Optimal. Leaf size=33 \[ -\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {a \cot (e+f x)}{f}+a x \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4141, 1802, 203} \[ -\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {a \cot (e+f x)}{f}+a x \]
Antiderivative was successfully verified.
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Rule 203
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \cot ^4(e+f x) \left (a+b \sec ^2(e+f x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \left (1+x^2\right )}{x^4 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a+b}{x^4}-\frac {a}{x^2}+\frac {a}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=a x+\frac {a \cot (e+f x)}{f}-\frac {(a+b) \cot ^3(e+f x)}{3 f}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 51, normalized size = 1.55 \[ -\frac {a \cot ^3(e+f x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(e+f x)\right )}{3 f}-\frac {b \cot ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 76, normalized size = 2.30 \[ \frac {{\left (4 \, a + b\right )} \cos \left (f x + e\right )^{3} - 3 \, a \cos \left (f x + e\right ) + 3 \, {\left (a f x \cos \left (f x + e\right )^{2} - a f x\right )} \sin \left (f x + e\right )}{3 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 119, normalized size = 3.61 \[ \frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 24 \, {\left (f x + e\right )} a - 15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {15 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.74, size = 48, normalized size = 1.45 \[ \frac {a \left (-\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}+\cot \left (f x +e \right )+f x +e \right )-\frac {b \left (\cos ^{3}\left (f x +e \right )\right )}{3 \sin \left (f x +e \right )^{3}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 41, normalized size = 1.24 \[ \frac {3 \, {\left (f x + e\right )} a + \frac {3 \, a \tan \left (f x + e\right )^{2} - a - b}{\tan \left (f x + e\right )^{3}}}{3 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.67, size = 35, normalized size = 1.06 \[ a\,x-\frac {-a\,{\mathrm {tan}\left (e+f\,x\right )}^2+\frac {a}{3}+\frac {b}{3}}{f\,{\mathrm {tan}\left (e+f\,x\right )}^3} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right ) \cot ^{4}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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