Optimal. Leaf size=237 \[ \frac {35 \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{1024 \sqrt {2} a^{9/2} d}-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 0.42, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4307, 2844,
3056, 3057, 12, 2861, 211} \begin {gather*} \frac {35 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \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 {853 \sin (c+d x)}{3072 a^3 d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {187 \sin (c+d x)}{768 a^2 d \sqrt {\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {19 \sin (c+d x)}{96 a d \sec ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^{7/2}}-\frac {\sin (c+d x)}{8 d \sec ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2844
Rule 2861
Rule 3056
Rule 3057
Rule 4307
Rubi steps
\begin {align*} \int \frac {1}{(a+a \cos (c+d x))^{9/2} \sec ^{\frac {7}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{(a+a \cos (c+d x))^{9/2}} \, dx\\ &=-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \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 {5 a}{2}-7 a \cos (c+d x)\right )}{(a+a \cos (c+d x))^{7/2}} \, dx}{8 a^2}\\ &=-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \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 {57 a^2}{4}-\frac {65}{2} a^2 \cos (c+d x)\right )}{(a+a \cos (c+d x))^{5/2}} \, dx}{48 a^4}\\ &=-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {187 a^3}{8}-\frac {333}{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 {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}-\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int -\frac {105 a^4}{16 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{384 a^8}\\ &=-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}+\frac {\left (35 \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}{2048 a^4}\\ &=-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}-\frac {\left (35 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \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 {35 \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)}}{1024 \sqrt {2} a^{9/2} d}-\frac {\sin (c+d x)}{8 d (a+a \cos (c+d x))^{9/2} \sec ^{\frac {5}{2}}(c+d x)}-\frac {19 \sin (c+d x)}{96 a d (a+a \cos (c+d x))^{7/2} \sec ^{\frac {3}{2}}(c+d x)}-\frac {187 \sin (c+d x)}{768 a^2 d (a+a \cos (c+d x))^{5/2} \sqrt {\sec (c+d x)}}+\frac {853 \sin (c+d x)}{3072 a^3 d (a+a \cos (c+d x))^{3/2} \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 6.10, size = 395, normalized size = 1.67 \begin {gather*} \frac {2 \cos ^9\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\frac {1}{1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}} \sqrt {1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )} \left (1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^{9/2} \left (\frac {35 \text {ArcSin}\left (\frac {\sin \left (\frac {c}{2}+\frac {d x}{2}\right )}{\sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )}}\right ) \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} \csc \left (\frac {c}{2}+\frac {d x}{2}\right )}{128 \left (1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^{9/2}}+\frac {1}{8} \left (\frac {35}{16 \left (1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^4}+\frac {35}{24 \left (1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {7}{6 \left (1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {1}{1-\sec ^2\left (\frac {1}{2} (c+d x)\right ) \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}\right )\right )}{d \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right )} (a (1+\cos (c+d x)))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 354, normalized size = 1.49
method | result | size |
default | \(-\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (-1+\cos \left (d x +c \right )\right )^{8} \cos \left (d x +c \right ) \left (853 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{4}\left (d x +c \right )\right )+105 \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right )-34 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+315 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-364 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+315 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-350 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )+105 \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sin \left (d x +c \right )-105 \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\right ) \sqrt {2}}{6144 d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \sin \left (d x +c \right )^{17} a^{5}}\) | \(354\) |
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 = 0.43, size = 237, normalized size = 1.00 \begin {gather*} -\frac {105 \, \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 {\cos \left (d x + c\right )}}{\sqrt {a} \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (853 \, \cos \left (d x + c\right )^{4} + 819 \, \cos \left (d x + c\right )^{3} + 455 \, \cos \left (d x + c\right )^{2} + 105 \, \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 )}}}{6144 \, {\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 {1}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}\,{\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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