Optimal. Leaf size=61 \[ \frac {(a d+b c) \tan (e+f x)}{f}+\frac {(2 a c+b d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f} \]
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Rubi [A] time = 0.08, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3997, 3787, 3770, 3767, 8} \[ \frac {(a d+b c) \tan (e+f x)}{f}+\frac {(2 a c+b d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3997
Rubi steps
\begin {align*} \int \sec (e+f x) (a+b \sec (e+f x)) (c+d \sec (e+f x)) \, dx &=\frac {b d \sec (e+f x) \tan (e+f x)}{2 f}+\frac {1}{2} \int \sec (e+f x) (2 a c+b d+2 (b c+a d) \sec (e+f x)) \, dx\\ &=\frac {b d \sec (e+f x) \tan (e+f x)}{2 f}+(b c+a d) \int \sec ^2(e+f x) \, dx+\frac {1}{2} (2 a c+b d) \int \sec (e+f x) \, dx\\ &=\frac {(2 a c+b d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {b d \sec (e+f x) \tan (e+f x)}{2 f}-\frac {(b c+a d) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=\frac {(2 a c+b d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {(b c+a d) \tan (e+f x)}{f}+\frac {b d \sec (e+f x) \tan (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 75, normalized size = 1.23 \[ \frac {a c \tanh ^{-1}(\sin (e+f x))}{f}+\frac {a d \tan (e+f x)}{f}+\frac {b c \tan (e+f x)}{f}+\frac {b d \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {b d \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 96, normalized size = 1.57 \[ \frac {{\left (2 \, a c + b d\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a c + b d\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (b d + 2 \, {\left (b c + a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \]
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 = 0.92, size = 86, normalized size = 1.41 \[ \frac {c a \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {d a \tan \left (f x +e \right )}{f}+\frac {c b \tan \left (f x +e \right )}{f}+\frac {b d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {b d \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 88, normalized size = 1.44 \[ -\frac {b d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 4 \, a c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 4 \, b c \tan \left (f x + e\right ) - 4 \, a d \tan \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.79, size = 104, normalized size = 1.70 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (2\,a\,c+b\,d\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a\,d+2\,b\,c+b\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a\,d+2\,b\,c-b\,d\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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 \[ \int \left (a + b \sec {\left (e + f x \right )}\right ) \left (c + d \sec {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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