Optimal. Leaf size=78 \[ -\frac {a^3 \tan (e+f x)}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac {a^3 x}{c} \]
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Rubi [A] time = 0.21, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {3904, 3886, 3473, 8, 2606, 3767, 2621, 321, 207, 2620, 14} \[ -\frac {a^3 \tan (e+f x)}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac {a^3 x}{c} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 207
Rule 321
Rule 2606
Rule 2620
Rule 2621
Rule 3473
Rule 3767
Rule 3886
Rule 3904
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^3}{c-c \sec (e+f x)} \, dx &=-\frac {\int \cot ^2(e+f x) (a+a \sec (e+f x))^4 \, dx}{a c}\\ &=-\frac {\int \left (a^4 \cot ^2(e+f x)+4 a^4 \cot (e+f x) \csc (e+f x)+6 a^4 \csc ^2(e+f x)+4 a^4 \csc ^2(e+f x) \sec (e+f x)+a^4 \csc ^2(e+f x) \sec ^2(e+f x)\right ) \, dx}{a c}\\ &=-\frac {a^3 \int \cot ^2(e+f x) \, dx}{c}-\frac {a^3 \int \csc ^2(e+f x) \sec ^2(e+f x) \, dx}{c}-\frac {\left (4 a^3\right ) \int \cot (e+f x) \csc (e+f x) \, dx}{c}-\frac {\left (4 a^3\right ) \int \csc ^2(e+f x) \sec (e+f x) \, dx}{c}-\frac {\left (6 a^3\right ) \int \csc ^2(e+f x) \, dx}{c}\\ &=\frac {a^3 \cot (e+f x)}{c f}+\frac {a^3 \int 1 \, dx}{c}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1+x^2}{x^2} \, dx,x,\tan (e+f x)\right )}{c f}+\frac {\left (4 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\csc (e+f x))}{c f}+\frac {\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{c f}+\frac {\left (6 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (e+f x))}{c f}\\ &=\frac {a^3 x}{c}+\frac {7 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {a^3 \operatorname {Subst}\left (\int \left (1+\frac {1}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{c f}+\frac {\left (4 a^3\right ) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\csc (e+f x)\right )}{c f}\\ &=\frac {a^3 x}{c}-\frac {4 a^3 \tanh ^{-1}(\sin (e+f x))}{c f}+\frac {8 a^3 \cot (e+f x)}{c f}+\frac {8 a^3 \csc (e+f x)}{c f}-\frac {a^3 \tan (e+f x)}{c f}\\ \end {align*}
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Mathematica [B] time = 2.58, size = 240, normalized size = 3.08 \[ \frac {a^3 \cos ^2(e+f x) \tan \left (\frac {1}{2} (e+f x)\right ) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (\sec (e+f x)+1)^3 \left (8 \csc \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right ) \sec \left (\frac {1}{2} (e+f x)\right )+\tan \left (\frac {1}{2} (e+f x)\right ) \left (\frac {\sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}-4 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+4 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-f x\right )\right )}{4 f (c-c \sec (e+f x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 125, normalized size = 1.60 \[ \frac {a^{3} f x \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} \cos \left (f x + e\right ) \log \left (\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 2 \, a^{3} \cos \left (f x + e\right ) \log \left (-\sin \left (f x + e\right ) + 1\right ) \sin \left (f x + e\right ) + 9 \, a^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{3} \cos \left (f x + e\right ) - a^{3}}{c f \cos \left (f x + e\right ) \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 137, normalized size = 1.76 \[ \frac {8 a^{3}}{f c \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}+\frac {a^{3}}{f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}+\frac {4 a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{f c}+\frac {a^{3}}{f c \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}-\frac {4 a^{3} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{f c}+\frac {2 a^{3} \arctan \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )\right )}{f c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 274, normalized size = 3.51 \[ -\frac {a^{3} {\left (\frac {\frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1}{\frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c}\right )} - a^{3} {\left (\frac {2 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} + \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} + 3 \, a^{3} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c} - \frac {\cos \left (f x + e\right ) + 1}{c \sin \left (f x + e\right )}\right )} - \frac {3 \, a^{3} {\left (\cos \left (f x + e\right ) + 1\right )}}{c \sin \left (f x + e\right )}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 85, normalized size = 1.09 \[ \frac {a^3\,x}{c}-\frac {10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-8\,a^3}{f\,\left (c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\right )}-\frac {8\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{c\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {a^{3} \left (\int \frac {3 \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {3 \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {\sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} - 1}\, dx + \int \frac {1}{\sec {\left (e + f x \right )} - 1}\, dx\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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