Optimal. Leaf size=46 \[ -\frac {(a+b) \cot ^3(e+f x)}{3 f}-\frac {(a+2 b) \cot (e+f x)}{f}+\frac {b \tan (e+f x)}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4132, 448} \[ -\frac {(a+b) \cot ^3(e+f x)}{3 f}-\frac {(a+2 b) \cot (e+f x)}{f}+\frac {b \tan (e+f x)}{f} \]
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 ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a+b+b x^2\right )}{x^4} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (b+\frac {a+b}{x^4}+\frac {a+2 b}{x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(a+2 b) \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 84, normalized size = 1.83 \[ -\frac {2 a \cot (e+f x)}{3 f}-\frac {a \cot (e+f x) \csc ^2(e+f x)}{3 f}+\frac {b \tan (e+f x)}{f}-\frac {5 b \cot (e+f x)}{3 f}-\frac {b \cot (e+f x) \csc ^2(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 66, normalized size = 1.43 \[ -\frac {2 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{4} - 3 \, {\left (a + 4 \, b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \, {\left (f \cos \left (f x + e\right )^{3} - f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 54, normalized size = 1.17 \[ \frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, a \tan \left (f x + e\right )^{2} + 6 \, b \tan \left (f x + e\right )^{2} + a + b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 73, normalized size = 1.59 \[ \frac {a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+b \left (-\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 43, normalized size = 0.93 \[ \frac {3 \, b \tan \left (f x + e\right ) - \frac {3 \, {\left (a + 2 \, b\right )} \tan \left (f x + e\right )^{2} + a + b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 46, normalized size = 1.00 \[ \frac {b\,\mathrm {tan}\left (e+f\,x\right )}{f}-\frac {\left (a+2\,b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{3}+\frac {b}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \csc ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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