Optimal. Leaf size=270 \[ -\frac {3 (121 A-21 B) \text {ArcTan}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 \sqrt {2} a^{7/2} d}-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}} \]
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Rubi [A]
time = 0.59, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3040, 3057,
3063, 12, 2861, 211} \begin {gather*} -\frac {3 (121 A-21 B) \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 )}{64 \sqrt {2} a^{7/2} d}+\frac {(691 A-103 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{192 a^3 d \sqrt {a \cos (c+d x)+a}}-\frac {(199 A-43 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{192 a^2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(19 A-7 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{48 a d (a \cos (c+d x)+a)^{5/2}}-\frac {(A-B) \sin (c+d x) \sqrt {\sec (c+d x)}}{6 d (a \cos (c+d x)+a)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 211
Rule 2861
Rule 3040
Rule 3057
Rule 3063
Rubi steps
\begin {align*} \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{(a+a \cos (c+d x))^{7/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx\\ &=-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} a (13 A-B)-3 a (A-B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{4} a^2 (41 A-5 B)-a^2 (19 A-7 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} a^3 (691 A-103 B)-\frac {1}{4} a^3 (199 A-43 B) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int -\frac {9 a^4 (121 A-21 B)}{16 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{24 a^7}\\ &=-\frac {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}}-\frac {\left (3 (121 A-21 B) \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 {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}}+\frac {\left (3 (121 A-21 B) \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 )}{64 a^2 d}\\ &=-\frac {3 (121 A-21 B) \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 {(A-B) \sqrt {\sec (c+d x)} \sin (c+d x)}{6 d (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a d (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^2 d (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{192 a^3 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.20, size = 242, normalized size = 0.90 \begin {gather*} \frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (-9 i (121 A-21 B) e^{-\frac {1}{2} i (c+d x)} \sqrt {\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \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 )+\frac {1}{16} (5284 A-532 B+9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))-103 B \cos (3 (c+d x))) \sec ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {\sec (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{24 d (a (1+\cos (c+d x)))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(594\) vs.
\(2(229)=458\).
time = 0.46, size = 595, normalized size = 2.20
method | result | size |
default | \(\frac {\left (-1+\cos \left (d x +c \right )\right )^{2} \left (1089 A \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 ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-189 B \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 ^{3}\left (d x +c \right )\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+3267 A \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 )-691 A \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-567 B \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \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 )+103 B \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}+3267 A \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-1183 A \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}-567 B \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+163 B \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}+1089 A \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )+275 A \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-189 B \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-71 B \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+1215 A \cos \left (d x +c \right ) \sqrt {2}-195 B \cos \left (d x +c \right ) \sqrt {2}+384 A \sqrt {2}\right ) \cos \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \sin \left (d x +c \right )^{5} \left (1+\cos \left (d x +c \right )\right ) a^{4}}\) | \(595\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 0.40, size = 260, normalized size = 0.96 \begin {gather*} \frac {9 \, \sqrt {2} {\left ({\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right ) + 121 \, A - 21 \, B\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 ({\left (691 \, A - 103 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (937 \, A - 133 \, B\right )} \cos \left (d x + c\right )^{2} + 39 \, {\left (41 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 384 \, A\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 )}} \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 {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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