Optimal. Leaf size=72 \[ -\frac {(2 a-b) \log (1-\cos (e+f x))}{4 f}-\frac {(2 a+b) \log (\cos (e+f x)+1)}{4 f}-\frac {\csc ^2(e+f x) (a+b \cos (e+f x))}{2 f} \]
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Rubi [A] time = 0.06, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4138, 1814, 633, 31} \[ -\frac {(2 a-b) \log (1-\cos (e+f x))}{4 f}-\frac {(2 a+b) \log (\cos (e+f x)+1)}{4 f}-\frac {\csc ^2(e+f x) (a+b \cos (e+f x))}{2 f} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 1814
Rule 4138
Rubi steps
\begin {align*} \int \cot ^3(e+f x) \left (a+b \sec ^3(e+f x)\right ) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {b+a x^3}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {(a+b \cos (e+f x)) \csc ^2(e+f x)}{2 f}+\frac {\operatorname {Subst}\left (\int \frac {-b+2 a x}{1-x^2} \, dx,x,\cos (e+f x)\right )}{2 f}\\ &=-\frac {(a+b \cos (e+f x)) \csc ^2(e+f x)}{2 f}+\frac {(2 a-b) \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\cos (e+f x)\right )}{4 f}+\frac {(2 a+b) \operatorname {Subst}\left (\int \frac {1}{-1-x} \, dx,x,\cos (e+f x)\right )}{4 f}\\ &=-\frac {(a+b \cos (e+f x)) \csc ^2(e+f x)}{2 f}-\frac {(2 a-b) \log (1-\cos (e+f x))}{4 f}-\frac {(2 a+b) \log (1+\cos (e+f x))}{4 f}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 114, normalized size = 1.58 \[ -\frac {a \left (\cot ^2(e+f x)+2 \log (\tan (e+f x))+2 \log (\cos (e+f x))\right )}{2 f}-\frac {b \csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}+\frac {b \sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}+\frac {b \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}-\frac {b \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 99, normalized size = 1.38 \[ \frac {2 \, b \cos \left (f x + e\right ) - {\left ({\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - b\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) - {\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} - 2 \, a + b\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 2 \, a}{4 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )}} \]
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 = 1.10, size = 69, normalized size = 0.96 \[ -\frac {a \left (\cot ^{2}\left (f x +e \right )\right )}{2 f}-\frac {a \ln \left (\sin \left (f x +e \right )\right )}{f}-\frac {b \csc \left (f x +e \right ) \cot \left (f x +e \right )}{2 f}+\frac {b \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 62, normalized size = 0.86 \[ -\frac {{\left (2 \, a + b\right )} \log \left (\cos \left (f x + e\right ) + 1\right ) + {\left (2 \, a - b\right )} \log \left (\cos \left (f x + e\right ) - 1\right ) - \frac {2 \, {\left (b \cos \left (f x + e\right ) + a\right )}}{\cos \left (f x + e\right )^{2} - 1}}{4 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.65, size = 86, normalized size = 1.19 \[ \frac {a\,\ln \left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a}{8}-\frac {b}{8}\right )}{f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a}{8}+\frac {b}{8}\right )}{f}-\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (a-\frac {b}{2}\right )}{f} \]
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 \[ \int \left (a + b \sec ^{3}{\left (e + f x \right )}\right ) \cot ^{3}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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