Optimal. Leaf size=356 \[ \frac{b \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{3/2} d \sqrt [4]{a^2-b^2}}-\frac{b \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{3/2} d \sqrt [4]{a^2-b^2}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^2 d \left (a-\sqrt{a^2-b^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^2 d \left (\sqrt{a^2-b^2}+a\right ) \sqrt{e \sin (c+d x)}}+\frac{2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}} \]
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Rubi [A] time = 0.76152, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.44, Rules used = {3872, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{b \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{3/2} d \sqrt [4]{a^2-b^2}}-\frac{b \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{a^2-b^2}}\right )}{a^{3/2} d \sqrt [4]{a^2-b^2}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^2 d \left (a-\sqrt{a^2-b^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{b^2 e \sqrt{\sin (c+d x)} \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{a^2 d \left (\sqrt{a^2-b^2}+a\right ) \sqrt{e \sin (c+d x)}}+\frac{2 E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{a d \sqrt{\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3872
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{\sqrt{e \sin (c+d x)}}{a+b \sec (c+d x)} \, dx &=-\int \frac{\cos (c+d x) \sqrt{e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx\\ &=\frac{\int \sqrt{e \sin (c+d x)} \, dx}{a}+\frac{b \int \frac{\sqrt{e \sin (c+d x)}}{-b-a \cos (c+d x)} \, dx}{a}\\ &=\frac{\left (b^2 e\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{2 a^2}-\frac{\left (b^2 e\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{2 a^2}+\frac{(b e) \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 )}{d}+\frac{\sqrt{e \sin (c+d x)} \int \sqrt{\sin (c+d x)} \, dx}{a \sqrt{\sin (c+d x)}}\\ &=\frac{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 d \sqrt{\sin (c+d x)}}+\frac{(2 b e) \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 )}{d}+\frac{\left (b^2 e \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 a^2 \sqrt{e \sin (c+d x)}}-\frac{\left (b^2 e \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 a^2 \sqrt{e \sin (c+d x)}}\\ &=-\frac{b^2 e \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^2 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^2 e \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^2 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{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 d \sqrt{\sin (c+d x)}}-\frac{(b e) \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 d}+\frac{(b e) \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 d}\\ &=\frac{b \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \sqrt [4]{a^2-b^2} d}-\frac{b \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \sqrt [4]{a^2-b^2} d}-\frac{b^2 e \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^2 \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}-\frac{b^2 e \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^2 \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{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 d \sqrt{\sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 20.0788, size = 351, normalized size = 0.99 \[ \frac{\sqrt{e \sin (c+d x)} \left (a \sqrt{\cos ^2(c+d x)}+b\right ) \left (8 a^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )+3 \sqrt{2} b \left (b^2-a^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right )+\log \left (\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+a \sin (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )\right )\right )}{12 a^{3/2} d \left (a^2-b^2\right ) \sqrt{\sin (c+d x)} (a \cos (c+d x)+b)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 2.525, size = 919, normalized size = 2.6 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e \sin \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a}\,{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{\sqrt{e \sin{\left (c + d x \right )}}}{a + b \sec{\left (c + d x \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{\sqrt{e \sin \left (d x + c\right )}}{b \sec \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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