Optimal. Leaf size=390 \[ \frac{3 a e^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{5/2} d \sqrt [4]{b^2-a^2}}-\frac{3 a e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{5/2} d \sqrt [4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^3 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^3 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.819143, antiderivative size = 390, 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 = {2693, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{3 a e^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{5/2} d \sqrt [4]{b^2-a^2}}-\frac{3 a e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{2 b^{5/2} d \sqrt [4]{b^2-a^2}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^3 d \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{3 a^2 e^3 \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{2 b^3 d \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{3 e^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{b^2 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2693
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))^2} \, dx &=-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{\left (3 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)} \sin (c+d x)}{a+b \sin (c+d x)} \, dx}{2 b}\\ &=-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{\left (3 e^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{2 b^2}+\frac{\left (3 a e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{2 b^2}\\ &=-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{\left (3 a^2 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}{4 b^3}+\frac{\left (3 a^2 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}{4 b^3}+\frac{\left (3 a 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 )}{2 b d}-\frac{\left (3 e^2 \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 b^2 \sqrt{\cos (c+d x)}}\\ &=-\frac{3 e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}+\frac{\left (3 a 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 )}{b d}-\frac{\left (3 a^2 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}{4 b^3 \sqrt{e \cos (c+d x)}}+\frac{\left (3 a^2 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}{4 b^3 \sqrt{e \cos (c+d x)}}\\ &=-\frac{3 e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}+\frac{3 a^2 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 )}{2 b^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{3 a^2 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 )}{2 b^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}-\frac{\left (3 a 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 )}{2 b^2 d}+\frac{\left (3 a 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 )}{2 b^2 d}\\ &=\frac{3 a e^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 b^{5/2} \sqrt [4]{-a^2+b^2} d}-\frac{3 a e^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{2 b^{5/2} \sqrt [4]{-a^2+b^2} d}-\frac{3 e^2 \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt{\cos (c+d x)}}+\frac{3 a^2 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 )}{2 b^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{3 a^2 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 )}{2 b^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}-\frac{e (e \cos (c+d x))^{3/2}}{b d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 38.8137, size = 371, normalized size = 0.95 \[ \frac{(e \cos (c+d x))^{5/2} \left (-\frac{\left (a+b \sqrt{\sin ^2(c+d x)}\right ) \left (8 b^{5/2} \cos ^{\frac{3}{2}}(c+d x) F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (-\log \left (-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right )+\log \left (\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}+b \cos (c+d x)\right )+2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )\right )\right )}{a^2-b^2}-8 b^{3/2} \cos ^{\frac{3}{2}}(c+d x)\right )}{8 b^{5/2} d \cos ^{\frac{5}{2}}(c+d x) (a+b \sin (c+d x))} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 6.415, size = 13221, normalized size = 33.9 \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 )}^{2}}\,{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(-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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