Optimal. Leaf size=579 \[ \frac{3 a b (e \cos (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 d e \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{3 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac{5 a \sqrt{e} \left (a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt{b} d \left (b^2-a^2\right )^{13/4}}+\frac{5 a \sqrt{e} \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt{b} d \left (b^2-a^2\right )^{13/4}}+\frac{\left (11 a^2+4 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{8 d \left (a^2-b^2\right )^3 \sqrt{\cos (c+d x)}}+\frac{5 a^2 e \left (a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b d \left (a^2-b^2\right )^3 \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{5 a^2 e \left (a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b d \left (a^2-b^2\right )^3 \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}} \]
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Rubi [A] time = 1.5317, antiderivative size = 579, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {2694, 2864, 2867, 2640, 2639, 2701, 2807, 2805, 329, 298, 205, 208} \[ \frac{3 a b (e \cos (c+d x))^{3/2}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 d e \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac{b (e \cos (c+d x))^{3/2}}{3 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac{5 a \sqrt{e} \left (a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt{b} d \left (b^2-a^2\right )^{13/4}}+\frac{5 a \sqrt{e} \left (a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{16 \sqrt{b} d \left (b^2-a^2\right )^{13/4}}+\frac{\left (11 a^2+4 b^2\right ) E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{e \cos (c+d x)}}{8 d \left (a^2-b^2\right )^3 \sqrt{\cos (c+d x)}}+\frac{5 a^2 e \left (a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b d \left (a^2-b^2\right )^3 \left (b-\sqrt{b^2-a^2}\right ) \sqrt{e \cos (c+d x)}}+\frac{5 a^2 e \left (a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{16 b d \left (a^2-b^2\right )^3 \left (\sqrt{b^2-a^2}+b\right ) \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2694
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{\sqrt{e \cos (c+d x)}}{(a+b \sin (c+d x))^4} \, dx &=\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}-\frac{\int \frac{\sqrt{e \cos (c+d x)} \left (-3 a+\frac{3}{2} b \sin (c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{\int \frac{\sqrt{e \cos (c+d x)} \left (3 \left (2 a^2+b^2\right )-\frac{9}{4} a b \sin (c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac{\int \frac{\sqrt{e \cos (c+d x)} \left (-\frac{3}{4} a \left (8 a^2+7 b^2\right )-\frac{3}{8} b \left (11 a^2+4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}+\frac{\left (5 a \left (a^2+2 b^2\right )\right ) \int \frac{\sqrt{e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3}+\frac{\left (11 a^2+4 b^2\right ) \int \sqrt{e \cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3}\\ &=\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac{\left (5 a^2 \left (a^2+2 b^2\right ) e\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3}+\frac{\left (5 a^2 \left (a^2+2 b^2\right ) e\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{32 b \left (a^2-b^2\right )^3}+\frac{\left (5 a b \left (a^2+2 b^2\right ) e\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 )}{16 \left (a^2-b^2\right )^3 d}+\frac{\left (\left (11 a^2+4 b^2\right ) \sqrt{e \cos (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{16 \left (a^2-b^2\right )^3 \sqrt{\cos (c+d x)}}\\ &=\frac{\left (11 a^2+4 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt{\cos (c+d x)}}+\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}+\frac{\left (5 a b \left (a^2+2 b^2\right ) e\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 )}{8 \left (a^2-b^2\right )^3 d}-\frac{\left (5 a^2 \left (a^2+2 b^2\right ) e \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}{32 b \left (a^2-b^2\right )^3 \sqrt{e \cos (c+d x)}}+\frac{\left (5 a^2 \left (a^2+2 b^2\right ) e \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}{32 b \left (a^2-b^2\right )^3 \sqrt{e \cos (c+d x)}}\\ &=\frac{\left (11 a^2+4 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt{\cos (c+d x)}}+\frac{5 a^2 \left (a^2+2 b^2\right ) e \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 )}{16 b \left (a^2-b^2\right )^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{5 a^2 \left (a^2+2 b^2\right ) e \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 )}{16 b \left (a^2-b^2\right )^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}-\frac{\left (5 a \left (a^2+2 b^2\right ) e\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 )}{16 \left (a^2-b^2\right )^3 d}+\frac{\left (5 a \left (a^2+2 b^2\right ) e\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 )}{16 \left (a^2-b^2\right )^3 d}\\ &=-\frac{5 a \left (a^2+2 b^2\right ) \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 \sqrt{b} \left (-a^2+b^2\right )^{13/4} d}+\frac{5 a \left (a^2+2 b^2\right ) \sqrt{e} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{16 \sqrt{b} \left (-a^2+b^2\right )^{13/4} d}+\frac{\left (11 a^2+4 b^2\right ) \sqrt{e \cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{8 \left (a^2-b^2\right )^3 d \sqrt{\cos (c+d x)}}+\frac{5 a^2 \left (a^2+2 b^2\right ) e \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 )}{16 b \left (a^2-b^2\right )^3 \left (b-\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{5 a^2 \left (a^2+2 b^2\right ) e \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 )}{16 b \left (a^2-b^2\right )^3 \left (b+\sqrt{-a^2+b^2}\right ) d \sqrt{e \cos (c+d x)}}+\frac{b (e \cos (c+d x))^{3/2}}{3 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^3}+\frac{3 a b (e \cos (c+d x))^{3/2}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))^2}+\frac{b \left (11 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{8 \left (a^2-b^2\right )^3 d e (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 6.73063, size = 900, normalized size = 1.55 \[ \frac{\sqrt{e \cos (c+d x)} \left (\frac{3 a b \cos (c+d x)}{4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{b \cos (c+d x)}{3 \left (a^2-b^2\right ) (a+b \sin (c+d x))^3}-\frac{-4 \cos (c+d x) b^3-11 a^2 \cos (c+d x) b}{8 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}\right )}{d}+\frac{\sqrt{e \cos (c+d x)} \left (-\frac{\left (4 b^3+11 a^2 b\right ) \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (8 F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac{3}{2}}(c+d x) b^{5/2}+3 \sqrt{2} a \left (a^2-b^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (b \cos (c+d x)-\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )+\log \left (b \cos (c+d x)+\sqrt{2} \sqrt{b} \sqrt [4]{a^2-b^2} \sqrt{\cos (c+d x)}+\sqrt{a^2-b^2}\right )\right )\right ) \sin ^2(c+d x)}{12 b^{3/2} \left (b^2-a^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac{2 \left (16 a^3+14 b^2 a\right ) \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{a F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \cos ^{\frac{3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac{\left (\frac{1}{8}+\frac{i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{b} \sqrt{\cos (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (i b \cos (c+d x)-(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )+\log \left (i b \cos (c+d x)+(1+i) \sqrt{b} \sqrt [4]{b^2-a^2} \sqrt{\cos (c+d x)}+\sqrt{b^2-a^2}\right )\right )}{\sqrt{b} \sqrt [4]{b^2-a^2}}\right ) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right )}{16 (a-b)^3 (a+b)^3 d \sqrt{\cos (c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 88.013, size = 112960, normalized size = 195.1 \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{\sqrt{e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}\,{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{\sqrt{e \cos \left (d x + c\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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