Optimal. Leaf size=222 \[ \frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}+\frac {472 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{315 a^3 d}-\frac {2 \sin ^4(c+d x) \cos (c+d x)}{9 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {38 \sin ^3(c+d x) \cos (c+d x)}{63 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {92 \sin ^2(c+d x) \cos (c+d x)}{105 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {2048 \cos (c+d x)}{315 a^2 d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 1.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, 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 206
Rule 2649
Rule 2751
Rule 2778
Rule 2880
Rule 2968
Rule 2983
Rule 3023
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*}
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Mathematica [C] time = 3.28, 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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fricas [A] time = 0.52, size = 280, normalized size = 1.26 \[ \frac {2 \, {\left (\frac {315 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - {\left (35 \, \cos \left (d x + c\right )^{5} + 130 \, \cos \left (d x + c\right )^{4} - 208 \, \cos \left (d x + c\right )^{3} - 634 \, \cos \left (d x + c\right )^{2} - {\left (35 \, \cos \left (d x + c\right )^{4} - 95 \, \cos \left (d x + c\right )^{3} - 303 \, \cos \left (d x + c\right )^{2} + 331 \, \cos \left (d x + c\right ) + 1292\right )} \sin \left (d x + c\right ) + 961 \, \cos \left (d x + c\right ) + 1292\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{315 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.91, size = 434, normalized size = 1.95 \[ \frac {8 \, {\left (\frac {\sqrt {2} {\left (315 \, a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 323 \, \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {-a} a^{3}} - \frac {315 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + \frac {{\left ({\left ({\left ({\left ({\left ({\left ({\left ({\left (\frac {197 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {315 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1044 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1470 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {2142 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {2142 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {1470 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {1044 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {315 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {197 \, a^{2}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {9}{2}}}\right )}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.28, size = 166, normalized size = 0.75 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (630 a^{\frac {9}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-35 \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}}+45 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}-63 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a^{2}-105 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{3}-630 a^{4} \sqrt {a -a \sin \left (d x +c \right )}\right )}{315 a^{7} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{3}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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