Optimal. Leaf size=127 \[ \frac {F_1\left (\frac {1-m}{2};1-\frac {m}{2},-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)} \]
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Rubi [A]
time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3762, 3748,
525, 524} \begin {gather*} \frac {\tan (e+f x) \sec ^2(e+f x)^{-m/2} (d \csc (e+f x))^m \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-m}{2};1-\frac {m}{2},-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3748
Rule 3762
Rubi steps
\begin {align*} \int (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\left ((d \csc (e+f x))^m \left (\frac {\sin (e+f x)}{d}\right )^m\right ) \int \left (\frac {\sin (e+f x)}{d}\right )^{-m} \left (a+b \tan ^2(e+f x)\right )^p \, dx\\ &=\frac {\left ((d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan ^m(e+f x)\right ) \text {Subst}\left (\int x^{-m} \left (1+x^2\right )^{-1+\frac {m}{2}} \left (a+b x^2\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \tan ^m(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}\right ) \text {Subst}\left (\int x^{-m} \left (1+x^2\right )^{-1+\frac {m}{2}} \left (1+\frac {b x^2}{a}\right )^p \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1-m}{2};1-\frac {m}{2},-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \csc (e+f x))^m \sec ^2(e+f x)^{-m/2} \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 (1-m)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(292\) vs. \(2(127)=254\).
time = 2.40, size = 292, normalized size = 2.30 \begin {gather*} -\frac {a (-3+m) F_1\left (\frac {1}{2}-\frac {m}{2};1-\frac {m}{2},-p;\frac {3}{2}-\frac {m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \cos ^2(e+f x) \cot (e+f x) (d \csc (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p}{f (-1+m) \left (-2 b p F_1\left (\frac {3}{2}-\frac {m}{2};1-\frac {m}{2},1-p;\frac {5}{2}-\frac {m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )-a (-2+m) F_1\left (\frac {3}{2}-\frac {m}{2};2-\frac {m}{2},-p;\frac {5}{2}-\frac {m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )+a (-3+m) F_1\left (\frac {1}{2}-\frac {m}{2};1-\frac {m}{2},-p;\frac {3}{2}-\frac {m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) \cot ^2(e+f x)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.41, size = 0, normalized size = 0.00 \[\int \left (d \csc \left (f x +e \right )\right )^{m} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,{\left (\frac {d}{\sin \left (e+f\,x\right )}\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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