Optimal. Leaf size=254 \[ -\frac {7 \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {637 \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {189 \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {259 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac {7 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.75, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2765, 2977, 2983, 2982, 2782, 205, 2774, 216} \[ -\frac {259 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {7 \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {189 \sin (c+d x) \sqrt {\cos (c+d x)}}{64 a^3 d \sqrt {a \cos (c+d x)+a}}+\frac {637 \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {\sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{6 d (a \cos (c+d x)+a)^{7/2}}-\frac {7 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{16 a d (a \cos (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 216
Rule 2765
Rule 2774
Rule 2782
Rule 2977
Rule 2982
Rule 2983
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {7 a}{2}-7 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {\int \frac {\cos ^{\frac {3}{2}}(c+d x) \left (\frac {105 a^2}{4}-\frac {77}{2} a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}-\frac {\int \frac {\sqrt {\cos (c+d x)} \left (\frac {777 a^3}{8}-\frac {567}{4} a^3 \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\int \frac {-\frac {567 a^4}{8}+168 a^4 \cos (c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^7}\\ &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {7 \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4}+\frac {637 \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {7 \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^4 d}-\frac {637 \operatorname {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 )}{64 a^2 d}\\ &=-\frac {7 \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{a^{7/2} d}+\frac {637 \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 )}{64 \sqrt {2} a^{7/2} d}-\frac {\cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {7 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2}}-\frac {259 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {189 \sqrt {\cos (c+d x)} \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 6.75, size = 448, normalized size = 1.76 \[ \frac {\sqrt {\cos (c+d x)} \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\frac {16 \sin \left (\frac {c}{2}\right ) \cos \left (\frac {d x}{2}\right )}{d}+\frac {16 \cos \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right )}{d}+\frac {\sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}+\frac {\tan \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right )}{3 d}-\frac {15 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{4 d}-\frac {15 \tan \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right )}{4 d}+\frac {523 \sec \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{24 d}+\frac {523 \tan \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right )}{24 d}\right )}{(a (\cos (c+d x)+1))^{7/2}}+\frac {7 i e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (64 \sinh ^{-1}\left (e^{i (c+d x)}\right )+91 \sqrt {2} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )-64 \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )\right )}{8 \sqrt {2} d \sqrt {1+e^{2 i (c+d x)}} (a (\cos (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 3.85, size = 280, normalized size = 1.10 \[ -\frac {1911 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2 \, {\left (192 \, \cos \left (d x + c\right )^{3} + 1099 \, \cos \left (d x + c\right )^{2} + 1442 \, \cos \left (d x + c\right ) + 567\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2688 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 464, normalized size = 1.83 \[ \frac {\left (\cos ^{\frac {9}{2}}\left (d x +c \right )\right ) \left (-1+\cos \left (d x +c \right )\right )^{7} \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (192 \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 )+907 \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 )}}+1344 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right ) \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+1911 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+343 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2688 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}\, \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right )+3822 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-875 \cos \left (d x +c \right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1344 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}{\cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right )+1911 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-567 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {2}}{384 d \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \sin \left (d x +c \right )^{15} a^{4}} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{9/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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