Integrand size = 26, antiderivative size = 64 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3255, 3284, 16, 45} \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f} \]
[In]
[Out]
Rule 16
Rule 45
Rule 3255
Rule 3284
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {a \cos ^2(e+f x)} \tan ^5(e+f x) \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {(1-x)^2 \sqrt {a x}}{x^3} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = -\frac {a^3 \text {Subst}\left (\int \frac {(1-x)^2}{(a x)^{5/2}} \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = -\frac {a^3 \text {Subst}\left (\int \left (\frac {1}{(a x)^{5/2}}-\frac {2}{a (a x)^{3/2}}+\frac {1}{a^2 \sqrt {a x}}\right ) \, dx,x,\cos ^2(e+f x)\right )}{2 f} \\ & = \frac {a^2}{3 f \left (a \cos ^2(e+f x)\right )^{3/2}}-\frac {2 a}{f \sqrt {a \cos ^2(e+f x)}}-\frac {\sqrt {a \cos ^2(e+f x)}}{f} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.66 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {a \left (-6-3 \cos ^2(e+f x)+\sec ^2(e+f x)\right )}{3 f \sqrt {a \cos ^2(e+f x)}} \]
[In]
[Out]
Time = 0.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {a \left (3 \left (\cos ^{4}\left (f x +e \right )\right )+6 \left (\cos ^{2}\left (f x +e \right )\right )-1\right )}{3 \cos \left (f x +e \right )^{2} \sqrt {a \left (\cos ^{2}\left (f x +e \right )\right )}\, f}\) | \(49\) |
risch | \(-\frac {\sqrt {\left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{2} a \,{\mathrm e}^{-2 i \left (f x +e \right )}}\, \left (3 \,{\mathrm e}^{8 i \left (f x +e \right )}+36 \,{\mathrm e}^{6 i \left (f x +e \right )}+50 \,{\mathrm e}^{4 i \left (f x +e \right )}+36 \,{\mathrm e}^{2 i \left (f x +e \right )}+3\right )}{6 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{4}}\) | \(91\) |
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.73 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=-\frac {{\left (3 \, \cos \left (f x + e\right )^{4} + 6 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \cos \left (f x + e\right )^{2}}}{3 \, f \cos \left (f x + e\right )^{4}} \]
[In]
[Out]
\[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\int \sqrt {- a \left (\sin {\left (e + f x \right )} - 1\right ) \left (\sin {\left (e + f x \right )} + 1\right )} \tan ^{5}{\left (e + f x \right )}\, dx \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.08 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=-\frac {3 \, \sqrt {-a \sin \left (f x + e\right )^{2} + a} a^{3} - \frac {6 \, {\left (a \sin \left (f x + e\right )^{2} - a\right )} a^{4} + a^{5}}{{\left (-a \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}}{3 \, a^{3} f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (56) = 112\).
Time = 2.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.00 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=\frac {2 \, \sqrt {a} {\left (\frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1} - \frac {3 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 12 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right ) \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 1\right )}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3}}\right )}}{3 \, f} \]
[In]
[Out]
Time = 17.39 (sec) , antiderivative size = 326, normalized size of antiderivative = 5.09 \[ \int \sqrt {a-a \sin ^2(e+f x)} \tan ^5(e+f x) \, dx=-\frac {\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{f}-\frac {8\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{f\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}+\frac {16\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{3\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )}-\frac {16\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a-a\,{\left (\frac {{\mathrm {e}}^{-e\,1{}\mathrm {i}-f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}}{3\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3\,\left ({\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}+{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\right )} \]
[In]
[Out]