Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))},x\right ) \]
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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 = {}
\begin {gather*} \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \end {gather*}
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) (a+a \sin (c+d x))} \, dx &=\int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end {align*}
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Mathematica [A]
time = 23.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\sec ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\sec ^{3}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^3\,\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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