Optimal. Leaf size=322 \[ \frac {32 a^3 (4184 A+4615 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d} \]
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Rubi [A]
time = 0.60, antiderivative size = 322, normalized size of antiderivative = 1.00, number of steps
used = 8, 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^3 (280 A+299 B) \sin (c+d x) \sec ^{\frac {9}{2}}(c+d x)}{1287 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{9009 d \sqrt {a \cos (c+d x)+a}}+\frac {4 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{15015 d \sqrt {a \cos (c+d x)+a}}+\frac {16 a^3 (4184 A+4615 B) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {32 a^3 (4184 A+4615 B) \sin (c+d x) \sqrt {\sec (c+d x)}}{45045 d \sqrt {a \cos (c+d x)+a}}+\frac {2 a^2 (16 A+13 B) \sin (c+d x) \sec ^{\frac {11}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}{143 d}+\frac {2 a A \sin (c+d x) \sec ^{\frac {13}{2}}(c+d x) (a \cos (c+d x)+a)^{3/2}}{13 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))^{5/2} (A+B \cos (c+d x)) \sec ^{\frac {15}{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))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {15}{2}}(c+d x)} \, dx\\ &=\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {1}{13} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (16 A+13 B)+\frac {1}{2} a (8 A+13 B) \cos (c+d x)\right )}{\cos ^{\frac {13}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {1}{143} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (280 A+299 B)+\frac {1}{4} a^2 (216 A+247 B) \cos (c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (a^2 (4184 A+4615 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}{1287}\\ &=\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (2 a^2 (4184 A+4615 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}{3003}\\ &=\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (8 a^2 (4184 A+4615 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}{15015}\\ &=\frac {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}+\frac {\left (16 a^2 (4184 A+4615 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}{45045}\\ &=\frac {32 a^3 (4184 A+4615 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a^3 (4184 A+4615 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45045 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a^3 (4184 A+4615 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{15015 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (4184 A+4615 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9009 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^3 (280 A+299 B) \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{1287 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a^2 (16 A+13 B) \sqrt {a+a \cos (c+d x)} \sec ^{\frac {11}{2}}(c+d x) \sin (c+d x)}{143 d}+\frac {2 a A (a+a \cos (c+d x))^{3/2} \sec ^{\frac {13}{2}}(c+d x) \sin (c+d x)}{13 d}\\ \end {align*}
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Mathematica [A]
time = 0.97, size = 171, normalized size = 0.53 \begin {gather*} \frac {a^2 \sqrt {a (1+\cos (c+d x))} (171806 A+162955 B+35 (5552 A+5083 B) \cos (c+d x)+14 (15167 A+15925 B) \cos (2 (c+d x))+62760 A \cos (3 (c+d x))+69225 B \cos (3 (c+d x))+62760 A \cos (4 (c+d x))+69225 B \cos (4 (c+d x))+8368 A \cos (5 (c+d x))+9230 B \cos (5 (c+d x))+8368 A \cos (6 (c+d x))+9230 B \cos (6 (c+d x))) \sec ^{\frac {13}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{90090 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 185, normalized size = 0.57
method | result | size |
default | \(-\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (66944 A \left (\cos ^{6}\left (d x +c \right )\right )+73840 B \left (\cos ^{6}\left (d x +c \right )\right )+33472 A \left (\cos ^{5}\left (d x +c \right )\right )+36920 B \left (\cos ^{5}\left (d x +c \right )\right )+25104 A \left (\cos ^{4}\left (d x +c \right )\right )+27690 B \left (\cos ^{4}\left (d x +c \right )\right )+20920 A \left (\cos ^{3}\left (d x +c \right )\right )+23075 B \left (\cos ^{3}\left (d x +c \right )\right )+18305 A \left (\cos ^{2}\left (d x +c \right )\right )+14560 B \left (\cos ^{2}\left (d x +c \right )\right )+11970 A \cos \left (d x +c \right )+4095 B \cos \left (d x +c \right )+3465 A \right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\cos \left (d x +c \right )\right )}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {15}{2}} a^{2}}{45045 d \sin \left (d x +c \right )}\) | \(185\) |
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. 763 vs.
\(2 (280) = 560\).
time = 0.62, size = 763, normalized size = 2.37 \begin {gather*} \frac {8 \, {\left (\frac {{\left (\frac {45045 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {165165 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {414414 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {604890 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {522665 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {289185 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {88980 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {11864 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}\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 {15}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {15}{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 {65 \, {\left (\frac {693 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {3003 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {6930 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {10098 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {9053 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {4875 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} + \frac {1500 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{13}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{13}} - \frac {200 \, \sqrt {2} a^{\frac {5}{2}} \sin \left (d x + c\right )^{15}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{15}}\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 {15}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {15}{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 )}}{45045 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 176, normalized size = 0.55 \begin {gather*} \frac {2 \, {\left (16 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{6} + 8 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + 6 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 5 \, {\left (4184 \, A + 4615 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 35 \, {\left (523 \, A + 416 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 315 \, {\left (38 \, A + 13 \, B\right )} a^{2} \cos \left (d x + c\right ) + 3465 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{45045 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\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 = 6.19, size = 789, normalized size = 2.45 \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 {a^2\,\sqrt {a+a\,\left (\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}\right )}\,\left (4184\,A+4615\,B\right )\,32{}\mathrm {i}}{45045\,d}-\frac {a^2\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (2\,A+5\,B\right )\,16{}\mathrm {i}}{5\,d}+\frac {a^2\,{\mathrm {e}}^{c\,8{}\mathrm {i}+d\,x\,8{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (2\,A+5\,B\right )\,16{}\mathrm {i}}{5\,d}+\frac {a^2\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (116\,A+115\,B\right )\,16{}\mathrm {i}}{35\,d}-\frac {a^2\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (116\,A+115\,B\right )\,16{}\mathrm {i}}{35\,d}+\frac {a^2\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (1046\,A+1075\,B\right )\,16{}\mathrm {i}}{315\,d}-\frac {a^2\,{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (1046\,A+1075\,B\right )\,16{}\mathrm {i}}{315\,d}+\frac {a^2\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (4184\,A+4615\,B\right )\,16{}\mathrm {i}}{3465\,d}-\frac {a^2\,{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (4184\,A+4615\,B\right )\,16{}\mathrm {i}}{3465\,d}-\frac {a^2\,{\mathrm {e}}^{c\,13{}\mathrm {i}+d\,x\,13{}\mathrm {i}}\,\sqrt {a+a\,\left (\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}\right )}\,\left (4184\,A+4615\,B\right )\,32{}\mathrm {i}}{45045\,d}\right )}{{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+6\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}+6\,{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}+15\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}+15\,{\mathrm {e}}^{c\,5{}\mathrm {i}+d\,x\,5{}\mathrm {i}}+20\,{\mathrm {e}}^{c\,6{}\mathrm {i}+d\,x\,6{}\mathrm {i}}+20\,{\mathrm {e}}^{c\,7{}\mathrm {i}+d\,x\,7{}\mathrm {i}}+15\,{\mathrm {e}}^{c\,8{}\mathrm {i}+d\,x\,8{}\mathrm {i}}+15\,{\mathrm {e}}^{c\,9{}\mathrm {i}+d\,x\,9{}\mathrm {i}}+6\,{\mathrm {e}}^{c\,10{}\mathrm {i}+d\,x\,10{}\mathrm {i}}+6\,{\mathrm {e}}^{c\,11{}\mathrm {i}+d\,x\,11{}\mathrm {i}}+{\mathrm {e}}^{c\,12{}\mathrm {i}+d\,x\,12{}\mathrm {i}}+{\mathrm {e}}^{c\,13{}\mathrm {i}+d\,x\,13{}\mathrm {i}}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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