Optimal. Leaf size=217 \[ \frac {45 \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{1024 \sqrt {2} a^{9/2} d}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {73 \sqrt {\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.37, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2844, 3056,
3057, 12, 2861, 211} \begin {gather*} \frac {45 \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{1024 \sqrt {2} a^{9/2} d}+\frac {73 \sin (c+d x) \sqrt {\cos (c+d x)}}{1024 a^3 d (a \cos (c+d x)+a)^{3/2}}+\frac {33 \sin (c+d x) \sqrt {\cos (c+d x)}}{256 a^2 d (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{8 d (a \cos (c+d x)+a)^{9/2}}-\frac {5 \sin (c+d x) \sqrt {\cos (c+d x)}}{32 a d (a \cos (c+d x)+a)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2844
Rule 2861
Rule 3056
Rule 3057
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {3 a}{2}-6 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{7/2}} \, dx}{8 a^2}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}-\frac {\int \frac {\frac {15 a^2}{4}-21 a^2 \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}} \, dx}{48 a^4}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}-\frac {\int \frac {\frac {21 a^3}{8}-\frac {99}{4} a^3 \cos (c+d x)}{\sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}} \, dx}{192 a^6}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {73 \sqrt {\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac {\int -\frac {135 a^4}{16 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{384 a^8}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {73 \sqrt {\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}+\frac {45 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{2048 a^4}\\ &=-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {73 \sqrt {\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}-\frac {45 \text {Subst}\left (\int \frac {1}{2 a^2+a x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{1024 a^3 d}\\ &=\frac {45 \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right )}{1024 \sqrt {2} a^{9/2} d}-\frac {\cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2}}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{32 a d (a+a \cos (c+d x))^{7/2}}+\frac {33 \sqrt {\cos (c+d x)} \sin (c+d x)}{256 a^2 d (a+a \cos (c+d x))^{5/2}}+\frac {73 \sqrt {\cos (c+d x)} \sin (c+d x)}{1024 a^3 d (a+a \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.22, size = 158, normalized size = 0.73 \begin {gather*} \frac {\sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (5760 \tanh ^{-1}\left (\sqrt {-\sec (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right )}\right ) \cos ^8\left (\frac {1}{2} (c+d x)\right )+(999+2466 \cos (c+d x)+1072 \cos (2 (c+d x))+702 \cos (3 (c+d x))+73 \cos (4 (c+d x))) \sqrt {2-2 \sec (c+d x)}\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{65536 \sqrt {2} a^4 d \sqrt {-1+\cos (c+d x)} \sqrt {a (1+\cos (c+d x))}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.22, size = 346, normalized size = 1.59
method | result | size |
default | \(-\frac {\left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{6} \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (73 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{4}\left (d x +c \right )\right )+45 \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right )+278 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+135 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-156 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+135 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-150 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+45 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-45 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {2}}{2048 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )^{13} a^{5}}\) | \(346\) |
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 [A]
time = 0.45, size = 248, normalized size = 1.14 \begin {gather*} \frac {45 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} + 10 \, \cos \left (d x + c\right )^{2} + 5 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) + 2 \, {\left (73 \, \cos \left (d x + c\right )^{3} + 351 \, \cos \left (d x + c\right )^{2} + 195 \, \cos \left (d x + c\right ) + 45\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2048 \, {\left (a^{5} d \cos \left (d x + c\right )^{5} + 5 \, a^{5} d \cos \left (d x + c\right )^{4} + 10 \, a^{5} d \cos \left (d x + c\right )^{3} + 10 \, a^{5} d \cos \left (d x + c\right )^{2} + 5 \, a^{5} d \cos \left (d x + c\right ) + a^{5} d\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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{5/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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