Optimal. Leaf size=222 \[ -\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{38 \sin ^3(c+d x) \cos (c+d x)}{63 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{92 \sin ^2(c+d x) \cos (c+d x)}{105 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{472 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a^3 d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d} \]
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Rubi [A] time = 1.07987, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2880, 2778, 2983, 2968, 3023, 2751, 2649, 206, 3046} \[ -\frac{2 \sin ^4(c+d x) \cos (c+d x)}{9 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{38 \sin ^3(c+d x) \cos (c+d x)}{63 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{92 \sin ^2(c+d x) \cos (c+d x)}{105 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{472 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{315 a^3 d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{5/2} d} \]
Antiderivative was successfully verified.
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Rule 2880
Rule 2778
Rule 2983
Rule 2968
Rule 3023
Rule 2751
Rule 2649
Rule 206
Rule 3046
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \sin ^3(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{\int \frac{\sin ^3(c+d x) \left (1+\sin ^2(c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}-\frac{2 \int \frac{\sin ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{4 \cos (c+d x) \sin ^3(c+d x)}{7 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \int \frac{\sin ^3(c+d x) \left (\frac{17 a}{2}-\frac{1}{2} a \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{9 a^3}+\frac{2 \int \frac{\sin ^2(c+d x) (-6 a+a \sin (c+d x))}{\sqrt{a+a \sin (c+d x)}} \, dx}{7 a^3}\\ &=-\frac{4 \cos (c+d x) \sin ^2(c+d x)}{35 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{4 \int \frac{\sin ^2(c+d x) \left (-\frac{3 a^2}{2}+30 a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{63 a^4}+\frac{4 \int \frac{\sin (c+d x) \left (2 a^2-\frac{31}{2} a^2 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{8 \int \frac{\sin (c+d x) \left (60 a^3-\frac{75}{4} a^3 \sin (c+d x)\right )}{\sqrt{a+a \sin (c+d x)}} \, dx}{315 a^5}+\frac{4 \int \frac{2 a^2 \sin (c+d x)-\frac{31}{2} a^2 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{35 a^4}\\ &=-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{124 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{105 a^3 d}+\frac{8 \int \frac{60 a^3 \sin (c+d x)-\frac{75}{4} a^3 \sin ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{315 a^5}+\frac{8 \int \frac{-\frac{31 a^3}{4}+\frac{37}{2} a^3 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{105 a^5}\\ &=-\frac{296 \cos (c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{472 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 a^3 d}+\frac{16 \int \frac{-\frac{75 a^4}{8}+\frac{435}{4} a^4 \sin (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{945 a^6}-\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{472 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 a^3 d}-\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{a^2 d}\\ &=\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{5/2} d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{472 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 a^3 d}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{a^2 d}\\ &=\frac{4 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{5/2} d}-\frac{2048 \cos (c+d x)}{315 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{92 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{38 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{2 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{472 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{315 a^3 d}\\ \end{align*}
Mathematica [C] time = 3.23577, size = 225, normalized size = 1.01 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (16380 \sin \left (\frac{1}{2} (c+d x)\right )+3150 \sin \left (\frac{3}{2} (c+d x)\right )-882 \sin \left (\frac{5}{2} (c+d x)\right )-225 \sin \left (\frac{7}{2} (c+d x)\right )+35 \sin \left (\frac{9}{2} (c+d x)\right )-16380 \cos \left (\frac{1}{2} (c+d x)\right )+3150 \cos \left (\frac{3}{2} (c+d x)\right )+882 \cos \left (\frac{5}{2} (c+d x)\right )-225 \cos \left (\frac{7}{2} (c+d x)\right )-35 \cos \left (\frac{9}{2} (c+d x)\right )+(20160+20160 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \sec \left (\frac{d x}{4}\right ) \left (\cos \left (\frac{1}{4} (2 c+d x)\right )-\sin \left (\frac{1}{4} (2 c+d x)\right )\right )\right )\right )}{2520 a^3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.003, size = 166, normalized size = 0.8 \begin{align*}{\frac{2+2\,\sin \left ( dx+c \right ) }{315\,{a}^{7}\cos \left ( dx+c \right ) d}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 630\,{a}^{9/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) -35\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{9/2}+45\,a \left ( a-a\sin \left ( dx+c \right ) \right ) ^{7/2}-63\,{a}^{2} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}-105\,{a}^{3} \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}-630\,{a}^{4}\sqrt{a-a\sin \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \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{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18415, size = 786, normalized size = 3.54 \begin{align*} \frac{2 \,{\left (\frac{315 \, \sqrt{2}{\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac{2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt{a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt{a}} -{\left (35 \, \cos \left (d x + c\right )^{5} + 130 \, \cos \left (d x + c\right )^{4} - 208 \, \cos \left (d x + c\right )^{3} - 634 \, \cos \left (d x + c\right )^{2} -{\left (35 \, \cos \left (d x + c\right )^{4} - 95 \, \cos \left (d x + c\right )^{3} - 303 \, \cos \left (d x + c\right )^{2} + 331 \, \cos \left (d x + c\right ) + 1292\right )} \sin \left (d x + c\right ) + 961 \, \cos \left (d x + c\right ) + 1292\right )} \sqrt{a \sin \left (d x + c\right ) + a}\right )}}{315 \,{\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \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 [B] time = 2.42919, size = 562, normalized size = 2.53 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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