Optimal. Leaf size=49 \[ \frac{(a-b) \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^4(e+f x)}{4 f} \]
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Rubi [A] time = 0.0488434, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4138, 446, 76} \[ \frac{(a-b) \sec ^2(e+f x)}{2 f}+\frac{a \log (\cos (e+f x))}{f}+\frac{b \sec ^4(e+f x)}{4 f} \]
Antiderivative was successfully verified.
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Rule 4138
Rule 446
Rule 76
Rubi steps
\begin{align*} \int \left (a+b \sec ^2(e+f x)\right ) \tan ^3(e+f x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )}{x^5} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x) (b+a x)}{x^3} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b}{x^3}+\frac{a-b}{x^2}-\frac{a}{x}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=\frac{a \log (\cos (e+f x))}{f}+\frac{(a-b) \sec ^2(e+f x)}{2 f}+\frac{b \sec ^4(e+f x)}{4 f}\\ \end{align*}
Mathematica [A] time = 0.080193, size = 43, normalized size = 0.88 \[ \frac{a \left (\tan ^2(e+f x)+2 \log (\cos (e+f x))\right )}{2 f}+\frac{b \tan ^4(e+f x)}{4 f} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 50, normalized size = 1. \begin{align*}{\frac{ \left ( \tan \left ( fx+e \right ) \right ) ^{2}a}{2\,f}}+{\frac{a\ln \left ( \cos \left ( fx+e \right ) \right ) }{f}}+{\frac{b \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{4\,f \left ( \cos \left ( fx+e \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03266, size = 86, normalized size = 1.76 \begin{align*} \frac{2 \, a \log \left (\sin \left (f x + e\right )^{2} - 1\right ) - \frac{2 \,{\left (a - b\right )} \sin \left (f x + e\right )^{2} - 2 \, a + b}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.525979, size = 128, normalized size = 2.61 \begin{align*} \frac{4 \, a \cos \left (f x + e\right )^{4} \log \left (-\cos \left (f x + e\right )\right ) + 2 \,{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{4 \, f \cos \left (f x + e\right )^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.35166, size = 80, normalized size = 1.63 \begin{align*} \begin{cases} - \frac{a \log{\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac{a \tan ^{2}{\left (e + f x \right )}}{2 f} + \frac{b \tan ^{2}{\left (e + f x \right )} \sec ^{2}{\left (e + f x \right )}}{4 f} - \frac{b \sec ^{2}{\left (e + f x \right )}}{4 f} & \text{for}\: f \neq 0 \\x \left (a + b \sec ^{2}{\left (e \right )}\right ) \tan ^{3}{\left (e \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.83268, size = 343, normalized size = 7. \begin{align*} -\frac{2 \, a \log \left (-\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2\right ) - 2 \, a \log \left (-\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} - \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} - 2\right ) + \frac{3 \, a{\left (\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )}^{2} + 20 \, a{\left (\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1}\right )} + 28 \, a - 16 \, b}{{\left (\frac{\cos \left (f x + e\right ) + 1}{\cos \left (f x + e\right ) - 1} + \frac{\cos \left (f x + e\right ) - 1}{\cos \left (f x + e\right ) + 1} + 2\right )}^{2}}}{4 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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