Optimal. Leaf size=78 \[ \frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f} \]
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Rubi [A]
time = 0.03, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 441, 440}
\begin {gather*} \frac {\tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1}{2};1,-p;\frac {3}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rule 3742
Rubi steps
\begin {align*} \int \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left (\left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1}{2};1,-p;\frac {3}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(192\) vs. \(2(78)=156\).
time = 0.30, size = 192, normalized size = 2.46 \begin {gather*} \frac {3 a F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \sin (2 (e+f x)) \left (a+b \tan ^2(e+f x)\right )^p}{6 a f F_1\left (\frac {1}{2};-p,1;\frac {3}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+4 f \left (b p F_1\left (\frac {3}{2};1-p,1;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )-a F_1\left (\frac {3}{2};-p,2;\frac {5}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )\right ) \tan ^2(e+f x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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 (a + b \tan ^{2}{\left (e + f x \right )}\right )^{p}\, 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 (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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