Integrand size = 15, antiderivative size = 117 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}+\frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3751, 1262, 755, 837, 12, 739, 212} \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}}-\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}+\frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}} \]
[In]
[Out]
Rule 12
Rule 212
Rule 739
Rule 755
Rule 837
Rule 1262
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \left (a+b x^4\right )^{5/2}} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(1+x) \left (a+b x^2\right )^{5/2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 a-2 b-2 b x}{(1+x) \left (a+b x^2\right )^{3/2}} \, dx,x,\tan ^2(x)\right )}{6 a (a+b)} \\ & = \frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}}+\frac {\text {Subst}\left (\int \frac {3 a^2 b}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{6 a^2 b (a+b)^2} \\ & = \frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}}+\frac {\text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )}{2 (a+b)^2} \\ & = \frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}}-\frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {a-b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^2} \\ & = -\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 (a+b)^{5/2}}+\frac {a+b \tan ^2(x)}{6 a (a+b) \left (a+b \tan ^4(x)\right )^{3/2}}+\frac {3 a^2+b (5 a+2 b) \tan ^2(x)}{6 a^2 (a+b)^2 \sqrt {a+b \tan ^4(x)}} \\ \end{align*}
Time = 0.88 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.97 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {1}{6} \left (-\frac {3 \text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{(a+b)^{5/2}}+\frac {a^2 (4 a+b)+3 a b (2 a+b) \tan ^2(x)+3 a^2 b \tan ^4(x)+b^2 (5 a+2 b) \tan ^6(x)}{a^2 (a+b)^2 \left (a+b \tan ^4(x)\right )^{3/2}}\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(585\) vs. \(2(101)=202\).
Time = 0.07 (sec) , antiderivative size = 586, normalized size of antiderivative = 5.01
method | result | size |
derivativedivides | \(-\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}+\frac {\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(586\) |
default | \(-\frac {b^{2} \ln \left (\frac {2 a +2 b -2 b \left (1+\tan \left (x \right )^{2}\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tan \left (x \right )^{2}\right )^{2}-2 b \left (1+\tan \left (x \right )^{2}\right )+a +b}}{1+\tan \left (x \right )^{2}}\right )}{2 \left (\sqrt {-a b}+b \right )^{2} \left (\sqrt {-a b}-b \right )^{2} \sqrt {a +b}}-\frac {-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right ) a}+\frac {\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}}-\frac {\sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{3 a \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right ) a}+\frac {\left (2 \sqrt {-a b}+b \right ) \sqrt {b \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )^{2}+2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}+b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}-\frac {\sqrt {-a b}}{b}\right )}-\frac {\left (2 \sqrt {-a b}-b \right ) \sqrt {b \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )^{2}-2 \sqrt {-a b}\, \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}}{8 \left (\sqrt {-a b}-b \right )^{2} a^{2} \left (\tan \left (x \right )^{2}+\frac {\sqrt {-a b}}{b}\right )}\) | \(586\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (103) = 206\).
Time = 0.46 (sec) , antiderivative size = 599, normalized size of antiderivative = 5.12 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a^{2} b^{2} \tan \left (x\right )^{8} + 2 \, a^{3} b \tan \left (x\right )^{4} + a^{4}\right )} \sqrt {a + b} \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} + 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + 2 \, {\left ({\left (5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (x\right )^{6} + 3 \, {\left (a^{3} b + a^{2} b^{2}\right )} \tan \left (x\right )^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )} \sqrt {b \tan \left (x\right )^{4} + a}}{12 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} \tan \left (x\right )^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (x\right )^{4}\right )}}, -\frac {3 \, {\left (a^{2} b^{2} \tan \left (x\right )^{8} + 2 \, a^{3} b \tan \left (x\right )^{4} + a^{4}\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) - {\left ({\left (5 \, a^{2} b^{2} + 7 \, a b^{3} + 2 \, b^{4}\right )} \tan \left (x\right )^{6} + 3 \, {\left (a^{3} b + a^{2} b^{2}\right )} \tan \left (x\right )^{4} + 4 \, a^{4} + 5 \, a^{3} b + a^{2} b^{2} + 3 \, {\left (2 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3}\right )} \tan \left (x\right )^{2}\right )} \sqrt {b \tan \left (x\right )^{4} + a}}{6 \, {\left ({\left (a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + a^{2} b^{5}\right )} \tan \left (x\right )^{8} + a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + 2 \, {\left (a^{6} b + 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (x\right )^{4}\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\tan {\left (x \right )}}{\left (a + b \tan ^{4}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (x\right )}{{\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 618, normalized size of antiderivative = 5.28 \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {{\left (5 \, a^{7} b^{3} + 32 \, a^{6} b^{4} + 87 \, a^{5} b^{5} + 130 \, a^{4} b^{6} + 115 \, a^{3} b^{7} + 60 \, a^{2} b^{8} + 17 \, a b^{9} + 2 \, b^{10}\right )} \tan \left (x\right )^{2}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}} + \frac {3 \, {\left (a^{8} b^{2} + 6 \, a^{7} b^{3} + 15 \, a^{6} b^{4} + 20 \, a^{5} b^{5} + 15 \, a^{4} b^{6} + 6 \, a^{3} b^{7} + a^{2} b^{8}\right )}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}\right )} \tan \left (x\right )^{2} + \frac {3 \, {\left (2 \, a^{8} b^{2} + 13 \, a^{7} b^{3} + 36 \, a^{6} b^{4} + 55 \, a^{5} b^{5} + 50 \, a^{4} b^{6} + 27 \, a^{3} b^{7} + 8 \, a^{2} b^{8} + a b^{9}\right )}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}\right )} \tan \left (x\right )^{2} + \frac {4 \, a^{9} b + 25 \, a^{8} b^{2} + 66 \, a^{7} b^{3} + 95 \, a^{6} b^{4} + 80 \, a^{5} b^{5} + 39 \, a^{4} b^{6} + 10 \, a^{3} b^{7} + a^{2} b^{8}}{a^{10} b + 8 \, a^{9} b^{2} + 28 \, a^{8} b^{3} + 56 \, a^{7} b^{4} + 70 \, a^{6} b^{5} + 56 \, a^{5} b^{6} + 28 \, a^{4} b^{7} + 8 \, a^{3} b^{8} + a^{2} b^{9}}}{6 \, {\left (b \tan \left (x\right )^{4} + a\right )}^{\frac {3}{2}}} - \frac {\arctan \left (\frac {\sqrt {b} \tan \left (x\right )^{2} - \sqrt {b \tan \left (x\right )^{4} + a} + \sqrt {b}}{\sqrt {-a - b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-a - b}} \]
[In]
[Out]
Timed out. \[ \int \frac {\tan (x)}{\left (a+b \tan ^4(x)\right )^{5/2}} \, dx=\int \frac {\mathrm {tan}\left (x\right )}{{\left (b\,{\mathrm {tan}\left (x\right )}^4+a\right )}^{5/2}} \,d x \]
[In]
[Out]