Optimal. Leaf size=39 \[ \text {Int}\left (\frac {\sec ^3(a+b x)}{(c+d x)^2},x\right )-\text {Int}\left (\frac {\sec (a+b x)}{(c+d x)^2},x\right ) \]
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Rubi [A] time = 0.09, 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 = {} \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sec (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx &=-\int \frac {\sec (a+b x)}{(c+d x)^2} \, dx+\int \frac {\sec ^3(a+b x)}{(c+d x)^2} \, dx\\ \end {align*}
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Mathematica [A] time = 29.12, size = 0, normalized size = 0.00 \[ \int \frac {\sec (a+b x) \tan ^2(a+b x)}{(c+d x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x + a\right ) \tan \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.60, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b x +a \right ) \left (\tan ^{2}\left (b x +a \right )\right )}{\left (d x +c \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\mathrm {tan}\left (a+b\,x\right )}^2}{\cos \left (a+b\,x\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\tan ^{2}{\left (a + b x \right )} \sec {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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