Optimal. Leaf size=50 \[ \frac {2 b (a+b) \tan (e+f x)}{f}-\frac {(a+b)^2 \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
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Rubi [A] time = 0.06, antiderivative size = 50, 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, 270} \[ \frac {2 b (a+b) \tan (e+f x)}{f}-\frac {(a+b)^2 \cot (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 270
Rule 4132
Rubi steps
\begin {align*} \int \csc ^2(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b+b x^2\right )^2}{x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 b (a+b)+\frac {(a+b)^2}{x^2}+b^2 x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(a+b)^2 \cot (e+f x)}{f}+\frac {2 b (a+b) \tan (e+f x)}{f}+\frac {b^2 \tan ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [B] time = 1.14, size = 109, normalized size = 2.18 \[ \frac {4 \sec ^3(e+f x) \left (a \cos ^2(e+f x)+b\right )^2 \left (\sin (f x) \cos ^2(e+f x) \left (3 (a+b)^2 \csc (e) \cot (e+f x)+b (6 a+5 b) \sec (e)\right )+b^2 \tan (e) \cos (e+f x)+b^2 \sec (e) \sin (f x)\right )}{3 f (a \cos (2 (e+f x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 71, normalized size = 1.42 \[ -\frac {{\left (3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - b^{2}}{3 \, f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 64, normalized size = 1.28 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 6 \, a b \tan \left (f x + e\right ) + 6 \, b^{2} \tan \left (f x + e\right ) - \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\tan \left (f x + e\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 96, normalized size = 1.92 \[ \frac {-a^{2} \cot \left (f x +e \right )+2 a b \left (\frac {1}{\sin \left (f x +e \right ) \cos \left (f x +e \right )}-2 \cot \left (f x +e \right )\right )+b^{2} \left (\frac {1}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )^{3}}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 54, normalized size = 1.08 \[ \frac {b^{2} \tan \left (f x + e\right )^{3} + 6 \, {\left (a b + b^{2}\right )} \tan \left (f x + e\right ) - \frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}}{\tan \left (f x + e\right )}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.40, size = 56, normalized size = 1.12 \[ \frac {b^2\,{\mathrm {tan}\left (e+f\,x\right )}^3}{3\,f}-\frac {a^2+2\,a\,b+b^2}{f\,\mathrm {tan}\left (e+f\,x\right )}+\frac {2\,b\,\mathrm {tan}\left (e+f\,x\right )\,\left (a+b\right )}{f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \csc ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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