Optimal. Leaf size=85 \[ -\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^3 x}{a^2} \]
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Rubi [A] time = 0.330986, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {3903, 3777, 3919, 3794, 3796, 3797, 3799, 3998, 3770} \[ -\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (\sec (e+f x)+1)^2}+\frac{c^3 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 3903
Rule 3777
Rule 3919
Rule 3794
Rule 3796
Rule 3797
Rule 3799
Rule 3998
Rule 3770
Rubi steps
\begin{align*} \int \frac{(c-c \sec (e+f x))^3}{(a+a \sec (e+f x))^2} \, dx &=\frac{\int \left (\frac{c^3}{(1+\sec (e+f x))^2}-\frac{3 c^3 \sec (e+f x)}{(1+\sec (e+f x))^2}+\frac{3 c^3 \sec ^2(e+f x)}{(1+\sec (e+f x))^2}-\frac{c^3 \sec ^3(e+f x)}{(1+\sec (e+f x))^2}\right ) \, dx}{a^2}\\ &=\frac{c^3 \int \frac{1}{(1+\sec (e+f x))^2} \, dx}{a^2}-\frac{c^3 \int \frac{\sec ^3(e+f x)}{(1+\sec (e+f x))^2} \, dx}{a^2}-\frac{\left (3 c^3\right ) \int \frac{\sec (e+f x)}{(1+\sec (e+f x))^2} \, dx}{a^2}+\frac{\left (3 c^3\right ) \int \frac{\sec ^2(e+f x)}{(1+\sec (e+f x))^2} \, dx}{a^2}\\ &=-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}-\frac{c^3 \int \frac{-3+\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}-\frac{c^3 \int \frac{\sec (e+f x) (-2+3 \sec (e+f x))}{1+\sec (e+f x)} \, dx}{3 a^2}-\frac{c^3 \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{a^2}+\frac{\left (2 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{a^2}\\ &=\frac{c^3 x}{a^2}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}+\frac{c^3 \tan (e+f x)}{a^2 f (1+\sec (e+f x))}-\frac{c^3 \int \sec (e+f x) \, dx}{a^2}-\frac{\left (4 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}+\frac{\left (5 c^3\right ) \int \frac{\sec (e+f x)}{1+\sec (e+f x)} \, dx}{3 a^2}\\ &=\frac{c^3 x}{a^2}-\frac{c^3 \tanh ^{-1}(\sin (e+f x))}{a^2 f}-\frac{8 c^3 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))^2}+\frac{4 c^3 \tan (e+f x)}{3 a^2 f (1+\sec (e+f x))}\\ \end{align*}
Mathematica [B] time = 1.09175, size = 216, normalized size = 2.54 \[ -\frac{c^3 (\cos (e+f x)-1)^3 \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right ) \left (4 \tan \left (\frac{e}{2}\right ) \cot \left (\frac{1}{2} (e+f x)\right ) \csc ^2\left (\frac{1}{2} (e+f x)\right )+4 \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \csc ^3\left (\frac{1}{2} (e+f x)\right )+3 \cot ^3\left (\frac{1}{2} (e+f x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )+f x\right )-4 \sec \left (\frac{e}{2}\right ) \sin \left (\frac{f x}{2}\right ) \cot ^2\left (\frac{1}{2} (e+f x)\right ) \csc \left (\frac{1}{2} (e+f x)\right )\right )}{6 a^2 f (\cos (e+f x)+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.086, size = 90, normalized size = 1.1 \begin{align*}{\frac{4\,{c}^{3}}{3\,f{a}^{2}} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}+2\,{\frac{{c}^{3}\arctan \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) }{f{a}^{2}}}-{\frac{{c}^{3}}{f{a}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +1 \right ) }+{\frac{{c}^{3}}{f{a}^{2}}\ln \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.58067, size = 362, normalized size = 4.26 \begin{align*} \frac{c^{3}{\left (\frac{\frac{9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac{6 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a^{2}} + \frac{6 \, \log \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a^{2}}\right )} - c^{3}{\left (\frac{\frac{9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a^{2}} - \frac{12 \, \arctan \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \frac{3 \, c^{3}{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}} - \frac{3 \, c^{3}{\left (\frac{3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a^{2}}}{6 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.07578, size = 425, normalized size = 5. \begin{align*} \frac{6 \, c^{3} f x \cos \left (f x + e\right )^{2} + 12 \, c^{3} f x \cos \left (f x + e\right ) + 6 \, c^{3} f x - 3 \,{\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \,{\left (c^{3} \cos \left (f x + e\right )^{2} + 2 \, c^{3} \cos \left (f x + e\right ) + c^{3}\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 8 \,{\left (c^{3} \cos \left (f x + e\right ) - c^{3}\right )} \sin \left (f x + e\right )}{6 \,{\left (a^{2} f \cos \left (f x + e\right )^{2} + 2 \, a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{c^{3} \left (\int \frac{3 \sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{3 \sec ^{2}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int \frac{\sec ^{3}{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx + \int - \frac{1}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33767, size = 113, normalized size = 1.33 \begin{align*} \frac{\frac{4 \, c^{3} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3}}{a^{2}} + \frac{3 \,{\left (f x + e\right )} c^{3}}{a^{2}} - \frac{3 \, c^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1 \right |}\right )}{a^{2}} + \frac{3 \, c^{3} \log \left ({\left | \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 1 \right |}\right )}{a^{2}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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