Optimal. Leaf size=76 \[ \frac {b (2 a+3 b) \tan (e+f x)}{f}-\frac {(a+b)^2 \cot ^3(e+f x)}{3 f}-\frac {(a+b) (a+3 b) \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.08, antiderivative size = 76, 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 = {4132, 448} \[ \frac {b (2 a+3 b) \tan (e+f x)}{f}-\frac {(a+b)^2 \cot ^3(e+f x)}{3 f}-\frac {(a+b) (a+3 b) \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 448
Rule 4132
Rubi steps
\begin {align*} \int \csc ^4(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b+b x^2\right )^2}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (b (2 a+3 b)+\frac {(a+b)^2}{x^4}+\frac {(a+b) (a+3 b)}{x^2}+b^2 x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(a+b) (a+3 b) \cot (e+f x)}{f}-\frac {(a+b)^2 \cot ^3(e+f x)}{3 f}+\frac {b (2 a+3 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 = 1.36, size = 151, normalized size = 1.99 \[ -\frac {\csc (2 e) \csc ^3(2 (e+f x)) \left (-3 a^2 \sin (2 (e+f x))+a^2 \sin (6 (e+f x))+3 a^2 \sin (4 e+2 f x)+a^2 \sin (4 e+6 f x)-6 a b \sin (2 (e+f x))+2 a b \sin (6 (e+f x))+8 a b \sin (4 e+6 f x)+8 a (a+2 b) \sin (2 e)-6 (a+2 b)^2 \sin (2 f x)+8 b^2 \sin (4 e+6 f x)\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 101, normalized size = 1.33 \[ -\frac {2 \, {\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{6} - 3 \, {\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 6 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}{3 \, {\left (f \cos \left (f x + e\right )^{5} - f \cos \left (f x + e\right )^{3}\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.80, size = 105, normalized size = 1.38 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 6 \, a b \tan \left (f x + e\right ) + 9 \, b^{2} \tan \left (f x + e\right ) - \frac {3 \, a^{2} \tan \left (f x + e\right )^{2} + 12 \, a b \tan \left (f x + e\right )^{2} + 9 \, b^{2} \tan \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.38, size = 144, normalized size = 1.89 \[ \frac {a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+2 a b \left (-\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )+b^{2} \left (\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )^{3}}-\frac {2}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {8}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {16 \cot \left (f x +e \right )}{3}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 80, normalized size = 1.05 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 3 \, {\left (2 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right ) - \frac {3 \, {\left (a^{2} + 4 \, a b + 3 \, b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 85, normalized size = 1.12 \[ \frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f}-\frac {\frac {2\,a\,b}{3}+{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (a^2+4\,a\,b+3\,b^2\right )+\frac {a^2}{3}+\frac {b^2}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3}+\frac {b\,\mathrm {tan}\left (e+f\,x\right )\,\left (2\,a+3\,b\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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