, {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f + {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )\right )} \int \frac {{\left (a^{2} d f x + a^{2} c f\right )} \cos \left (2 \, f x + 2 \, e\right ) \cos \left (f x + e\right ) + {\left (a^{2} d f x + a^{2} c f\right )} \cos \left (f x + e\right ) + {\left (a^{2} d + {\left (a^{2} d f x + a^{2} c f\right )} \sin \left (f x + e\right )\right )} \sin \left (2 \, f x + 2 \, e\right )}{d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f + {\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{3} f x^{3} + 3 \, c d^{2} f x^{2} + 3 \, c^{2} d f x + c^{3} f\right )} \cos \left (2 \, f x + 2 \, e\right )}\,{d x}}{d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f + {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (d^{3} f x^{2} + 2 \, c d^{2} f x + c^{2} d f\right )} \cos \left (2 \, f x + 2 \, e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \frac {2 \sec {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\sec ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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