Integrand size = 25, antiderivative size = 131 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-4 b) \tan (e+f x)}{3 (a-b)^2 b f \sqrt {a+b \tan ^2(e+f x)}} \]
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Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3751, 481, 541, 12, 385, 209} \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{f (a-b)^{5/2}}+\frac {(a-4 b) \tan (e+f x)}{3 b f (a-b)^2 \sqrt {a+b \tan ^2(e+f x)}}-\frac {a \tan (e+f x)}{3 b f (a-b) \left (a+b \tan ^2(e+f x)\right )^{3/2}} \]
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Rule 12
Rule 209
Rule 385
Rule 481
Rule 541
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 )^{5/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {a+(a-3 b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 (a-b) b f} \\ & = -\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-4 b) \tan (e+f x)}{3 (a-b)^2 b f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {3 a b}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{3 a (a-b)^2 b f} \\ & = -\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-4 b) \tan (e+f x)}{3 (a-b)^2 b f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{(a-b)^2 f} \\ & = -\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-4 b) \tan (e+f x)}{3 (a-b)^2 b f \sqrt {a+b \tan ^2(e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^2 f} \\ & = \frac {\arctan \left (\frac {\sqrt {a-b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{(a-b)^{5/2} f}-\frac {a \tan (e+f x)}{3 (a-b) b f \left (a+b \tan ^2(e+f x)\right )^{3/2}}+\frac {(a-4 b) \tan (e+f x)}{3 (a-b)^2 b f \sqrt {a+b \tan ^2(e+f x)}} \\ \end{align*}
Time = 6.07 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.98 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\frac {\tan ^5(e+f x) \left (1+\frac {b \tan ^2(e+f x)}{a}\right ) \left (\frac {\text {arctanh}\left (\frac {\sqrt {-\tan ^2(e+f x)+\frac {b \tan ^2(e+f x)}{a}}}{\sqrt {1+\frac {b \tan ^2(e+f x)}{a}}}\right ) \sqrt {-\tan ^2(e+f x)+\frac {b \tan ^2(e+f x)}{a}}}{\sqrt {1+\frac {b \tan ^2(e+f x)}{a}}}-\frac {-\tan ^2(e+f x)+\frac {b \tan ^2(e+f x)}{a}}{1+\frac {b \tan ^2(e+f x)}{a}}-\frac {\left (-\tan ^2(e+f x)+\frac {b \tan ^2(e+f x)}{a}\right )^2}{3 \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^2}\right )}{a^2 f \sqrt {a+b \tan ^2(e+f x)} \left (-\tan ^2(e+f x)+\frac {b \tan ^2(e+f x)}{a}\right )^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs. \(2(117)=234\).
Time = 0.07 (sec) , antiderivative size = 291, normalized size of antiderivative = 2.22
method | result | size |
derivativedivides | \(-\frac {\tan \left (f x +e \right )}{3 f a \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \tan \left (f x +e \right )}{3 f \,a^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}-\frac {\tan \left (f x +e \right )}{3 f b \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}+\frac {\tan \left (f x +e \right )}{3 f a b \sqrt {a +b \tan \left (f x +e \right )^{2}}}-\frac {b \tan \left (f x +e \right )}{f \left (a -b \right )^{2} a \sqrt {a +b \tan \left (f x +e \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )^{3} b^{2}}-\frac {b \tan \left (f x +e \right )}{3 a \left (a -b \right ) f \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 b \tan \left (f x +e \right )}{3 f \left (a -b \right ) a^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}\) | \(291\) |
default | \(-\frac {\tan \left (f x +e \right )}{3 f a \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \tan \left (f x +e \right )}{3 f \,a^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}-\frac {\tan \left (f x +e \right )}{3 f b \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}+\frac {\tan \left (f x +e \right )}{3 f a b \sqrt {a +b \tan \left (f x +e \right )^{2}}}-\frac {b \tan \left (f x +e \right )}{f \left (a -b \right )^{2} a \sqrt {a +b \tan \left (f x +e \right )^{2}}}+\frac {\sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (f x +e \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (f x +e \right )^{2}}}\right )}{f \left (a -b \right )^{3} b^{2}}-\frac {b \tan \left (f x +e \right )}{3 a \left (a -b \right ) f \left (a +b \tan \left (f x +e \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 b \tan \left (f x +e \right )}{3 f \left (a -b \right ) a^{2} \sqrt {a +b \tan \left (f x +e \right )^{2}}}\) | \(291\) |
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Time = 0.33 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.80 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\left [-\frac {3 \, {\left (b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (f x + e\right )^{2} - 2 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} \sqrt {-a + b} \tan \left (f x + e\right ) - a}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} - a b\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{6 \, {\left ({\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} f\right )}}, \frac {3 \, {\left (b^{2} \tan \left (f x + e\right )^{4} + 2 \, a b \tan \left (f x + e\right )^{2} + a^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (f x + e\right )^{2} + a}}{\sqrt {a - b} \tan \left (f x + e\right )}\right ) + {\left ({\left (a^{2} - 5 \, a b + 4 \, b^{2}\right )} \tan \left (f x + e\right )^{3} - 3 \, {\left (a^{2} - a b\right )} \tan \left (f x + e\right )\right )} \sqrt {b \tan \left (f x + e\right )^{2} + a}}{3 \, {\left ({\left (a^{3} b^{2} - 3 \, a^{2} b^{3} + 3 \, a b^{4} - b^{5}\right )} f \tan \left (f x + e\right )^{4} + 2 \, {\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} f \tan \left (f x + e\right )^{2} + {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} f\right )}}\right ] \]
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\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\left (a + b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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