Optimal. Leaf size=221 \[ \frac {283 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d} \]
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Rubi [A]
time = 0.35, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2844, 3056,
3062, 3047, 3102, 2830, 2728, 212} \begin {gather*} \frac {283 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {787 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{240 a^3 d}-\frac {157 \sin ^2(c+d x) \cos (c+d x)}{80 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {\sin ^4(c+d x) \cos (c+d x)}{4 d (a \sin (c+d x)+a)^{5/2}}+\frac {21 \sin ^3(c+d x) \cos (c+d x)}{16 a d (a \sin (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2830
Rule 2844
Rule 3047
Rule 3056
Rule 3062
Rule 3102
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}-\frac {\int \frac {\sin ^3(c+d x) \left (4 a-\frac {13}{2} a \sin (c+d x)\right )}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {\sin ^2(c+d x) \left (\frac {63 a^2}{2}-\frac {157}{4} a^2 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{8 a^4}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\int \frac {\sin (c+d x) \left (-\frac {157 a^3}{2}+\frac {787}{8} a^3 \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{20 a^5}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {\int \frac {-\frac {157}{2} a^3 \sin (c+d x)+\frac {787}{8} a^3 \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{20 a^5}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d}-\frac {\int \frac {\frac {787 a^4}{16}-\frac {1729}{8} a^4 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{30 a^6}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d}-\frac {283 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{32 a^2}\\ &=\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d}+\frac {283 \text {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 )}{16 a^2 d}\\ &=\frac {283 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}+\frac {\cos (c+d x) \sin ^4(c+d x)}{4 d (a+a \sin (c+d x))^{5/2}}+\frac {21 \cos (c+d x) \sin ^3(c+d x)}{16 a d (a+a \sin (c+d x))^{3/2}}-\frac {1729 \cos (c+d x)}{120 a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {157 \cos (c+d x) \sin ^2(c+d x)}{80 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {787 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{240 a^3 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.37, size = 221, normalized size = 1.00 \begin {gather*} -\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (2547 \cos \left (\frac {1}{2} (c+d x)\right )+3603 \cos \left (\frac {3}{2} (c+d x)\right )-872 \cos \left (\frac {5}{2} (c+d x)\right )+52 \cos \left (\frac {7}{2} (c+d x)\right )+12 \cos \left (\frac {9}{2} (c+d x)\right )-2547 \sin \left (\frac {1}{2} (c+d x)\right )+(8490+8490 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4+3603 \sin \left (\frac {3}{2} (c+d x)\right )+872 \sin \left (\frac {5}{2} (c+d x)\right )+52 \sin \left (\frac {7}{2} (c+d x)\right )-12 \sin \left (\frac {9}{2} (c+d x)\right )\right )}{480 d (a (1+\sin (c+d x)))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.48, size = 323, normalized size = 1.46
method | result | size |
default | \(-\frac {\left (\sin \left (d x +c \right ) \left (384 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}+640 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+7680 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}-8490 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right )+\left (-192 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}-320 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}-3840 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}+4245 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+384 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a}-470 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}}+9780 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {5}{2}}-8490 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{480 a^{\frac {11}{2}} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(323\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 381 vs.
\(2 (190) = 380\).
time = 0.36, size = 381, normalized size = 1.72 \begin {gather*} \frac {4245 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) - 4\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\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 ) + 4 \, {\left (96 \, \cos \left (d x + c\right )^{5} + 256 \, \cos \left (d x + c\right )^{4} - 1760 \, \cos \left (d x + c\right )^{3} + 2475 \, \cos \left (d x + c\right )^{2} - {\left (96 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 1920 \, \cos \left (d x + c\right )^{2} - 4395 \, \cos \left (d x + c\right ) - 60\right )} \sin \left (d x + c\right ) + 4335 \, \cos \left (d x + c\right ) - 60\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{960 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 241, normalized size = 1.09 \begin {gather*} -\frac {\frac {4245 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {4245 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {30 \, \sqrt {2} {\left (37 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {256 \, \sqrt {2} {\left (6 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, a^{\frac {25}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{960 \, d} \end {gather*}
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 \begin {gather*} \int \frac {{\sin \left (c+d\,x\right )}^5}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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