Optimal. Leaf size=65 \[ \frac {1}{2 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {\log \left (a \cos ^2(e+f x)+b \sin ^2(e+f x)\right )}{2 f (a-b)^2} \]
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Rubi [A] time = 0.07, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3670, 444, 44} \[ \frac {1}{2 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {\log \left (a \cos ^2(e+f x)+b \sin ^2(e+f x)\right )}{2 f (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 444
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tan (e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{(1+x) (a+b x)^2} \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{(a-b)^2 (1+x)}-\frac {b}{(a-b) (a+b x)^2}-\frac {b}{(a-b)^2 (a+b x)}\right ) \, dx,x,\tan ^2(e+f x)\right )}{2 f}\\ &=-\frac {\log \left (a \cos ^2(e+f x)+b \sin ^2(e+f x)\right )}{2 (a-b)^2 f}+\frac {1}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 57, normalized size = 0.88 \[ -\frac {\frac {b-a}{a+b \tan ^2(e+f x)}+\log \left (a+b \tan ^2(e+f x)\right )+2 \log (\cos (e+f x))}{2 f (a-b)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 98, normalized size = 1.51 \[ -\frac {b \tan \left (f x + e\right )^{2} + {\left (b \tan \left (f x + e\right )^{2} + a\right )} \log \left (\frac {b \tan \left (f x + e\right )^{2} + a}{\tan \left (f x + e\right )^{2} + 1}\right ) + b}{2 \, {\left ({\left (a^{2} b - 2 \, a b^{2} + b^{3}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} f\right )}} \]
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 [A] time = 0.21, size = 104, normalized size = 1.60 \[ -\frac {\ln \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}{2 f \left (a -b \right )^{2}}+\frac {a}{2 f \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}-\frac {b}{2 f \left (a -b \right )^{2} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {\ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 f \left (a -b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 88, normalized size = 1.35 \[ -\frac {\frac {b}{a^{3} - 2 \, a^{2} b + a b^{2} - {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \sin \left (f x + e\right )^{2}} + \frac {\log \left (-{\left (a - b\right )} \sin \left (f x + e\right )^{2} + a\right )}{a^{2} - 2 \, a b + b^{2}}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.64, size = 195, normalized size = 3.00 \[ -\frac {b\,\left (1+{\mathrm {tan}\left (e+f\,x\right )}^2\,\mathrm {atan}\left (\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-b\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}}{2\,a+a\,{\mathrm {tan}\left (e+f\,x\right )}^2+b\,{\mathrm {tan}\left (e+f\,x\right )}^2}\right )\,2{}\mathrm {i}\right )+a\,\left (-1+\mathrm {atan}\left (\frac {a\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}-b\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}}{2\,a+a\,{\mathrm {tan}\left (e+f\,x\right )}^2+b\,{\mathrm {tan}\left (e+f\,x\right )}^2}\right )\,2{}\mathrm {i}\right )}{f\,\left (2\,a^3+2\,a^2\,b\,{\mathrm {tan}\left (e+f\,x\right )}^2-4\,a^2\,b-4\,a\,b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2+2\,a\,b^2+2\,b^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.39, size = 816, normalized size = 12.55 \[ \begin {cases} \frac {\tilde {\infty } x}{\tan ^{3}{\relax (e )}} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\- \frac {1}{4 b^{2} f \tan ^{4}{\left (e + f x \right )} + 8 b^{2} f \tan ^{2}{\left (e + f x \right )} + 4 b^{2} f} & \text {for}\: a = b \\\frac {x \tan {\relax (e )}}{\left (a + b \tan ^{2}{\relax (e )}\right )^{2}} & \text {for}\: f = 0 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 a^{2} f} & \text {for}\: b = 0 \\- \frac {a \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} - \frac {a \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} + \frac {a \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} + \frac {a}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} - \frac {b \log {\left (- i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )} \tan ^{2}{\left (e + f x \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} - \frac {b \log {\left (i \sqrt {a} \sqrt {\frac {1}{b}} + \tan {\left (e + f x \right )} \right )} \tan ^{2}{\left (e + f x \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} + \frac {b \log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )} \tan ^{2}{\left (e + f x \right )}}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} - \frac {b}{2 a^{3} f + 2 a^{2} b f \tan ^{2}{\left (e + f x \right )} - 4 a^{2} b f - 4 a b^{2} f \tan ^{2}{\left (e + f x \right )} + 2 a b^{2} f + 2 b^{3} f \tan ^{2}{\left (e + f x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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