Optimal. Leaf size=68 \[ \frac {d (2 c-d) \tanh ^{-1}(\sin (e+f x))}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac {d^2 \tan (e+f x)}{a f} \]
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Rubi [A] time = 0.15, antiderivative size = 125, normalized size of antiderivative = 1.84, number of steps used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3987, 89, 80, 63, 217, 203} \[ \frac {2 d (2 c-d) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {(c-d)^2 \tan (e+f x)}{f (a \sec (e+f x)+a)}+\frac {d^2 \tan (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 89
Rule 203
Rule 217
Rule 3987
Rubi steps
\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^2}{a+a \sec (e+f x)} \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^2}{\sqrt {a-a x} (a+a x)^{3/2}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {a^3 (2 c-d) d+a^3 d^2 x}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{a^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}-\frac {(a (2 c-d) d \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {(2 (2 c-d) d \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {(2 (2 c-d) d \tan (e+f x)) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d^2 \tan (e+f x)}{a f}+\frac {(c-d)^2 \tan (e+f x)}{f (a+a \sec (e+f x))}+\frac {2 (2 c-d) d \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 1.73, size = 237, normalized size = 3.49 \[ \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) \cos (e+f x) (c+d \sec (e+f x))^2 \left ((c-d)^2 \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+d \cos \left (\frac {1}{2} (e+f x)\right ) \left (\frac {d \sin (f x)}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}-(2 c-d) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )\right )}{a f (\sec (e+f x)+1) (c \cos (e+f x)+d)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 155, normalized size = 2.28 \[ \frac {{\left ({\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left ({\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, c d - d^{2}\right )} \cos \left (f x + e\right )\right )} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (d^{2} + {\left (c^{2} - 2 \, c d + 2 \, d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right )^{2} + a f \cos \left (f x + e\right )\right )}} \]
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 [B] time = 0.64, size = 196, normalized size = 2.88 \[ \frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) c^{2}}{a f}-\frac {2 c d \tan \left (\frac {e}{2}+\frac {f x}{2}\right )}{a f}+\frac {\tan \left (\frac {e}{2}+\frac {f x}{2}\right ) d^{2}}{a f}-\frac {d^{2}}{a f \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}-\frac {2 d \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right ) c}{a f}+\frac {d^{2} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )-1\right )}{a f}-\frac {d^{2}}{a f \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}+\frac {2 d \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right ) c}{a f}-\frac {d^{2} \ln \left (\tan \left (\frac {e}{2}+\frac {f x}{2}\right )+1\right )}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 223, normalized size = 3.28 \[ -\frac {d^{2} {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (f x + e\right )}{{\left (a - \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - 2 \, c d {\left (\frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{a} - \frac {\sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}\right )} - \frac {c^{2} \sin \left (f x + e\right )}{a {\left (\cos \left (f x + e\right ) + 1\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.82, size = 85, normalized size = 1.25 \[ \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,{\left (c-d\right )}^2}{a\,f}+\frac {2\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a-a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\right )}+\frac {2\,d\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (2\,c-d\right )}{a\,f} \]
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 \[ \frac {\int \frac {c^{2} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d^{2} \sec ^{3}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {2 c d \sec ^{2}{\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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