Optimal. Leaf size=294 \[ -\frac {7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {637 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {259 \sin (c+d x)}{192 a^2 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac {7 \sin (c+d x)}{16 a d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x)}{6 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
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Rubi [A] time = 0.82, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4222, 2765, 2977, 2983, 2982, 2782, 205, 2774, 216} \[ -\frac {259 \sin (c+d x)}{192 a^2 d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}-\frac {7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sin ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{a^{7/2} d}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {\sec (c+d x)} \sqrt {a \cos (c+d x)+a}}+\frac {637 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {7 \sin (c+d x)}{16 a d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x)}{6 d \sec ^{\frac {7}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 216
Rule 2765
Rule 2774
Rule 2782
Rule 2977
Rule 2982
Rule 2983
Rule 4222
Rubi steps
\begin {align*} \int \frac {1}{(a+a \cos (c+d x))^{7/2} \sec ^{\frac {9}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {9}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx\\ &=-\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}-\frac {\left (7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \, dx}{2 a^4}+\frac {\left (637 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}+\frac {\left (7 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {\left (637 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{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 ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}-\frac {\sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {7}{2}}(c+d x)}-\frac {7 \sin (c+d x)}{16 a d (a+a \cos (c+d x))^{5/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {259 \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2} \sec ^{\frac {3}{2}}(c+d x)}+\frac {189 \sin (c+d x)}{64 a^3 d \sqrt {a+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 3.48, size = 460, normalized size = 1.56 \[ \frac {e^{-\frac {1}{2} i (c+d x)} \sqrt {a (\cos (c+d x)+1)} \left (672 i \sqrt {2} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {1+e^{2 i (c+d x)}} \cos ^7\left (\frac {1}{2} (c+d x)\right ) \sinh ^{-1}\left (e^{i (c+d x)}\right )-\frac {1}{64} i e^{-4 i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \cos \left (\frac {1}{2} (c+d x)\right ) \left (\sqrt {2} \left (-1003 e^{i (c+d x)}-2169 e^{2 i (c+d x)}-2297 e^{3 i (c+d x)}-779 e^{4 i (c+d x)}+779 e^{5 i (c+d x)}+2297 e^{6 i (c+d x)}+2169 e^{7 i (c+d x)}+1003 e^{8 i (c+d x)}+96 e^{9 i (c+d x)}+672 e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (1+e^{i (c+d x)}\right )^6 \tanh ^{-1}\left (\sqrt {1+e^{2 i (c+d x)}}\right )-96\right )-1911 e^{i (c+d x)} \left (1+e^{i (c+d x)}\right )^6 \sqrt {1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )\right )\right )}{24 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.38, size = 289, normalized size = 0.98 \[ -\frac {1911 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - 2688 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (192 \, \cos \left (d x + c\right )^{4} + 1099 \, \cos \left (d x + c\right )^{3} + 1442 \, \cos \left (d x + c\right )^{2} + 567 \, \cos \left (d x + c\right )\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 472, normalized size = 1.61 \[ -\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (-1+\cos \left (d x +c \right )\right )^{8} \cos \left (d x +c \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 {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {11}{2}} \sin \left (d x +c \right )^{17} 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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \sec \left (d x + c\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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