Integrand size = 25, antiderivative size = 449 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {21 a^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \]
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Time = 0.29 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2663, 2665, 2654, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \arctan \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}+1\right )}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)} \log \left (\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)+\sqrt {a}\right )}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \]
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2654
Rule 2663
Rule 2665
Rubi steps \begin{align*} \text {integral}& = -\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{8} \left (7 a^2\right ) \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{5/2} \, dx \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{32} \left (21 a^4\right ) \int \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \, dx \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {1}{32} \left (21 a^4 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \int \frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}} \, dx \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 a^5 b \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{16 f} \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}-\frac {\left (21 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {a-b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{32 f}+\frac {\left (21 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {a+b x^2}{a^2+b^2 x^4} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{32 f} \\ & = -\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{64 b f}+\frac {\left (21 a^5 \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}+x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{64 b f}+\frac {\left (21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}+2 x}{-\frac {a}{b}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{64 \sqrt {2} \sqrt {b} f}+\frac {\left (21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {b}}-2 x}{-\frac {a}{b}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {b}}-x^2} \, dx,x,\frac {\sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}\right )}{64 \sqrt {2} \sqrt {b} f} \\ & = \frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}}+\frac {\left (21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{32 \sqrt {2} \sqrt {b} f}-\frac {\left (21 a^{9/2} \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right )}{32 \sqrt {2} \sqrt {b} f} \\ & = -\frac {21 a^{9/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {a} \sqrt {b \cos (e+f x)}}\right ) \sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}{32 \sqrt {2} \sqrt {b} f}+\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}-\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {21 a^{9/2} \sqrt {b \cos (e+f x)} \log \left (\sqrt {a}+\frac {\sqrt {2} \sqrt {b} \sqrt {a \sin (e+f x)}}{\sqrt {b \cos (e+f x)}}+\sqrt {a} \tan (e+f x)\right ) \sqrt {b \sec (e+f x)}}{64 \sqrt {2} \sqrt {b} f}-\frac {7 a^3 b (a \sin (e+f x))^{3/2}}{16 f \sqrt {b \sec (e+f x)}}-\frac {a b (a \sin (e+f x))^{7/2}}{4 f \sqrt {b \sec (e+f x)}} \\ \end{align*}
Time = 2.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.38 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\frac {a^4 \cot (e+f x) \sqrt {b \sec (e+f x)} \sqrt {a \sin (e+f x)} \left (4 (-9+2 \cos (2 (e+f x))) \sin ^2(e+f x)+21 \sqrt {2} \arctan \left (\frac {-1+\sqrt {\tan ^2(e+f x)}}{\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}-21 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{\tan ^2(e+f x)}}{1+\sqrt {\tan ^2(e+f x)}}\right ) \sqrt [4]{\tan ^2(e+f x)}\right )}{64 f} \]
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Time = 3.49 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {\sqrt {2}\, \left (16 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )+16 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-44 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sqrt {2}\, \sin \left (f x +e \right ) \cos \left (f x +e \right )-44 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+21 \ln \left (-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )-2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )-21 \ln \left (2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cot \left (f x +e \right )+2 \sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \csc \left (f x +e \right )+2-2 \cot \left (f x +e \right )\right )+42 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )+42 \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )\right ) \sqrt {b \sec \left (f x +e \right )}\, \sqrt {a \sin \left (f x +e \right )}\, a^{4} \cos \left (f x +e \right )}{128 f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}}\) | \(537\) |
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 1145, normalized size of antiderivative = 2.55 \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\text {Timed out} \]
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\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \]
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\[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int { \sqrt {b \sec \left (f x + e\right )} \left (a \sin \left (f x + e\right )\right )^{\frac {9}{2}} \,d x } \]
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Timed out. \[ \int \sqrt {b \sec (e+f x)} (a \sin (e+f x))^{9/2} \, dx=\int {\left (a\,\sin \left (e+f\,x\right )\right )}^{9/2}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \]
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