Optimal. Leaf size=57 \[ -\frac {\left (a^2-b^2\right ) \log (\sin (e+f x))}{f}-\frac {(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac {b^2 \log (\cos (e+f x))}{f} \]
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Rubi [A] time = 0.08, antiderivative size = 57, 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 = {4138, 446, 88} \[ -\frac {\left (a^2-b^2\right ) \log (\sin (e+f x))}{f}-\frac {(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac {b^2 \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \cot ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^2 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (b+a x^2\right )^2}{x \left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^2}{(1-x)^2 x} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {(a+b)^2}{(-1+x)^2}+\frac {a^2-b^2}{-1+x}+\frac {b^2}{x}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {(a+b)^2 \csc ^2(e+f x)}{2 f}-\frac {b^2 \log (\cos (e+f x))}{f}-\frac {\left (a^2-b^2\right ) \log (\sin (e+f x))}{f}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 81, normalized size = 1.42 \[ -\frac {2 \left (a \cos ^2(e+f x)+b\right )^2 \left (2 \left (a^2-b^2\right ) \log (\sin (e+f x))+(a+b)^2 \csc ^2(e+f x)+2 b^2 \log (\cos (e+f x))\right )}{f (a \cos (2 (e+f x))+a+2 b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 100, normalized size = 1.75 \[ \frac {a^{2} + 2 \, a b + b^{2} - {\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \log \left (\cos \left (f x + e\right )^{2}\right ) - {\left ({\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} + b^{2}\right )} \log \left (-\frac {1}{4} \, \cos \left (f x + e\right )^{2} + \frac {1}{4}\right )}{2 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.18, size = 78, normalized size = 1.37 \[ -\frac {a^{2} \left (\cot ^{2}\left (f x +e \right )\right )}{2 f}-\frac {a^{2} \ln \left (\sin \left (f x +e \right )\right )}{f}-\frac {a b}{f \sin \left (f x +e \right )^{2}}-\frac {b^{2}}{2 f \sin \left (f x +e \right )^{2}}+\frac {b^{2} \ln \left (\tan \left (f x +e \right )\right )}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 60, normalized size = 1.05 \[ -\frac {b^{2} \log \left (\sin \left (f x + e\right )^{2} - 1\right ) + {\left (a^{2} - b^{2}\right )} \log \left (\sin \left (f x + e\right )^{2}\right ) + \frac {a^{2} + 2 \, a b + b^{2}}{\sin \left (f x + e\right )^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.58, size = 68, normalized size = 1.19 \[ \frac {a^2\,\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a^2-b^2\right )}{f}-\frac {{\mathrm {cot}\left (e+f\,x\right )}^2\,\left (\frac {a^2}{2}+a\,b+\frac {b^2}{2}\right )}{f} \]
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 \[ \int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{2} \cot ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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