Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sec ^3(c+d x)}{(e+f x)^2 (a \sin (c+d x)+a)},x\right ) \]
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Rubi [A] time = 0.07, 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 = {} \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx &=\int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx\\ \end {align*}
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Mathematica [A] time = 54.48, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^3(c+d x)}{(e+f x)^2 (a+a \sin (c+d x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sec \left (d x + c\right )^{3}}{a f^{2} x^{2} + 2 \, a e f x + a e^{2} + {\left (a f^{2} x^{2} + 2 \, a e f x + a e^{2}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 = 2.46, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}\left (d x +c \right )}{\left (f x +e \right )^{2} \left (a +a \sin \left (d x +c \right )\right )}\, dx \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (e+f\,x\right )}^2\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]
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 \[ \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{e^{2} \sin {\left (c + d x \right )} + e^{2} + 2 e f x \sin {\left (c + d x \right )} + 2 e f x + f^{2} x^{2} \sin {\left (c + d x \right )} + f^{2} x^{2}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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