Optimal. Leaf size=850 \[ \frac{3 e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac{3 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac{e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{7 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)} b^2}{a^4 d \sqrt{\sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac{2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{9/2} d}-\frac{2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{9/2} d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}} \]
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Rubi [A] time = 2.12648, antiderivative size = 850, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 15, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3872, 2912, 2635, 2640, 2639, 2693, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2695} \[ \frac{3 e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac{3 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac{e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac{7 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)} b^2}{a^4 d \sqrt{\sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac{2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{9/2} d}-\frac{2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{9/2} d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2912
Rule 2635
Rule 2640
Rule 2639
Rule 2693
Rule 2867
Rule 2701
Rule 2807
Rule 2805
Rule 329
Rule 298
Rule 205
Rule 208
Rule 2695
Rubi steps
\begin{align*} \int \frac{(e \sin (c+d x))^{5/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) (e \sin (c+d x))^{5/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac{(e \sin (c+d x))^{5/2}}{a^2}+\frac{b^2 (e \sin (c+d x))^{5/2}}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b (e \sin (c+d x))^{5/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac{\int (e \sin (c+d x))^{5/2} \, dx}{a^2}-\frac{(2 b) \int \frac{(e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac{b^2 \int \frac{(e \sin (c+d x))^{5/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (3 e^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{5 a^2}+\frac{\left (2 b e^2\right ) \int \frac{(-a-b \cos (c+d x)) \sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac{\left (3 b^2 e^2\right ) \int \frac{\cos (c+d x) \sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^3}\\ &=\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (3 b^2 e^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{2 a^4}-\frac{\left (2 b^2 e^2\right ) \int \sqrt{e \sin (c+d x)} \, dx}{a^4}+\frac{\left (3 b^3 e^2\right ) \int \frac{\sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^4}-\frac{\left (2 b \left (a^2-b^2\right ) e^2\right ) \int \frac{\sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^4}+\frac{\left (3 e^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{5 a^2 \sqrt{\sin (c+d x)}}\\ &=\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}+\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (3 b^4 e^3\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^5}+\frac{\left (3 b^4 e^3\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^5}+\frac{\left (b^2 \left (a^2-b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^5}-\frac{\left (b^2 \left (a^2-b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^5}-\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{2 a^3 d}+\frac{\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{a^3 d}-\frac{\left (3 b^2 e^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{2 a^4 \sqrt{\sin (c+d x)}}-\frac{\left (2 b^2 e^2 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{a^4 \sqrt{\sin (c+d x)}}\\ &=\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a^4 d \sqrt{\sin (c+d x)}}+\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}+\frac{\left (4 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^3 d}-\frac{\left (3 b^4 e^3 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^5 \sqrt{e \sin (c+d x)}}+\frac{\left (3 b^4 e^3 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^5 \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \left (a^2-b^2\right ) e^3 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^5 \sqrt{e \sin (c+d x)}}-\frac{\left (b^2 \left (a^2-b^2\right ) e^3 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^5 \sqrt{e \sin (c+d x)}}\\ &=\frac{3 b^4 e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 b^4 e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a^4 d \sqrt{\sin (c+d x)}}+\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a^4 d}-\frac{\left (3 b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a^4 d}-\frac{\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^4 d}+\frac{\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a^4 d}\\ &=-\frac{3 b^3 e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac{2 b \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{9/2} d}+\frac{3 b^3 e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}-\frac{2 b \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{9/2} d}+\frac{3 b^4 e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^5 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 b^4 e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 b^2 \left (a^2-b^2\right ) e^3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^5 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{6 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{5 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 e^2 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a^4 d \sqrt{\sin (c+d x)}}+\frac{4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 14.9118, size = 886, normalized size = 1.04 \[ \frac{(b+a \cos (c+d x))^2 \csc ^2(c+d x) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\frac{\sin (c+d x) b^2}{a^3 (b+a \cos (c+d x))}+\frac{4 \sin (c+d x) b}{3 a^3}-\frac{\sin (2 (c+d x))}{5 a^2}\right )}{d (a+b \sec (c+d x))^2}-\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\frac{\left (35 b^2-6 a^2\right ) \left (8 F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left (b^2-a^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )+\log \left (a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )\right )\right ) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \cos ^2(c+d x)}{12 a^{3/2} \left (a^2-b^2\right ) (b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac{28 a b \left (\frac{b F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac{3}{2}}(c+d x)}{3 \left (b^2-a^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )\right )}{\sqrt{a} \sqrt [4]{a^2-b^2}}\right ) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right )}{10 a^3 d (a+b \sec (c+d x))^2 \sin ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 7.117, size = 2540, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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