Optimal. Leaf size=430 \[ \frac{\sqrt{a} b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{3/2} \left (a^2-b^2\right )^{5/4}}-\frac{\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{3/2} \left (a^2-b^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \left (a^2-b^2\right ) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (a-\sqrt{a^2-b^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (\sqrt{a^2-b^2}+a\right ) \sqrt{e \sin (c+d x)}} \]
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Rubi [A] time = 1.03685, antiderivative size = 430, 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, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{\sqrt{a} b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{3/2} \left (a^2-b^2\right )^{5/4}}-\frac{\sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{d e^{3/2} \left (a^2-b^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \left (a^2-b^2\right ) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (a-\sqrt{a^2-b^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (\sqrt{a^2-b^2}+a\right ) \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
Rule 2866
Rule 2867
Rule 2640
Rule 2639
Rule 2701
Rule 2807
Rule 2805
Rule 329
Rule 298
Rule 205
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \, dx &=-\int \frac{\cos (c+d x)}{(-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}} \, dx\\ &=\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}+\frac{2 \int \frac{\left (a b+\frac{1}{2} a^2 \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}\\ &=\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}-\frac{a \int \sqrt{e \sin (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}+\frac{(a b) \int \frac{\sqrt{e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2}\\ &=\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}+\frac{b^2 \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 ) e}-\frac{b^2 \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 ) e}+\frac{\left (a^2 b\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 )}{\left (a^2-b^2\right ) d e}-\frac{\left (a \sqrt{e \sin (c+d x)}\right ) \int \sqrt{\sin (c+d x)} \, dx}{\left (a^2-b^2\right ) e^2 \sqrt{\sin (c+d x)}}\\ &=\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}+\frac{\left (2 a^2 b\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 )}{\left (a^2-b^2\right ) d e}+\frac{\left (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 ) e \sqrt{e \sin (c+d x)}}-\frac{\left (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 ) e \sqrt{e \sin (c+d x)}}\\ &=\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}-\frac{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 ) \left (a-\sqrt{a^2-b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{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 ) \left (a+\sqrt{a^2-b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}-\frac{(a b) \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 ) d e}+\frac{(a b) \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 ) d e}\\ &=\frac{\sqrt{a} 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 )^{5/4} d e^{3/2}}-\frac{\sqrt{a} 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 )^{5/4} d e^{3/2}}+\frac{2 (b-a \cos (c+d x))}{\left (a^2-b^2\right ) d e \sqrt{e \sin (c+d x)}}-\frac{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 ) \left (a-\sqrt{a^2-b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{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 ) \left (a+\sqrt{a^2-b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{\left (a^2-b^2\right ) d e^2 \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 14.2727, size = 834, normalized size = 1.94 \[ -\frac{a (b+a \cos (c+d x)) \sec (c+d x) \left (\frac{\left (8 F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac{3}{2}}(c+d x) a^{5/2}+3 \sqrt{2} b \left (b^2-a^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )+\log \left (a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )\right )\right ) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \cos ^2(c+d x)}{12 \sqrt{a} \left (a^2-b^2\right ) (b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac{4 b \left (\frac{b F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac{3}{2}}(c+d x)}{3 \left (b^2-a^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )\right )}{\sqrt{a} \sqrt [4]{a^2-b^2}}\right ) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right ) \sin ^{\frac{3}{2}}(c+d x)}{(a-b) (a+b) d (a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}}-\frac{2 (b-a \cos (c+d x)) (b+a \cos (c+d x)) \tan (c+d x)}{\left (b^2-a^2\right ) d (a+b \sec (c+d x)) (e \sin (c+d x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 3.08, size = 1083, normalized size = 2.5 \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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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