Optimal. Leaf size=56 \[ \frac {a (c+d) \tan (e+f x)}{f}+\frac {a (2 c+d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a d \tan (e+f x) \sec (e+f x)}{2 f} \]
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Rubi [A] time = 0.07, antiderivative size = 56, 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} \[ \frac {a (c+d) \tan (e+f x)}{f}+\frac {a (2 c+d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a d \tan (e+f x) \sec (e+f x)}{2 f} \]
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+a \sec (e+f x)) (c+d \sec (e+f x)) \, dx &=\frac {a d \sec (e+f x) \tan (e+f x)}{2 f}+\frac {1}{2} \int \sec (e+f x) (a (2 c+d)+2 a (c+d) \sec (e+f x)) \, dx\\ &=\frac {a d \sec (e+f x) \tan (e+f x)}{2 f}+(a (c+d)) \int \sec ^2(e+f x) \, dx+\frac {1}{2} (a (2 c+d)) \int \sec (e+f x) \, dx\\ &=\frac {a (2 c+d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a d \sec (e+f x) \tan (e+f x)}{2 f}-\frac {(a (c+d)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{f}\\ &=\frac {a (2 c+d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a (c+d) \tan (e+f x)}{f}+\frac {a d \sec (e+f x) \tan (e+f x)}{2 f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 1.34 \[ \frac {a c \tan (e+f x)}{f}+\frac {a c \tanh ^{-1}(\sin (e+f x))}{f}+\frac {a d \tan (e+f x)}{f}+\frac {a d \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a d \tan (e+f x) \sec (e+f x)}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 96, normalized size = 1.71 \[ \frac {{\left (2 \, a c + a d\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (2 \, a c + a d\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (a d + 2 \, {\left (a c + a d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, f \cos \left (f x + e\right )^{2}} \]
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 = 0.96, size = 86, normalized size = 1.54 \[ \frac {c a \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {d a \tan \left (f x +e \right )}{f}+\frac {a c \tan \left (f x +e \right )}{f}+\frac {a d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {d a \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 88, normalized size = 1.57 \[ -\frac {a d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 4 \, a c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) - 4 \, a c \tan \left (f x + e\right ) - 4 \, a d \tan \left (f x + e\right )}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.57, size = 111, normalized size = 1.98 \[ \frac {a\,\mathrm {atanh}\left (\frac {2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c+d\right )}{4\,c+2\,d}\right )\,\left (2\,c+d\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,a\,c+a\,d\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a\,c+3\,a\,d\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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 \[ a \left (\int c \sec {\left (e + f x \right )}\, dx + \int c \sec ^{2}{\left (e + f x \right )}\, dx + \int d \sec ^{2}{\left (e + f x \right )}\, dx + \int d \sec ^{3}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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