Optimal. Leaf size=260 \[ -\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}-\frac {1048 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{693 a^3 d}-\frac {2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {4496 \cos (c+d x)}{693 a^2 d \sqrt {a \sin (c+d x)+a}} \]
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Rubi [A] time = 1.36, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 18, 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} \[ -\frac {2 \sin ^5(c+d x) \cos (c+d x)}{11 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {46 \sin ^4(c+d x) \cos (c+d x)}{99 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {424 \sin ^3(c+d x) \cos (c+d x)}{693 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {200 \sin ^2(c+d x) \cos (c+d x)}{231 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {1048 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{693 a^3 d}+\frac {4496 \cos (c+d x)}{693 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d} \]
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 ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\int \frac {\sin ^4(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 ^5(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac {4 \cos (c+d x) \sin ^4(c+d x)}{9 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {2 \int \frac {\sin ^4(c+d x) \left (\frac {21 a}{2}-\frac {1}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{11 a^3}+\frac {2 \int \frac {\sin ^3(c+d x) (-8 a+a \sin (c+d x))}{\sqrt {a+a \sin (c+d x)}} \, dx}{9 a^3}\\ &=-\frac {4 \cos (c+d x) \sin ^3(c+d x)}{63 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {\sin ^3(c+d x) \left (-2 a^2+\frac {95}{2} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{99 a^4}+\frac {4 \int \frac {\sin ^2(c+d x) \left (3 a^2-\frac {57}{2} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{63 a^4}\\ &=\frac {76 \cos (c+d x) \sin ^2(c+d x)}{105 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {8 \int \frac {\sin ^2(c+d x) \left (\frac {285 a^3}{2}-\frac {123}{4} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{693 a^5}+\frac {8 \int \frac {\sin (c+d x) \left (-57 a^3+\frac {87}{4} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{315 a^5}\\ &=\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {16 \int \frac {\sin (c+d x) \left (-\frac {123 a^4}{2}+\frac {2973}{8} a^4 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{3465 a^6}+\frac {8 \int \frac {-57 a^3 \sin (c+d x)+\frac {87}{4} a^3 \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{315 a^5}\\ &=\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {116 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{315 a^3 d}+\frac {16 \int \frac {-\frac {123}{2} a^4 \sin (c+d x)+\frac {2973}{8} a^4 \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{3465 a^6}+\frac {16 \int \frac {\frac {87 a^4}{8}-\frac {429}{4} a^4 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{945 a^6}\\ &=\frac {1144 \cos (c+d x)}{315 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {1048 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 a^3 d}+\frac {32 \int \frac {\frac {2973 a^5}{16}-\frac {3711}{8} a^5 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{10395 a^7}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac {4496 \cos (c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {1048 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 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 {4496 \cos (c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {1048 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 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 {4496 \cos (c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {200 \cos (c+d x) \sin ^2(c+d x)}{231 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {424 \cos (c+d x) \sin ^3(c+d x)}{693 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {46 \cos (c+d x) \sin ^4(c+d x)}{99 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {2 \cos (c+d x) \sin ^5(c+d x)}{11 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {1048 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{693 a^3 d}\\ \end {align*}
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Mathematica [C] time = 1.08, size = 224, normalized size = 0.86 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (-73458 \sin \left (\frac {1}{2} (c+d x)\right )-15246 \sin \left (\frac {3}{2} (c+d x)\right )+4851 \sin \left (\frac {5}{2} (c+d x)\right )+1485 \sin \left (\frac {7}{2} (c+d x)\right )-385 \sin \left (\frac {9}{2} (c+d x)\right )-63 \sin \left (\frac {11}{2} (c+d x)\right )+73458 \cos \left (\frac {1}{2} (c+d x)\right )-15246 \cos \left (\frac {3}{2} (c+d x)\right )-4851 \cos \left (\frac {5}{2} (c+d x)\right )+1485 \cos \left (\frac {7}{2} (c+d x)\right )+385 \cos \left (\frac {9}{2} (c+d x)\right )-63 \cos \left (\frac {11}{2} (c+d x)\right )+(88704+88704 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac {1}{4} (c+d x)\right )-1\right )\right )\right )}{11088 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 299, normalized size = 1.15 \[ \frac {2 \, {\left (\frac {693 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - {\left (63 \, \cos \left (d x + c\right )^{6} - 161 \, \cos \left (d x + c\right )^{5} - 562 \, \cos \left (d x + c\right )^{4} + 622 \, \cos \left (d x + c\right )^{3} + 1759 \, \cos \left (d x + c\right )^{2} + {\left (63 \, \cos \left (d x + c\right )^{5} + 224 \, \cos \left (d x + c\right )^{4} - 338 \, \cos \left (d x + c\right )^{3} - 960 \, \cos \left (d x + c\right )^{2} + 799 \, \cos \left (d x + c\right ) + 2984\right )} \sin \left (d x + c\right ) - 2185 \, \cos \left (d x + c\right ) - 2984\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{693 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 504, normalized size = 1.94 \[ -\frac {8 \, {\left (\frac {\sqrt {2} {\left (693 \, a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 746 \, \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 {693 \, \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 {\frac {431 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {693 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {2717 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {3927 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {7326 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {8778 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {8778 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {7326 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {3927 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - {\left (\frac {2717 \, a^{3}}{\mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} + {\left (\frac {431 \, a^{3} \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 {693 \, a^{3}}{\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 )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {11}{2}}}\right )}}{693 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.23, size = 166, normalized size = 0.64 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-1386 a^{\frac {11}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )+63 \left (a -a \sin \left (d x +c \right )\right )^{\frac {11}{2}}-154 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {9}{2}}+198 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}} a^{2}+231 a^{4} \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}+1386 a^{5} \sqrt {a -a \sin \left (d x +c \right )}\right )}{693 a^{8} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^4}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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