Optimal. Leaf size=275 \[ \frac {32 a^2 (168 A+187 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (168 A+187 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^2 (168 A+187 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (168 A+187 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d} \]
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Rubi [A]
time = 0.46, antiderivative size = 275, 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 = {3040, 3054,
3059, 2851, 2850} \begin {gather*} \frac {2 a^2 (12 A+11 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{99 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{693 d \sqrt {a \cos (c+d x)+a}}+\frac {4 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{1155 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^2 (168 A+187 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {32 a^2 (168 A+187 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{3465 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a A \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{11 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2850
Rule 2851
Rule 3040
Rule 3054
Rule 3059
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \sec ^{\frac {13}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{2} a (12 A+11 B)+\frac {1}{2} a (8 A+11 B) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{99} \left (a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (168 A+187 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{231} \left (2 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {4 a^2 (168 A+187 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (168 A+187 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (8 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{1155}\\ &=\frac {16 a^2 (168 A+187 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^2 (168 A+187 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (168 A+187 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {\left (16 a (168 A+187 B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{3465}\\ &=\frac {32 a^2 (168 A+187 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^2 (168 A+187 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{3465 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^2 (168 A+187 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{1155 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (168 A+187 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{693 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (12 A+11 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{99 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [A]
time = 0.85, size = 146, normalized size = 0.53 \begin {gather*} \frac {a \sqrt {a (1+\cos (c+d x))} (2478 A+2057 B+(6342 A+6193 B) \cos (c+d x)+13 (168 A+187 B) \cos (2 (c+d x))+2184 A \cos (3 (c+d x))+2431 B \cos (3 (c+d x))+336 A \cos (4 (c+d x))+374 B \cos (4 (c+d x))+336 A \cos (5 (c+d x))+374 B \cos (5 (c+d x))) \sec ^{\frac {11}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{3465 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 161, normalized size = 0.59
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (2688 A \left (\cos ^{5}\left (d x +c \right )\right )+2992 B \left (\cos ^{5}\left (d x +c \right )\right )+1344 A \left (\cos ^{4}\left (d x +c \right )\right )+1496 B \left (\cos ^{4}\left (d x +c \right )\right )+1008 A \left (\cos ^{3}\left (d x +c \right )\right )+1122 B \left (\cos ^{3}\left (d x +c \right )\right )+840 A \left (\cos ^{2}\left (d x +c \right )\right )+935 B \left (\cos ^{2}\left (d x +c \right )\right )+735 A \cos \left (d x +c \right )+385 B \cos \left (d x +c \right )+315 A \right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {13}{2}} a}{3465 d \sin \left (d x +c \right )}\) | \(161\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 712 vs.
\(2 (239) = 478\).
time = 0.60, size = 712, normalized size = 2.59 \begin {gather*} \frac {4 \, {\left (\frac {21 \, {\left (\frac {165 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {495 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1056 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {1254 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {781 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {299 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {46 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}\right )} A {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (\frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {\sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}} + \frac {11 \, {\left (\frac {315 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1155 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2184 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2586 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {1759 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {611 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {94 \, \sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}}\right )} B {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5}}{{\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {13}{2}} {\left (\frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {\sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + 1\right )}}\right )}}{3465 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 144, normalized size = 0.52 \begin {gather*} \frac {2 \, {\left (16 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{5} + 8 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{4} + 6 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{3} + 5 \, {\left (168 \, A + 187 \, B\right )} a \cos \left (d x + c\right )^{2} + 35 \, {\left (21 \, A + 11 \, B\right )} a \cos \left (d x + c\right ) + 315 \, A a\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{3465 \, {\left (d \cos \left (d x + c\right )^{6} + d \cos \left (d x + c\right )^{5}\right )} \sqrt {\cos \left (d x + c\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 [B]
time = 5.14, size = 348, normalized size = 1.27 \begin {gather*} \frac {\sqrt {\frac {1}{\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}}}\,\left (-\frac {32\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,A+3\,B\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{5\,d}+\frac {64\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,\left (21\,A+19\,B\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{35\,d}+\frac {32\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,\left (168\,A+187\,B\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{315\,d}+\frac {64\,a\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,\left (168\,A+187\,B\right )\,\sqrt {a+a\,\cos \left (c+d\,x\right )}}{3465\,d}\right )}{20\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+20\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )+10\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+10\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )+2\,{\mathrm {e}}^{\frac {c\,11{}\mathrm {i}}{2}+\frac {d\,x\,11{}\mathrm {i}}{2}}\,\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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