Optimal. Leaf size=1070 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.7946, antiderivative size = 1070, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 16, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.64, Rules used = {3872, 2912, 2635, 2640, 2639, 2693, 2865, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2695} \[ -\frac{2 b^2 \left (a^2-b^2\right )^2 \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)} e^5}{a^7 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left (a^2-b^2\right ) \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)} e^5}{2 a^7 \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 )^2 \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)} e^5}{a^7 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left (a^2-b^2\right ) \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)} e^5}{2 a^7 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b \left (a^2-b^2\right )^{7/4} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{9/2}}{a^{13/2} d}-\frac{7 b^3 \left (a^2-b^2\right )^{3/4} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac{2 b \left (a^2-b^2\right )^{7/4} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{9/2}}{a^{13/2} d}+\frac{7 b^3 \left (a^2-b^2\right )^{3/4} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac{7 b^2 \left (3 a^2-5 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left (8 a^2-5 b^2\right ) E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)} e^4}{5 a^6 d \sqrt{\sin (c+d x)}}+\frac{14 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)} e^4}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac{7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac{4 b \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac{2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac{4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac{b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3872
Rule 2912
Rule 2635
Rule 2640
Rule 2639
Rule 2693
Rule 2865
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))^{9/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac{\cos ^2(c+d x) (e \sin (c+d x))^{9/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac{(e \sin (c+d x))^{9/2}}{a^2}+\frac{b^2 (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))^2}-\frac{2 b (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac{\int (e \sin (c+d x))^{9/2} \, dx}{a^2}-\frac{(2 b) \int \frac{(e \sin (c+d x))^{9/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac{b^2 \int \frac{(e \sin (c+d x))^{9/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (7 e^2\right ) \int (e \sin (c+d x))^{5/2} \, dx}{9 a^2}+\frac{\left (2 b e^2\right ) \int \frac{(-a-b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac{\left (7 b^2 e^2\right ) \int \frac{\cos (c+d x) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{2 a^3}\\ &=-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (7 e^4\right ) \int \sqrt{e \sin (c+d x)} \, dx}{15 a^2}+\frac{\left (4 b e^4\right ) \int \frac{\left (-\frac{1}{2} a \left (5 a^2-2 b^2\right )-\frac{1}{2} b \left (8 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5}-\frac{\left (7 b^2 e^4\right ) \int \frac{\left (-a b+\frac{1}{2} \left (3 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5}\\ &=-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4\right ) \int \sqrt{e \sin (c+d x)} \, dx}{10 a^6}-\frac{\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4\right ) \int \sqrt{e \sin (c+d x)} \, dx}{5 a^6}+\frac{\left (7 b^3 \left (a^2-b^2\right ) e^4\right ) \int \frac{\sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^6}-\frac{\left (2 b \left (a^2-b^2\right )^2 e^4\right ) \int \frac{\sqrt{e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^6}+\frac{\left (7 e^4 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{15 a^2 \sqrt{\sin (c+d x)}}\\ &=\frac{14 e^4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (7 b^4 \left (a^2-b^2\right ) e^5\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^7}+\frac{\left (7 b^4 \left (a^2-b^2\right ) e^5\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^7}+\frac{\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^7}-\frac{\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^7}-\frac{\left (7 b^3 \left (a^2-b^2\right ) e^5\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^5 d}+\frac{\left (2 b \left (a^2-b^2\right )^2 e^5\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^5 d}-\frac{\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{10 a^6 \sqrt{\sin (c+d x)}}-\frac{\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4 \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{5 a^6 \sqrt{\sin (c+d x)}}\\ &=\frac{14 e^4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 \left (3 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left (8 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac{\left (7 b^3 \left (a^2-b^2\right ) e^5\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^5 d}+\frac{\left (4 b \left (a^2-b^2\right )^2 e^5\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^5 d}-\frac{\left (7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \sqrt{e \sin (c+d x)}}+\frac{\left (7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \left (a^2-b^2\right )^2 e^5 \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^7 \sqrt{e \sin (c+d x)}}-\frac{\left (b^2 \left (a^2-b^2\right )^2 e^5 \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^7 \sqrt{e \sin (c+d x)}}\\ &=\frac{7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \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 )^2 e^5 \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^7 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \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 )^2 e^5 \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^7 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{14 e^4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 \left (3 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left (8 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac{\left (7 b^3 \left (a^2-b^2\right ) e^5\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^6 d}-\frac{\left (7 b^3 \left (a^2-b^2\right ) e^5\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^6 d}-\frac{\left (2 b \left (a^2-b^2\right )^2 e^5\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^6 d}+\frac{\left (2 b \left (a^2-b^2\right )^2 e^5\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^6 d}\\ &=-\frac{7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{13/2} d}+\frac{2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{13/2} d}+\frac{7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{13/2} d}-\frac{2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{13/2} d}+\frac{7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \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 )^2 e^5 \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^7 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{7 b^4 \left (a^2-b^2\right ) e^5 \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^7 \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 )^2 e^5 \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^7 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{14 e^4 E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{15 a^2 d \sqrt{\sin (c+d x)}}-\frac{7 b^2 \left (3 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{4 b^2 \left (8 a^2-5 b^2\right ) e^4 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^6 d \sqrt{\sin (c+d x)}}-\frac{14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac{7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac{4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac{2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac{b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}\\ \end{align*}
Mathematica [C] time = 15.3528, size = 974, normalized size = 0.91 \[ \frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left (\frac{\left (14 a^4-159 b^2 a^2+165 b^4\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{2 \left (66 a b^3-46 a^3 b\right ) \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 ) (e \sin (c+d x))^{9/2}}{30 a^5 d (a+b \sec (c+d x))^2 \sin ^{\frac{9}{2}}(c+d x)}+\frac{(b+a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) \left (-\frac{b \left (56 b^2-37 a^2\right ) \sin (c+d x)}{21 a^5}+\frac{a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)}{a^5 (b+a \cos (c+d x))}-\frac{\left (19 a^2-54 b^2\right ) \sin (2 (c+d x))}{90 a^4}-\frac{b \sin (3 (c+d x))}{7 a^3}+\frac{\sin (4 (c+d x))}{36 a^2}\right ) (e \sin (c+d x))^{9/2}}{d (a+b \sec (c+d x))^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 9.805, size = 3808, normalized size = 3.6 \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(-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{9}{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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