Optimal. Leaf size=64 \[ -\frac {(2 a+b) \cot ^3(e+f x)}{3 f}-\frac {(a+2 b) \cot (e+f x)}{f}-\frac {a \cot ^5(e+f x)}{5 f}+\frac {b \tan (e+f x)}{f} \]
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Rubi [A] time = 0.05, antiderivative size = 64, 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 = {3663, 448} \[ -\frac {(2 a+b) \cot ^3(e+f x)}{3 f}-\frac {(a+2 b) \cot (e+f x)}{f}-\frac {a \cot ^5(e+f x)}{5 f}+\frac {b \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
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Rule 448
Rule 3663
Rubi steps
\begin {align*} \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2 \left (a+b x^2\right )}{x^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (b+\frac {a}{x^6}+\frac {2 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 {(2 a+b) \cot ^3(e+f x)}{3 f}-\frac {a \cot ^5(e+f x)}{5 f}+\frac {b \tan (e+f x)}{f}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 106, normalized size = 1.66 \[ -\frac {8 a \cot (e+f x)}{15 f}-\frac {a \cot (e+f x) \csc ^4(e+f x)}{5 f}-\frac {4 a \cot (e+f x) \csc ^2(e+f x)}{15 f}+\frac {b \tan (e+f x)}{f}-\frac {5 b \cot (e+f x)}{3 f}-\frac {b \cot (e+f x) \csc ^2(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 91, normalized size = 1.42 \[ -\frac {8 \, {\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{6} - 20 \, {\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{4} + 15 \, {\left (a + 5 \, b\right )} \cos \left (f x + e\right )^{2} - 15 \, b}{15 \, {\left (f \cos \left (f x + e\right )^{5} - 2 \, f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 79, normalized size = 1.23 \[ \frac {15 \, b \tan \left (f x + e\right ) - \frac {15 \, a \tan \left (f x + e\right )^{4} + 30 \, b \tan \left (f x + e\right )^{4} + 10 \, a \tan \left (f x + e\right )^{2} + 5 \, b \tan \left (f x + e\right )^{2} + 3 \, a}{\tan \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 83, normalized size = 1.30 \[ \frac {a \left (-\frac {8}{15}-\frac {\left (\csc ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\csc ^{2}\left (f x +e \right )\right )}{15}\right ) \cot \left (f x +e \right )+b \left (-\frac {1}{3 \sin \left (f x +e \right )^{3} \cos \left (f x +e \right )}+\frac {4}{3 \sin \left (f x +e \right ) \cos \left (f x +e \right )}-\frac {8 \cot \left (f x +e \right )}{3}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 59, normalized size = 0.92 \[ \frac {15 \, b \tan \left (f x + e\right ) - \frac {15 \, {\left (a + 2 \, b\right )} \tan \left (f x + e\right )^{4} + 5 \, {\left (2 \, a + b\right )} \tan \left (f x + e\right )^{2} + 3 \, a}{\tan \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.41, size = 59, normalized size = 0.92 \[ \frac {b\,\mathrm {tan}\left (e+f\,x\right )}{f}-\frac {\left (a+2\,b\right )\,{\mathrm {tan}\left (e+f\,x\right )}^4+\left (\frac {2\,a}{3}+\frac {b}{3}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{5}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^5} \]
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 \[ \int \left (a + b \tan ^{2}{\left (e + f x \right )}\right ) \csc ^{6}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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