Optimal. Leaf size=70 \[ -\frac {a^2 \cot ^3(e+f x)}{3 f}+\frac {b (2 a+b) \tan (e+f x)}{f}-\frac {a (a+2 b) \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.07, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3663, 448} \[ -\frac {a^2 \cot ^3(e+f x)}{3 f}+\frac {b (2 a+b) \tan (e+f x)}{f}-\frac {a (a+2 b) \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 448
Rule 3663
Rubi steps
\begin {align*} \int \csc ^4(e+f x) \left (a+b \tan ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b x^2\right )^2}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (b (2 a+b)+\frac {a^2}{x^4}+\frac {a (a+2 b)}{x^2}+b^2 x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a (a+2 b) \cot (e+f x)}{f}-\frac {a^2 \cot ^3(e+f x)}{3 f}+\frac {b (2 a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.47, size = 59, normalized size = 0.84 \[ \frac {b \tan (e+f x) \left (6 a+b \sec ^2(e+f x)+2 b\right )-a \cot (e+f x) \left (a \csc ^2(e+f x)+2 a+6 b\right )}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 92, normalized size = 1.31 \[ -\frac {2 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 6 \, a b \cos \left (f x + e\right )^{2} + b^{2}}{3 \, {\left (f \cos \left (f x + e\right )^{5} - f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.75, size = 84, normalized size = 1.20 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 6 \, a b \tan \left (f x + e\right ) + 3 \, b^{2} \tan \left (f x + e\right ) - \frac {3 \, a^{2} \tan \left (f x + e\right )^{2} + 6 \, a b \tan \left (f x + e\right )^{2} + a^{2}}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 81, normalized size = 1.16 \[ \frac {a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+2 a b \left (\frac {1}{\sin \left (f x +e \right ) \cos \left (f x +e \right )}-2 \cot \left (f x +e \right )\right )-b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 66, normalized size = 0.94 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 3 \, {\left (2 \, a b + b^{2}\right )} \tan \left (f x + e\right ) - \frac {3 \, {\left (a^{2} + 2 \, a b\right )} \tan \left (f x + e\right )^{2} + a^{2}}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.82, size = 69, normalized size = 0.99 \[ \frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a^2+2\,b\,a\right )+\frac {a^2}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3}+\frac {b\,\mathrm {tan}\left (e+f\,x\right )\,\left (2\,a+b\right )}{f} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{2} \csc ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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