Optimal. Leaf size=426 \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{d e^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{d e^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \left (a^2-b^2\right ) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \sin (c+d x)}} \]
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Rubi [A] time = 0.964438, antiderivative size = 426, 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 = {2696, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ -\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{d e^{3/2} \left (b^2-a^2\right )^{5/4}}+\frac{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \sin (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{d e^{3/2} \left (b^2-a^2\right )^{5/4}}-\frac{2 a E\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{e \sin (c+d x)}}{d e^2 \left (a^2-b^2\right ) \sqrt{\sin (c+d x)}}+\frac{2 (b-a \cos (c+d x))}{d e \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \sin (c+d x)}}-\frac{a b \sqrt{\sin (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right )}{d e \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2696
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 \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 (\frac{a^2}{2}+\frac{b^2}{2}+\frac{1}{2} a b \cos (c+d x)\right ) \sqrt{e \sin (c+d x)}}{a+b \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{b^2 \int \frac{\sqrt{e \sin (c+d x)}}{a+b \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 b) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right ) e}-\frac{(a b) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right ) e}+\frac{b^3 \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (a^2-b^2\right ) e^2+b^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 b^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac{\left (a b \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{2 \left (a^2-b^2\right ) e \sqrt{e \sin (c+d x)}}-\frac{\left (a b \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{-a^2+b^2}+b \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{a b \Pi \left (\frac{2 b}{b-\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 (b-\sqrt{-a^2+b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{a b \Pi \left (\frac{2 b}{b+\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 (b+\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{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e-b x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e+b x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{\left (a^2-b^2\right ) d e}\\ &=-\frac{b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} \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{b^{3/2} \tanh ^{-1}\left (\frac{\sqrt{b} \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{a b \Pi \left (\frac{2 b}{b-\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 (b-\sqrt{-a^2+b^2}\right ) d e \sqrt{e \sin (c+d x)}}-\frac{a b \Pi \left (\frac{2 b}{b+\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 (b+\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 = 14.6561, size = 791, normalized size = 1.86 \[ -\frac{2 \sin (c+d x) (a \cos (c+d x)-b)}{d \left (a^2-b^2\right ) (e \sin (c+d x))^{3/2}}-\frac{\sin ^{\frac{3}{2}}(c+d x) \left (\frac{a \cos ^2(c+d x) \left (a+b \sqrt{1-\sin ^2(c+d x)}\right ) \left (8 b^{5/2} \sin ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right )+\log \left (\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}+b \sin (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{12 \sqrt{b} \left (b^2-a^2\right ) \left (1-\sin ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{2 \left (a^2+b^2\right ) \cos (c+d x) \left (a+b \sqrt{1-\sin ^2(c+d x)}\right ) \left (\frac{a \sin ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\sin ^2(c+d x),\frac{b^2 \sin ^2(c+d x)}{b^2-a^2}\right )}{3 \left (a^2-b^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (-\log \left (-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right )+\log \left ((1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}+i b \sin (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (1+\frac{(1+i) \sqrt{b} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )\right )}{\sqrt{b} \sqrt [4]{b^2-a^2}}\right )}{\sqrt{1-\sin ^2(c+d x)} (a+b \cos (c+d x))}\right )}{d (a-b) (a+b) (e \sin (c+d x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 5.633, size = 1248, normalized size = 2.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (e \sin{\left (c + d x \right )}\right )^{\frac{3}{2}} \left (a + b \cos{\left (c + d x \right )}\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{1}{{\left (b \cos \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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