Optimal. Leaf size=65 \[ \frac {\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac {a^2 \cot (e+f x)}{f}-a^2 x-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f} \]
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Rubi [A] time = 0.09, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4141, 1802, 203} \[ \frac {\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac {a^2 \cot (e+f x)}{f}-a^2 x-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1802
Rule 4141
Rubi steps
\begin {align*} \int \cot ^6(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \left (1+x^2\right )\right )^2}{x^6 \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {(a+b)^2}{x^6}+\frac {-a^2+b^2}{x^4}+\frac {a^2}{x^2}-\frac {a^2}{1+x^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {a^2 \cot (e+f x)}{f}+\frac {\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-a^2 x-\frac {a^2 \cot (e+f x)}{f}+\frac {\left (a^2-b^2\right ) \cot ^3(e+f x)}{3 f}-\frac {(a+b)^2 \cot ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [B] time = 1.03, size = 256, normalized size = 3.94 \[ \frac {\csc (e) \csc ^5(e+f x) \left (180 a^2 \sin (2 e+f x)-140 a^2 \sin (2 e+3 f x)-90 a^2 \sin (4 e+3 f x)+46 a^2 \sin (4 e+5 f x)+150 a^2 f x \cos (2 e+f x)+75 a^2 f x \cos (2 e+3 f x)-75 a^2 f x \cos (4 e+3 f x)-15 a^2 f x \cos (4 e+5 f x)+15 a^2 f x \cos (6 e+5 f x)+280 a^2 \sin (f x)-150 a^2 f x \cos (f x)-60 a b \sin (4 e+3 f x)+12 a b \sin (4 e+5 f x)+120 a b \sin (f x)-60 b^2 \sin (2 e+f x)+20 b^2 \sin (2 e+3 f x)-4 b^2 \sin (4 e+5 f x)+20 b^2 \sin (f x)\right )}{480 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 136, normalized size = 2.09 \[ -\frac {{\left (23 \, a^{2} + 6 \, a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{5} - 5 \, {\left (7 \, a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{3} + 15 \, a^{2} \cos \left (f x + e\right ) + 15 \, {\left (a^{2} f x \cos \left (f x + e\right )^{4} - 2 \, a^{2} f x \cos \left (f x + e\right )^{2} + a^{2} f x\right )} \sin \left (f x + e\right )}{15 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )} \sin \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.01, size = 290, normalized size = 4.46 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 30 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 480 \, {\left (f x + e\right )} a^{2} + 330 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 60 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 30 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - \frac {330 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 60 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 30 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 30 \, a b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5}}}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 107, normalized size = 1.65 \[ \frac {a^{2} \left (-\frac {\left (\cot ^{5}\left (f x +e \right )\right )}{5}+\frac {\left (\cot ^{3}\left (f x +e \right )\right )}{3}-\cot \left (f x +e \right )-f x -e \right )-\frac {2 a b \left (\cos ^{5}\left (f x +e \right )\right )}{5 \sin \left (f x +e \right )^{5}}+b^{2} \left (-\frac {\cos ^{3}\left (f x +e \right )}{5 \sin \left (f x +e \right )^{5}}-\frac {2 \left (\cos ^{3}\left (f x +e \right )\right )}{15 \sin \left (f x +e \right )^{3}}\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 72, normalized size = 1.11 \[ -\frac {15 \, {\left (f x + e\right )} a^{2} + \frac {15 \, a^{2} \tan \left (f x + e\right )^{4} - 5 \, {\left (a^{2} - b^{2}\right )} \tan \left (f x + e\right )^{2} + 3 \, a^{2} + 6 \, a b + 3 \, b^{2}}{\tan \left (f x + e\right )^{5}}}{15 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.84, size = 68, normalized size = 1.05 \[ -a^2\,x-\frac {\frac {2\,a\,b}{5}+\frac {a^2}{5}+\frac {b^2}{5}-{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^2}{3}-\frac {b^2}{3}\right )+a^2\,{\mathrm {tan}\left (e+f\,x\right )}^4}{f\,{\mathrm {tan}\left (e+f\,x\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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