Optimal. Leaf size=217 \[ -\frac {363 \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {691 \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]
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Rubi [A] time = 0.55, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2766, 2978, 2984, 12, 2782, 205} \[ \frac {691 \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{3/2}}-\frac {363 \tan ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )}{64 \sqrt {2} a^{7/2} d}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{5/2}}-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^{7/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 205
Rule 2766
Rule 2782
Rule 2978
Rule 2984
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{7/2}} \, dx &=-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}+\frac {\int \frac {\frac {13 a}{2}-3 a \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{5/2}} \, dx}{6 a^2}\\ &=-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}+\frac {\int \frac {\frac {123 a^2}{4}-19 a^2 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^{3/2}} \, dx}{24 a^4}\\ &=-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {199 \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 {691 a^3}{8}-\frac {199}{4} a^3 \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \cos (c+d x)}} \, dx}{48 a^6}\\ &=-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {691 \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {\int -\frac {1089 a^4}{16 \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}} \, dx}{24 a^7}\\ &=-\frac {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {691 \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}-\frac {363 \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 \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {691 \sin (c+d x)}{192 a^3 d \sqrt {\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}+\frac {363 \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 {363 \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 {\sin (c+d x)}{6 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{7/2}}-\frac {19 \sin (c+d x)}{48 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{5/2}}-\frac {199 \sin (c+d x)}{192 a^2 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^{3/2}}+\frac {691 \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 = 8.44, size = 559, normalized size = 2.58 \[ \frac {2 \sin \left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (\frac {16 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \cos ^8\left (\frac {1}{2} (c+d x)\right ) \, _5F_4\left (2,2,2,2,\frac {5}{2};1,1,1,\frac {13}{2};\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}\right )}{3465 \left (2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1\right )}-\frac {\left (1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2 \sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}} \csc ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}} \left (1144608 \sin ^{14}\left (\frac {c}{2}+\frac {d x}{2}\right )-6712984 \sin ^{12}\left (\frac {c}{2}+\frac {d x}{2}\right )+16548816 \sin ^{10}\left (\frac {c}{2}+\frac {d x}{2}\right )-22251094 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )+17646926 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )-8267707 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )+2120790 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-229635\right )+105 \left (8752 \sin ^8\left (\frac {c}{2}+\frac {d x}{2}\right )-26380 \sin ^6\left (\frac {c}{2}+\frac {d x}{2}\right )+27986 \sin ^4\left (\frac {c}{2}+\frac {d x}{2}\right )-12908 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )+2187\right ) \cos ^6\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sqrt {\frac {\sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )-1}}\right )\right )}{1680}\right )}{d \left (1-2 \sin ^2\left (\frac {c}{2}+\frac {d x}{2}\right )\right )^{3/2} (a (\cos (c+d x)+1))^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.00, size = 239, normalized size = 1.10 \[ -\frac {1089 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, {\left (a \cos \left (d x + c\right )^{2} + a \cos \left (d x + c\right )\right )}}\right ) - 2 \, {\left (691 \, \cos \left (d x + c\right )^{3} + 1874 \, \cos \left (d x + c\right )^{2} + 1599 \, \cos \left (d x + c\right ) + 384\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{384 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 377, normalized size = 1.74 \[ \frac {\left (-1+\cos \left (d x +c \right )\right ) \left (-1089 \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}-4356 \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-6534 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+691 \sqrt {2}\, \left (\cos ^{5}\left (d x +c \right )\right )-4356 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}+1183 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-1089 \sin \left (d x +c \right ) \arcsin \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right ) \left (\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {3}{2}}-275 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {2}-1215 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-384 \cos \left (d x +c \right ) \sqrt {2}\right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \sqrt {2}}{384 d \sin \left (d x +c \right )^{3} \left (1+\cos \left (d x +c \right )\right )^{2} \cos \left (d x +c \right )^{\frac {3}{2}} a^{4}} \]
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 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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