Optimal. Leaf size=614 \[ -\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}-\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac{a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 d e^2 \left (a^2-b^2\right )^3 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{11 a b}{4 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 d e \left (a^2-b^2\right )^3 (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 1.72145, antiderivative size = 614, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2694, 2864, 2866, 2867, 2642, 2641, 2702, 2807, 2805, 329, 212, 208, 205} \[ -\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}-\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt{e} \sqrt [4]{b^2-a^2}}\right )}{8 d e^{5/2} \left (b^2-a^2\right )^{15/4}}+\frac{a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 d e^2 \left (a^2-b^2\right )^3 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b-\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (b-\sqrt{b^2-a^2}\right )\right ) \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{b+\sqrt{b^2-a^2}};\left .\frac{1}{2} (c+d x)\right |2\right )}{8 d e^2 \left (a^2-b^2\right )^3 \left (a^2-b \left (\sqrt{b^2-a^2}+b\right )\right ) \sqrt{e \cos (c+d x)}}+\frac{11 a b}{4 d e \left (a^2-b^2\right )^2 (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{b}{2 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 d e \left (a^2-b^2\right )^3 (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2694
Rule 2864
Rule 2866
Rule 2867
Rule 2642
Rule 2641
Rule 2702
Rule 2807
Rule 2805
Rule 329
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3} \, dx &=\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}-\frac{\int \frac{-2 a+\frac{7}{2} b \sin (c+d x)}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}+\frac{\int \frac{\frac{1}{2} \left (4 a^2+7 b^2\right )-\frac{55}{4} a b \sin (c+d x)}{(e \cos (c+d x))^{5/2} (a+b \sin (c+d x))} \, dx}{2 \left (a^2-b^2\right )^2}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}-\frac{\int \frac{\frac{1}{4} \left (-4 a^4+60 a^2 b^2+21 b^4\right )-\frac{1}{8} a b \left (8 a^2+69 b^2\right ) \sin (c+d x)}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{3 \left (a^2-b^2\right )^3 e^2}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}-\frac{\left (7 b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{8 \left (a^2-b^2\right )^3 e^2}+\frac{\left (a \left (8 a^2+69 b^2\right )\right ) \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}-\frac{\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}-\frac{\left (7 a b^2 \left (9 a^2+2 b^2\right )\right ) \int \frac{1}{\sqrt{e \cos (c+d x)} \left (\sqrt{-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{7/2} e^2}-\frac{\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d e}+\frac{\left (a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 \left (a^2-b^2\right )^3 e^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}-\frac{\left (7 b^3 \left (9 a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{4 \left (a^2-b^2\right )^3 d e}-\frac{\left (7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt{e \cos (c+d x)}}-\frac{\left (7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} e^2 \sqrt{e \cos (c+d x)}}\\ &=\frac{a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \cos (c+d x)}}+\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}-\frac{\left (7 b^3 \left (9 a^2+2 b^2\right )\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 \left (-a^2+b^2\right )^{7/2} d e^2}-\frac{\left (7 b^3 \left (9 a^2+2 b^2\right )\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 \left (-a^2+b^2\right )^{7/2} d e^2}\\ &=-\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}-\frac{7 b^{5/2} \left (9 a^2+2 b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt{e}}\right )}{8 \left (-a^2+b^2\right )^{15/4} d e^{5/2}}+\frac{a \left (8 a^2+69 b^2\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 \left (a^2-b^2\right )^3 d e^2 \sqrt{e \cos (c+d x)}}+\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} \left (b-\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \cos (c+d x)}}-\frac{7 a b^2 \left (9 a^2+2 b^2\right ) \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 \left (-a^2+b^2\right )^{7/2} \left (b+\sqrt{-a^2+b^2}\right ) d e^2 \sqrt{e \cos (c+d x)}}+\frac{b}{2 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}+\frac{11 a b}{4 \left (a^2-b^2\right )^2 d e (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}-\frac{7 b \left (9 a^2+2 b^2\right )-a \left (8 a^2+69 b^2\right ) \sin (c+d x)}{12 \left (a^2-b^2\right )^3 d e (e \cos (c+d x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 24.0007, size = 1308, normalized size = 2.13 \[ \frac{\left (-\frac{15 a b^3}{4 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac{b^3}{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac{2 \sec ^2(c+d x) \left (\sin (c+d x) a^3-3 b a^2+3 b^2 \sin (c+d x) a-b^3\right )}{3 \left (a^2-b^2\right )^3}\right ) \cos ^3(c+d x)}{d (e \cos (c+d x))^{5/2}}+\frac{\left (-\frac{2 \left (8 b a^3+69 b^3 a\right ) \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{5 b \left (a^2-b^2\right ) \sqrt{\cos (c+d x)} \sqrt{1-\cos ^2(c+d x)} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \cos ^2(c+d x)-5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \left (a^2+b^2 \left (\cos ^2(c+d x)-1\right )\right )}+\frac{a \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 )}{4 \sqrt{2} \sqrt{b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(c+d x)}{\left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}-\frac{2 \left (8 a^4-120 b^2 a^2-42 b^4\right ) \left (a+b \sqrt{1-\cos ^2(c+d x)}\right ) \left (\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) \sqrt{\cos (c+d x)}}{\sqrt{1-\cos ^2(c+d x)} \left (5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )-2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right ) b^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\cos ^2(c+d x),\frac{b^2 \cos ^2(c+d x)}{b^2-a^2}\right )\right ) \cos ^2(c+d x)\right ) \left (a^2+b^2 \left (\cos ^2(c+d x)-1\right )\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{b} \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 )}{\left (b^2-a^2\right )^{3/4}}\right ) \sin (c+d x)}{\sqrt{1-\cos ^2(c+d x)} (a+b \sin (c+d x))}\right ) \cos ^{\frac{5}{2}}(c+d x)}{24 (a-b)^3 (a+b)^3 d (e \cos (c+d x))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 62.991, size = 32645, normalized size = 53.2 \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(-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(-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{1}{\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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