Optimal. Leaf size=250 \[ -\frac {3 (121 A-21 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]
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Rubi [A] time = 0.80, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2978, 2984, 12, 2782, 205} \[ \frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {3 (121 A-21 B) \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2782
Rule 2978
Rule 2984
Rubi steps
\begin {align*} \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) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}+\frac {\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) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}+\frac {\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) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {\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) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {\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) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}-\frac {(3 (121 A-21 B)) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{128 a^3}\\ &=-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {(3 (121 A-21 B)) \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 {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 )}{64 \sqrt {2} a^{7/2} d}-\frac {(A-B) \sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {(19 A-7 B) \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {(199 A-43 B) \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {(691 A-103 B) \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 2.92, size = 240, normalized size = 0.96 \[ \frac {\cos ^7\left (\frac {1}{2} (c+d x)\right ) \left (\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right ) (9 (941 A-121 B) \cos (c+d x)+4 (937 A-133 B) \cos (2 (c+d x))+691 A \cos (3 (c+d x))+5284 A-103 B \cos (3 (c+d x))-532 B)}{16 \sqrt {\cos (c+d x)}}-\frac {9 i (121 A-21 B) e^{\frac {1}{2} i (c+d x)} \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \tanh ^{-1}\left (\frac {1-e^{i (c+d x)}}{\sqrt {2} \sqrt {1+e^{2 i (c+d x)}}}\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right )}{24 d (a (\cos (c+d x)+1))^{7/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 298, normalized size = 1.19 \[ -\frac {9 \, \sqrt {2} {\left ({\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{5} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )^{2} + {\left (121 \, A - 21 \, B\right )} \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \, {\left ({\left (691 \, A - 103 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (937 \, A - 133 \, B\right )} \cos \left (d x + c\right )^{2} + 39 \, {\left (41 \, A - 5 \, B\right )} \cos \left (d x + c\right ) + 384 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 581, normalized size = 2.32 \[ -\frac {\sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (-1+\cos \left (d x +c \right )\right )^{2} \left (-1089 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \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 ) \sqrt {2}+189 B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \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 ) \sqrt {2}-3267 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+567 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-3267 A \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 ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+567 B \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 ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+1382 A \left (\cos ^{4}\left (d x +c \right )\right )-1089 A \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-206 B \left (\cos ^{4}\left (d x +c \right )\right )+189 B \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \sqrt {2}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+2366 A \left (\cos ^{3}\left (d x +c \right )\right )-326 B \left (\cos ^{3}\left (d x +c \right )\right )-550 A \left (\cos ^{2}\left (d x +c \right )\right )+142 B \left (\cos ^{2}\left (d x +c \right )\right )-2430 A \cos \left (d x +c \right )+390 B \cos \left (d x +c \right )-768 A \right )}{384 d \,a^{4} \sin \left (d x +c \right )^{5} \left (1+\cos \left (d x +c \right )\right ) \sqrt {\cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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