Integrand size = 23, antiderivative size = 63 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\frac {x}{a-b}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{(a-b) b^{3/2} f}+\frac {\tan (e+f x)}{b f} \]
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Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3751, 490, 536, 209, 211} \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{b^{3/2} f (a-b)}+\frac {x}{a-b}+\frac {\tan (e+f x)}{b f} \]
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Rule 209
Rule 211
Rule 490
Rule 536
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\tan (e+f x)}{b f}-\frac {\text {Subst}\left (\int \frac {a+(a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{b f} \\ & = \frac {\tan (e+f x)}{b f}+\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{(a-b) f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{(a-b) b f} \\ & = \frac {x}{a-b}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{(a-b) b^{3/2} f}+\frac {\tan (e+f x)}{b f} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.11 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\frac {e+f x}{(a-b) f}-\frac {a^{3/2} \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{(a-b) b^{3/2} f}+\frac {\tan (e+f x)}{b f} \]
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Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(\frac {\frac {\tan \left (f x +e \right )}{b}-\frac {a^{2} \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{b \left (a -b \right ) \sqrt {a b}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a -b}}{f}\) | \(65\) |
default | \(\frac {\frac {\tan \left (f x +e \right )}{b}-\frac {a^{2} \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {a b}}\right )}{b \left (a -b \right ) \sqrt {a b}}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{a -b}}{f}\) | \(65\) |
risch | \(\frac {x}{a -b}+\frac {2 i}{f b \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {\sqrt {-a b}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b}+a +b}{a -b}\right )}{2 b^{2} \left (a -b \right ) f}-\frac {\sqrt {-a b}\, a \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b}-a -b}{a -b}\right )}{2 b^{2} \left (a -b \right ) f}\) | \(144\) |
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Time = 0.29 (sec) , antiderivative size = 220, normalized size of antiderivative = 3.49 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\left [\frac {4 \, b f x - a \sqrt {-\frac {a}{b}} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{4} - 6 \, a b \tan \left (f x + e\right )^{2} + a^{2} + 4 \, {\left (b^{2} \tan \left (f x + e\right )^{3} - a b \tan \left (f x + e\right )\right )} \sqrt {-\frac {a}{b}}}{b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}}\right ) + 4 \, {\left (a - b\right )} \tan \left (f x + e\right )}{4 \, {\left (a b - b^{2}\right )} f}, \frac {2 \, b f x - a \sqrt {\frac {a}{b}} \arctan \left (\frac {{\left (b \tan \left (f x + e\right )^{2} - a\right )} \sqrt {\frac {a}{b}}}{2 \, a \tan \left (f x + e\right )}\right ) + 2 \, {\left (a - b\right )} \tan \left (f x + e\right )}{2 \, {\left (a b - b^{2}\right )} f}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 427 vs. \(2 (48) = 96\).
Time = 3.65 (sec) , antiderivative size = 427, normalized size of antiderivative = 6.78 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\begin {cases} \tilde {\infty } x \tan ^{2}{\left (e \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge f = 0 \\\frac {x + \frac {\tan ^{3}{\left (e + f x \right )}}{3 f} - \frac {\tan {\left (e + f x \right )}}{f}}{a} & \text {for}\: b = 0 \\\frac {- x + \frac {\tan {\left (e + f x \right )}}{f}}{b} & \text {for}\: a = 0 \\- \frac {3 f x \tan ^{2}{\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} - \frac {3 f x}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} + \frac {2 \tan ^{3}{\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} + \frac {3 \tan {\left (e + f x \right )}}{2 b f \tan ^{2}{\left (e + f x \right )} + 2 b f} & \text {for}\: a = b \\\frac {x \tan ^{4}{\left (e \right )}}{a + b \tan ^{2}{\left (e \right )}} & \text {for}\: f = 0 \\- \frac {a^{2} \log {\left (- \sqrt {- \frac {a}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b^{2} f \sqrt {- \frac {a}{b}} - 2 b^{3} f \sqrt {- \frac {a}{b}}} + \frac {a^{2} \log {\left (\sqrt {- \frac {a}{b}} + \tan {\left (e + f x \right )} \right )}}{2 a b^{2} f \sqrt {- \frac {a}{b}} - 2 b^{3} f \sqrt {- \frac {a}{b}}} + \frac {2 a b \sqrt {- \frac {a}{b}} \tan {\left (e + f x \right )}}{2 a b^{2} f \sqrt {- \frac {a}{b}} - 2 b^{3} f \sqrt {- \frac {a}{b}}} + \frac {2 b^{2} f x \sqrt {- \frac {a}{b}}}{2 a b^{2} f \sqrt {- \frac {a}{b}} - 2 b^{3} f \sqrt {- \frac {a}{b}}} - \frac {2 b^{2} \sqrt {- \frac {a}{b}} \tan {\left (e + f x \right )}}{2 a b^{2} f \sqrt {- \frac {a}{b}} - 2 b^{3} f \sqrt {- \frac {a}{b}}} & \text {otherwise} \end {cases} \]
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Time = 0.35 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\frac {a^{2} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a b - b^{2}\right )} \sqrt {a b}} - \frac {f x + e}{a - b} - \frac {\tan \left (f x + e\right )}{b}}{f} \]
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Time = 0.98 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.30 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=-\frac {\frac {{\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} a^{2}}{{\left (a b - b^{2}\right )} \sqrt {a b}} - \frac {f x + e}{a - b} - \frac {\tan \left (f x + e\right )}{b}}{f} \]
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Time = 11.62 (sec) , antiderivative size = 1212, normalized size of antiderivative = 19.24 \[ \int \frac {\tan ^4(e+f x)}{a+b \tan ^2(e+f x)} \, dx=\text {Too large to display} \]
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