Optimal. Leaf size=863 \[ \frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right ) b^2}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right ) b^2}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{\log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right ) b^2}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{\log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right ) b^2}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2} b}{\left (a^2-b^2\right ) d e^3}+\frac{2 \cos (c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right ) \sqrt{e \tan (c+d x)} b}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (2 c+2 d x)}}+\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}} \]
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Rubi [A] time = 1.24995, antiderivative size = 863, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 23, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.92, Rules used = {3893, 3882, 3884, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2613, 2615, 2572, 2639, 3890, 2733, 2730, 2906, 2905, 490, 1213, 537} \[ \frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} \sqrt{\cos (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)+1}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)} b^3}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right ) b^2}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right ) b^2}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{\log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right ) b^2}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{\log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right ) b^2}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 \cos (c+d x) (e \tan (c+d x))^{3/2} b}{\left (a^2-b^2\right ) d e^3}+\frac{2 \cos (c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right ) \sqrt{e \tan (c+d x)} b}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (2 c+2 d x)}}+\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \log \left (\sqrt{e} \tan (c+d x)+\sqrt{e}+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3893
Rule 3882
Rule 3884
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 2613
Rule 2615
Rule 2572
Rule 2639
Rule 3890
Rule 2733
Rule 2730
Rule 2906
Rule 2905
Rule 490
Rule 1213
Rule 537
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sec (c+d x)) (e \tan (c+d x))^{3/2}} \, dx &=\frac{\int \frac{a-b \sec (c+d x)}{(e \tan (c+d x))^{3/2}} \, dx}{a^2-b^2}+\frac{b^2 \int \frac{\sqrt{e \tan (c+d x)}}{a+b \sec (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}\\ &=-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}+\frac{2 \int \left (-\frac{a}{2}-\frac{1}{2} b \sec (c+d x)\right ) \sqrt{e \tan (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}+\frac{b^2 \int \sqrt{e \tan (c+d x)} \, dx}{a \left (a^2-b^2\right ) e^2}-\frac{b^3 \int \frac{\sqrt{e \tan (c+d x)}}{b+a \cos (c+d x)} \, dx}{a \left (a^2-b^2\right ) e^2}\\ &=-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}-\frac{a \int \sqrt{e \tan (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}-\frac{b \int \sec (c+d x) \sqrt{e \tan (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{a \left (a^2-b^2\right ) d e}-\frac{\left (b^3 \sqrt{e \cot (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{1}{(b+a \cos (c+d x)) \sqrt{e \cot (c+d x)}} \, dx}{a \left (a^2-b^2\right ) e^2}\\ &=-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}+\frac{(2 b) \int \cos (c+d x) \sqrt{e \tan (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{\left (a^2-b^2\right ) d e}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a \left (a^2-b^2\right ) d e}-\frac{\left (b^3 \sqrt{-\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{\sqrt{\sin (c+d x)}}{\sqrt{-\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a \left (a^2-b^2\right ) e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{\left (a^2-b^2\right ) d e}-\frac{b^2 \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a \left (a^2-b^2\right ) d e}+\frac{b^2 \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{a \left (a^2-b^2\right ) d e}+\frac{\left (2 b \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \sqrt{\sin (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2 \sqrt{\sin (c+d x)}}-\frac{\left (b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \int \frac{\sqrt{\sin (c+d x)}}{\sqrt{\cos (c+d x)} (b+a \cos (c+d x))} \, dx}{a \left (a^2-b^2\right ) e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{e-x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{\left (a^2-b^2\right ) d e}-\frac{a \operatorname{Subst}\left (\int \frac{e+x^2}{e^2+x^4} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 a \left (a^2-b^2\right ) d e}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 a \left (a^2-b^2\right ) d e}-\frac{\left (4 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4} \left (a+b+(-a+b) x^4\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left (2 b \cos (c+d x) \sqrt{e \tan (c+d x)}\right ) \int \sqrt{\sin (2 c+2 d x)} \, dx}{\left (a^2-b^2\right ) e^2 \sqrt{\sin (2 c+2 d x)}}\\ &=\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}+\frac{2 b \cos (c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{e \tan (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}+2 x}{-e-\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{e}-2 x}{-e+\sqrt{2} \sqrt{e} x-x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \operatorname{Subst}\left (\int \frac{1}{e-\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \left (a^2-b^2\right ) d e}-\frac{a \operatorname{Subst}\left (\int \frac{1}{e+\sqrt{2} \sqrt{e} x+x^2} \, dx,x,\sqrt{e \tan (c+d x)}\right )}{2 \left (a^2-b^2\right ) d e}-\frac{\left (2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b}-\sqrt{a-b} x^2\right ) \sqrt{1-x^4}} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} \left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left (2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\sqrt{a+b}+\sqrt{a-b} x^2\right ) \sqrt{1-x^4}} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} \left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}\\ &=-\frac{b^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}+\frac{2 b \cos (c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{e \tan (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{\left (2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a+b}-\sqrt{a-b} x^2\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} \left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left (2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \sqrt{e \tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+x^2} \left (\sqrt{a+b}+\sqrt{a-b} x^2\right )} \, dx,x,\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )}{a \sqrt{a-b} \left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}\\ &=\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{b^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{a \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}+\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}+\frac{a \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} \left (a^2-b^2\right ) d e^{3/2}}-\frac{b^2 \log \left (\sqrt{e}+\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}\right )}{2 \sqrt{2} a \left (a^2-b^2\right ) d e^{3/2}}-\frac{2 (a-b \sec (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \tan (c+d x)}}+\frac{2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \Pi \left (-\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}-\frac{2 \sqrt{2} b^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{\sqrt{a-b}}{\sqrt{a+b}};\left .\sin ^{-1}\left (\frac{\sqrt{\sin (c+d x)}}{\sqrt{1+\cos (c+d x)}}\right )\right |-1\right ) \sqrt{e \tan (c+d x)}}{a (a-b)^{3/2} (a+b)^{3/2} d e^2 \sqrt{\sin (c+d x)}}+\frac{2 b \cos (c+d x) E\left (\left .c-\frac{\pi }{4}+d x\right |2\right ) \sqrt{e \tan (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (2 c+2 d x)}}-\frac{2 b \cos (c+d x) (e \tan (c+d x))^{3/2}}{\left (a^2-b^2\right ) d e^3}\\ \end{align*}
Mathematica [C] time = 27.1128, size = 1571, normalized size = 1.82 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.251, size = 6426, normalized size = 7.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \tan{\left (c + d x \right )}\right )^{\frac{3}{2}} \left (a + b \sec{\left (c + d x \right )}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \tan \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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