Optimal. Leaf size=74 \[ -\frac {b^2 \log \left (a \cos ^2(e+f x)+b\right )}{2 a f (a+b)^2}-\frac {\csc ^2(e+f x)}{2 f (a+b)}-\frac {(a+2 b) \log (\sin (e+f x))}{f (a+b)^2} \]
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Rubi [A] time = 0.11, antiderivative size = 74, 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} \[ -\frac {b^2 \log \left (a \cos ^2(e+f x)+b\right )}{2 a f (a+b)^2}-\frac {\csc ^2(e+f x)}{2 f (a+b)}-\frac {(a+2 b) \log (\sin (e+f x))}{f (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 4138
Rubi steps
\begin {align*} \int \frac {\cot ^3(e+f x)}{a+b \sec ^2(e+f x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^5}{\left (1-x^2\right )^2 \left (b+a x^2\right )} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {x^2}{(1-x)^2 (b+a x)} \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {1}{(a+b) (-1+x)^2}+\frac {a+2 b}{(a+b)^2 (-1+x)}+\frac {b^2}{(a+b)^2 (b+a x)}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f}\\ &=-\frac {\csc ^2(e+f x)}{2 (a+b) f}-\frac {b^2 \log \left (b+a \cos ^2(e+f x)\right )}{2 a (a+b)^2 f}-\frac {(a+2 b) \log (\sin (e+f x))}{(a+b)^2 f}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 100, normalized size = 1.35 \[ -\frac {\sec ^2(e+f x) (a \cos (2 (e+f x))+a+2 b) \left (b^2 \log \left (-a \sin ^2(e+f x)+a+b\right )+a (a+b) \csc ^2(e+f x)+2 a (a+2 b) \log (\sin (e+f x))\right )}{4 a f (a+b)^2 \left (a+b \sec ^2(e+f x)\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 126, normalized size = 1.70 \[ \frac {a^{2} + a b - {\left (b^{2} \cos \left (f x + e\right )^{2} - b^{2}\right )} \log \left (a \cos \left (f x + e\right )^{2} + b\right ) - 2 \, {\left ({\left (a^{2} + 2 \, a b\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b\right )} \log \left (\frac {1}{2} \, \sin \left (f x + e\right )\right )}{2 \, {\left ({\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f \cos \left (f x + e\right )^{2} - {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} f\right )}} \]
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 [B] time = 1.03, size = 158, normalized size = 2.14 \[ -\frac {b^{2} \ln \left (b +a \left (\cos ^{2}\left (f x +e \right )\right )\right )}{2 a \left (a +b \right )^{2} f}+\frac {1}{f \left (4 a +4 b \right ) \left (-1+\cos \left (f x +e \right )\right )}-\frac {\ln \left (-1+\cos \left (f x +e \right )\right ) a}{2 f \left (a +b \right )^{2}}-\frac {\ln \left (-1+\cos \left (f x +e \right )\right ) b}{f \left (a +b \right )^{2}}-\frac {1}{f \left (4 a +4 b \right ) \left (1+\cos \left (f x +e \right )\right )}-\frac {\ln \left (1+\cos \left (f x +e \right )\right ) a}{2 f \left (a +b \right )^{2}}-\frac {\ln \left (1+\cos \left (f x +e \right )\right ) b}{f \left (a +b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 87, normalized size = 1.18 \[ -\frac {\frac {b^{2} \log \left (a \sin \left (f x + e\right )^{2} - a - b\right )}{a^{3} + 2 \, a^{2} b + a b^{2}} + \frac {{\left (a + 2 \, b\right )} \log \left (\sin \left (f x + e\right )^{2}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {1}{{\left (a + b\right )} \sin \left (f x + e\right )^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.94, size = 98, normalized size = 1.32 \[ \frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,a\,f}-\frac {{\mathrm {cot}\left (e+f\,x\right )}^2}{2\,f\,\left (a+b\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )\right )\,\left (a+2\,b\right )}{f\,\left (a^2+2\,a\,b+b^2\right )}-\frac {b^2\,\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a+b\right )}{2\,a\,f\,{\left (a+b\right )}^2} \]
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 \[ \int \frac {\cot ^{3}{\left (e + f x \right )}}{a + b \sec ^{2}{\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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