Optimal. Leaf size=505 \[ \frac{3 e^{5/2} \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{5/2} d \left (b^2-a^2\right )^{5/4}}-\frac{3 e^{5/2} \left (a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{5/2} d \left (b^2-a^2\right )^{5/4}}+\frac{3 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{4 b^2 d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)}}-\frac{3 a e^3 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 d \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}-\frac{3 a e^3 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 d \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 1.10473, antiderivative size = 505, 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 = {2693, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{3 e^{5/2} \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{5/2} d \left (b^2-a^2\right )^{5/4}}-\frac{3 e^{5/2} \left (a^2-2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 b^{5/2} d \left (b^2-a^2\right )^{5/4}}+\frac{3 a e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{4 b^2 d \left (a^2-b^2\right ) \sqrt{\cos (c+d x)}}-\frac{3 a e^3 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 d \left (a^2-b^2\right ) \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}-\frac{3 a e^3 \left (a^2-2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 d \left (a^2-b^2\right ) \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2693
Rule 2864
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{(e \cos (c+d x))^{5/2}}{(a+b \sin (c+d x))^3} \, dx &=-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}-\frac{\left (3 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)} \sin (c+d x)}{(a+b \sin (c+d x))^2} \, dx}{4 b}\\ &=-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (3 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)} \left (b+\frac{1}{2} a \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 b \left (a^2-b^2\right )}\\ &=-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (3 a e^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )}-\frac{\left (3 \left (a^2-2 b^2\right ) e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right )}\\ &=-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (3 a \left (a^2-2 b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b^3 \left (a^2-b^2\right )}-\frac{\left (3 a \left (a^2-2 b^2\right ) e^3\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b^3 \left (a^2-b^2\right )}-\frac{\left (3 \left (a^2-2 b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{8 b \left (a^2-b^2\right ) d}+\frac{\left (3 a e^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 b^2 \left (a^2-b^2\right ) \sqrt{\cos (c+d x)}}\\ &=\frac{3 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac{\left (3 \left (a^2-2 b^2\right ) e^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 \cos (c+d x)}\right )}{4 b \left (a^2-b^2\right ) d}+\frac{\left (3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 b^3 \left (a^2-b^2\right ) \sqrt{e \cos (c+d x)}}-\frac{\left (3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 b^3 \left (a^2-b^2\right ) \sqrt{e \cos (c+d x)}}\\ &=\frac{3 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 \left (a^2-b^2\right ) \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 \left (a^2-b^2\right ) \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{\left (3 \left (a^2-2 b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e-b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{8 b^2 \left (a^2-b^2\right ) d}-\frac{\left (3 \left (a^2-2 b^2\right ) e^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a^2+b^2} e+b x^2} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{8 b^2 \left (a^2-b^2\right ) d}\\ &=\frac{3 \left (a^2-2 b^2\right ) e^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{8 b^{5/2} \left (-a^2+b^2\right )^{5/4} d}-\frac{3 \left (a^2-2 b^2\right ) e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{8 b^{5/2} \left (-a^2+b^2\right )^{5/4} d}+\frac{3 a e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 b^2 \left (a^2-b^2\right ) d \sqrt{\cos (c+d x)}}-\frac{3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 \left (a^2-b^2\right ) \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{3 a \left (a^2-2 b^2\right ) e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{-a^2+b^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 b^3 \left (a^2-b^2\right ) \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{2 b d (a+b \sin (c+d x))^2}+\frac{3 a e (e \cos (c+d x))^{3/2}}{4 b \left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 24.1576, size = 831, normalized size = 1.65 \[ \frac{\sec ^2(c+d x) \left (-\frac{3 a \cos (c+d x)}{4 b \left (b^2-a^2\right ) (a+b \sin (c+d x))}-\frac{\cos (c+d x)}{2 b (a+b \sin (c+d x))^2}\right ) (e \cos (c+d x))^{5/2}}{d}+\frac{3 \left (-\frac{a \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac{4 b \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{a F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac{3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )\right )}{\sqrt{b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right ) (e \cos (c+d x))^{5/2}}{8 (a-b) b (a+b) d \cos ^{\frac{5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 26.309, size = 63272, normalized size = 125.3 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{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(-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 \cos \left (d x + c\right )\right )^{\frac{5}{2}}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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