Optimal. Leaf size=147 \[ \frac {a b (2+m) (d \sec (e+f x))^m}{f m (1+m)}+\frac {d \left (b^2-a^2 (1+m)\right ) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^{-1+m} \sin (e+f x)}{f (1-m) (1+m) \sqrt {\sin ^2(e+f x)}}+\frac {b (d \sec (e+f x))^m (a+b \tan (e+f x))}{f (1+m)} \]
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Rubi [A]
time = 0.12, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3589, 3567,
3857, 2722} \begin {gather*} \frac {d \left (b^2-a^2 (m+1)\right ) \sin (e+f x) (d \sec (e+f x))^{m-1} \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right )}{f (1-m) (m+1) \sqrt {\sin ^2(e+f x)}}+\frac {a b (m+2) (d \sec (e+f x))^m}{f m (m+1)}+\frac {b (a+b \tan (e+f x)) (d \sec (e+f x))^m}{f (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2722
Rule 3567
Rule 3589
Rule 3857
Rubi steps
\begin {align*} \int (d \sec (e+f x))^m (a+b \tan (e+f x))^2 \, dx &=\frac {b (d \sec (e+f x))^m (a+b \tan (e+f x))}{f (1+m)}+\frac {\int (d \sec (e+f x))^m \left (-b^2+a^2 (1+m)+a b (2+m) \tan (e+f x)\right ) \, dx}{1+m}\\ &=\frac {a b (2+m) (d \sec (e+f x))^m}{f m (1+m)}+\frac {b (d \sec (e+f x))^m (a+b \tan (e+f x))}{f (1+m)}+\left (a^2-\frac {b^2}{1+m}\right ) \int (d \sec (e+f x))^m \, dx\\ &=\frac {a b (2+m) (d \sec (e+f x))^m}{f m (1+m)}+\frac {b (d \sec (e+f x))^m (a+b \tan (e+f x))}{f (1+m)}+\left (\left (a^2-\frac {b^2}{1+m}\right ) \left (\frac {\cos (e+f x)}{d}\right )^m (d \sec (e+f x))^m\right ) \int \left (\frac {\cos (e+f x)}{d}\right )^{-m} \, dx\\ &=\frac {a b (2+m) (d \sec (e+f x))^m}{f m (1+m)}-\frac {\left (a^2-\frac {b^2}{1+m}\right ) \cos (e+f x) \, _2F_1\left (\frac {1}{2},\frac {1-m}{2};\frac {3-m}{2};\cos ^2(e+f x)\right ) (d \sec (e+f x))^m \sin (e+f x)}{f (1-m) \sqrt {\sin ^2(e+f x)}}+\frac {b (d \sec (e+f x))^m (a+b \tan (e+f x))}{f (1+m)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 26.49, size = 11095, normalized size = 75.48 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.28, size = 0, normalized size = 0.00 \[\int \left (d \sec \left (f x +e \right )\right )^{m} \left (a +b \tan \left (f x +e \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (d \sec {\left (e + f x \right )}\right )^{m} \left (a + b \tan {\left (e + f x \right )}\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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