Optimal. Leaf size=452 \[ \frac{2 a \sqrt{\sin (c+d x)} \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right )}{3 d e^2 \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}-\frac{a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{5/2} \left (a^2-b^2\right )^{7/4}}-\frac{a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{5/2} \left (a^2-b^2\right )^{7/4}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e^2 \left (a^2-b^2\right ) \left (-a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e^2 \left (a^2-b^2\right ) \left (a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 1.05037, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3872, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ -\frac{a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{5/2} \left (a^2-b^2\right )^{7/4}}-\frac{a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{5/2} \left (a^2-b^2\right )^{7/4}}+\frac{2 a \sqrt{\sin (c+d x)} F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{3 d e^2 \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e^2 \left (a^2-b^2\right ) \left (-a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{a b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e^2 \left (a^2-b^2\right ) \left (a \sqrt{a^2-b^2}+a^2-b^2\right ) \sqrt{e \sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{3 d e \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2866
Rule 2867
Rule 2642
Rule 2641
Rule 2702
Rule 2807
Rule 2805
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}} \, dx &=-\int \frac{\cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{5/2}} \, dx\\ &=\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{2 \int \frac{a b-\frac{1}{2} a^2 \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right ) e^2}\\ &=\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{a \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right ) e^2}+\frac{(a b) \int \frac{1}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{\left (a^2-b^2\right ) e^2}\\ &=\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{\left (a b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^{3/2} e^2}+\frac{\left (a b^2\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right )^{3/2} e^2}+\frac{\left (a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{\left (a^2-b^2\right ) d e}+\frac{\left (a \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right ) e^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right ) d e^2 \sqrt{e \sin (c+d x)}}+\frac{\left (2 a^2 b\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac{\left (a b^2 \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}{2 \left (a^2-b^2\right )^{3/2} e^2 \sqrt{e \sin (c+d x)}}+\frac{\left (a b^2 \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}{2 \left (a^2-b^2\right )^{3/2} e^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right ) d e^2 \sqrt{e \sin (c+d x)}}-\frac{a b^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a-\sqrt{a^2-b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}+\frac{a b^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a+\sqrt{a^2-b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}-\frac{\left (a^2 b\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 )}{\left (a^2-b^2\right )^{3/2} d e^2}-\frac{\left (a^2 b\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 )}{\left (a^2-b^2\right )^{3/2} d e^2}\\ &=-\frac{a^{3/2} b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}-\frac{a^{3/2} b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{\left (a^2-b^2\right )^{7/4} d e^{5/2}}+\frac{2 (b-a \cos (c+d x))}{3 \left (a^2-b^2\right ) d e (e \sin (c+d x))^{3/2}}+\frac{2 a F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{3 \left (a^2-b^2\right ) d e^2 \sqrt{e \sin (c+d x)}}-\frac{a b^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a-\sqrt{a^2-b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}+\frac{a b^2 \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)}}{\left (a^2-b^2\right )^{3/2} \left (a+\sqrt{a^2-b^2}\right ) d e^2 \sqrt{e \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 12.2987, size = 1233, normalized size = 2.73 \[ -\frac{a (b+a \cos (c+d x)) \sec (c+d x) \left (\frac{4 b \cos (c+d x) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \left (\frac{5 b \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left (2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{a} \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 )}{\left (a^2-b^2\right )^{3/4}}\right )}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}-\frac{2 a \cos ^2(c+d x) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \left (\frac{b \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 )}{4 \sqrt{2} \sqrt{a} \left (b^2-a^2\right )^{3/4}}-\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}\right )}{(b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}\right ) \sin ^{\frac{5}{2}}(c+d x)}{3 (a-b) (a+b) d (a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}}-\frac{2 (b-a \cos (c+d x)) (b+a \cos (c+d x)) \tan (c+d x)}{3 \left (b^2-a^2\right ) d (a+b \sec (c+d x)) (e \sin (c+d x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 4.983, size = 681, normalized size = 1.5 \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{1}{{\left (b \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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