Optimal. Leaf size=50 \[ \frac {a \log \left (a+b \tan ^2(e+f x)\right )}{2 b f (a-b)}+\frac {\log (\cos (e+f x))}{f (a-b)} \]
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Rubi [A] time = 0.09, antiderivative size = 50, 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 = {3670, 446, 72} \[ \frac {a \log \left (a+b \tan ^2(e+f x)\right )}{2 b f (a-b)}+\frac {\log (\cos (e+f x))}{f (a-b)} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan ^3(e+f x)}{a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x}{(1+x) (a+b x)} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{(a-b) (1+x)}+\frac {a}{(a-b) (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\log (\cos (e+f x))}{(a-b) f}+\frac {a \log \left (a+b \tan ^2(e+f x)\right )}{2 (a-b) b f}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 0.82 \[ \frac {a \log \left (a+b \tan ^2(e+f x)\right )+2 b \log (\cos (e+f x))}{2 a b f-2 b^2 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 65, normalized size = 1.30 \[ \frac {a \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right ) - {\left (a - b\right )} \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right )}{2 \, {\left (a b - b^{2}\right )} f} \]
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 = 0.14, size = 54, normalized size = 1.08 \[ \frac {a \ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 \left (a -b \right ) b f}-\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a -b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 53, normalized size = 1.06 \[ \frac {\frac {a \log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a b - b^{2}} - \frac {\log \left (\sin \left (f x + e\right )^{2} - 1\right )}{b}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.78, size = 54, normalized size = 1.08 \[ \frac {a\,\ln \left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}{2\,f\,\left (a\,b-b^2\right )}-\frac {\ln \left ({\mathrm {tan}\left (e+f\,x\right )}^2+1\right )}{2\,f\,\left (a-b\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.67, size = 240, normalized size = 4.80 \[ \begin {cases} \tilde {\infty } x \tan {\relax (e )} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\\frac {- \frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} + \frac {\tan ^{2}{\left (e + f x \right )}}{2 f}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan ^{2}{\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} + \frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} + \frac {1}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} & \text {for}\: a = b \\\frac {x \tan ^{3}{\relax (e )}}{a + b \tan ^{2}{\relax (e )}} & \text {for}\: f = 0 \\\frac {a \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b f - 2 b^{2} f} + \frac {a \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b f - 2 b^{2} f} - \frac {b \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 a b f - 2 b^{2} f} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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