Optimal. Leaf size=317 \[ \frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.51, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2841, 3059,
2851, 2850} \begin {gather*} -\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^5 \sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 f (c+d)^4 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 f (c+d)^3 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{5/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{7/2}}+\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{9 d f (c+d) (c+d \sin (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2841
Rule 2850
Rule 2851
Rule 3059
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^{11/2}} \, dx &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}-\frac {(2 a) \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-19 d)-\frac {3}{2} a (c+5 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^{9/2}} \, dx}{9 d (c+d)}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}+\frac {\left (a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{7/2}} \, dx}{21 d^2 (c+d)^2}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}+\frac {\left (4 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{5/2}} \, dx}{105 d^2 (c+d)^3}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}+\frac {\left (8 a^2 \left (c^2+10 c d+73 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^{3/2}} \, dx}{315 d^2 (c+d)^4}\\ &=\frac {2 a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{9 d (c+d) f (c+d \sin (e+f x))^{9/2}}+\frac {2 a^3 (c-d) (3 c+19 d) \cos (e+f x)}{63 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}-\frac {2 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{105 d^2 (c+d)^3 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}-\frac {8 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^4 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}-\frac {16 a^3 \left (c^2+10 c d+73 d^2\right ) \cos (e+f x)}{315 d^2 (c+d)^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 5.21, size = 304, normalized size = 0.96 \begin {gather*} -\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (1869 c^4+2088 c^3 d+5776 c^2 d^2+1804 c d^3+727 d^4-\left (63 c^4+648 c^3 d+4790 c^2 d^2+1424 c d^3+803 d^4\right ) \cos (2 (e+f x))+2 d^2 \left (c^2+10 c d+73 d^2\right ) \cos (4 (e+f x))+588 c^4 \sin (e+f x)+7326 c^3 d \sin (e+f x)+4370 c^2 d^2 \sin (e+f x)+5498 c d^3 \sin (e+f x)+698 d^4 \sin (e+f x)-18 c^3 d \sin (3 (e+f x))-182 c^2 d^2 \sin (3 (e+f x))-1334 c d^3 \sin (3 (e+f x))-146 d^4 \sin (3 (e+f x))\right )}{315 (c+d)^5 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sin (e+f x))^{9/2}} \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. \(1729\) vs.
\(2(287)=574\).
time = 7.51, size = 1730, normalized size = 5.46
method | result | size |
default | \(\text {Expression too large to display}\) | \(1730\) |
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. 1018 vs.
\(2 (302) = 604\).
time = 0.84, size = 1018, normalized size = 3.21 \begin {gather*} -\frac {2 \, {\left ({\left (903 \, c^{5} + 720 \, c^{4} d + 494 \, c^{3} d^{2} + 200 \, c^{2} d^{3} + 35 \, c d^{4}\right )} a^{\frac {5}{2}} - \frac {{\left (315 \, c^{5} - 8358 \, c^{4} d - 4770 \, c^{3} d^{2} - 2284 \, c^{2} d^{3} - 625 \, c d^{4} - 70 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {{\left (4179 \, c^{5} - 1710 \, c^{4} d + 30878 \, c^{3} d^{2} + 11540 \, c^{2} d^{3} + 3383 \, c d^{4} + 450 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {3 \, {\left (805 \, c^{5} - 9912 \, c^{4} d + 2330 \, c^{3} d^{2} - 18504 \, c^{2} d^{3} - 3895 \, c d^{4} - 504 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {6 \, {\left (1239 \, c^{5} - 3100 \, c^{4} d + 12918 \, c^{3} d^{2} - 3560 \, c^{2} d^{3} + 8043 \, c d^{4} + 700 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {42 \, {\left (149 \, c^{5} - 894 \, c^{4} d + 1402 \, c^{3} d^{2} - 2052 \, c^{2} d^{3} + 745 \, c d^{4} - 390 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {42 \, {\left (149 \, c^{5} - 894 \, c^{4} d + 1402 \, c^{3} d^{2} - 2052 \, c^{2} d^{3} + 745 \, c d^{4} - 390 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {6 \, {\left (1239 \, c^{5} - 3100 \, c^{4} d + 12918 \, c^{3} d^{2} - 3560 \, c^{2} d^{3} + 8043 \, c d^{4} + 700 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {3 \, {\left (805 \, c^{5} - 9912 \, c^{4} d + 2330 \, c^{3} d^{2} - 18504 \, c^{2} d^{3} - 3895 \, c d^{4} - 504 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {{\left (4179 \, c^{5} - 1710 \, c^{4} d + 30878 \, c^{3} d^{2} + 11540 \, c^{2} d^{3} + 3383 \, c d^{4} + 450 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {{\left (315 \, c^{5} - 8358 \, c^{4} d - 4770 \, c^{3} d^{2} - 2284 \, c^{2} d^{3} - 625 \, c d^{4} - 70 \, d^{5}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {{\left (903 \, c^{5} + 720 \, c^{4} d + 494 \, c^{3} d^{2} + 200 \, c^{2} d^{3} + 35 \, c d^{4}\right )} a^{\frac {5}{2}} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}}\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{3}}{315 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5} + \frac {3 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, {\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {{\left (c^{5} + 5 \, c^{4} d + 10 \, c^{3} d^{2} + 10 \, c^{2} d^{3} + 5 \, c d^{4} + d^{5}\right )} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )} {\left (c + \frac {2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac {11}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1516 vs.
\(2 (302) = 604\).
time = 0.43, size = 1516, normalized size = 4.78 \begin {gather*} \frac {2 \, {\left (672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left (a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{5} - 4 \, {\left (9 \, a^{2} c^{3} d + 89 \, a^{2} c^{2} d^{2} + 647 \, a^{2} c d^{3} - 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{4} - {\left (63 \, a^{2} c^{4} + 648 \, a^{2} c^{3} d + 4798 \, a^{2} c^{2} d^{2} + 1504 \, a^{2} c d^{3} + 1387 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} + {\left (231 \, a^{2} c^{4} + 3060 \, a^{2} c^{3} d - 2158 \, a^{2} c^{2} d^{2} + 4580 \, a^{2} c d^{3} - 673 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (483 \, a^{2} c^{4} + 684 \, a^{2} c^{3} d + 2642 \, a^{2} c^{2} d^{2} + 812 \, a^{2} c d^{3} + 419 \, a^{2} d^{4}\right )} \cos \left (f x + e\right ) - {\left (672 \, a^{2} c^{4} - 2304 \, a^{2} c^{3} d + 3008 \, a^{2} c^{2} d^{2} - 1792 \, a^{2} c d^{3} + 416 \, a^{2} d^{4} + 8 \, {\left (a^{2} c^{2} d^{2} + 10 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{4} + 4 \, {\left (9 \, a^{2} c^{3} d + 91 \, a^{2} c^{2} d^{2} + 667 \, a^{2} c d^{3} + 73 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (21 \, a^{2} c^{4} + 204 \, a^{2} c^{3} d + 1478 \, a^{2} c^{2} d^{2} - 388 \, a^{2} c d^{3} + 365 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (147 \, a^{2} c^{4} + 1836 \, a^{2} c^{3} d + 1138 \, a^{2} c^{2} d^{2} + 1708 \, a^{2} c d^{3} + 211 \, a^{2} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right ) + c}}{315 \, {\left ({\left (c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{6} - 5 \, {\left (c^{6} d^{4} + 5 \, c^{5} d^{5} + 10 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right )} f \cos \left (f x + e\right )^{5} - {\left (10 \, c^{7} d^{3} + 55 \, c^{6} d^{4} + 128 \, c^{5} d^{5} + 165 \, c^{4} d^{6} + 130 \, c^{3} d^{7} + 65 \, c^{2} d^{8} + 20 \, c d^{9} + 3 \, d^{10}\right )} f \cos \left (f x + e\right )^{4} + 10 \, {\left (c^{8} d^{2} + 5 \, c^{7} d^{3} + 11 \, c^{6} d^{4} + 15 \, c^{5} d^{5} + 15 \, c^{4} d^{6} + 11 \, c^{3} d^{7} + 5 \, c^{2} d^{8} + c d^{9}\right )} f \cos \left (f x + e\right )^{3} + {\left (5 \, c^{9} d + 35 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 260 \, c^{6} d^{4} + 378 \, c^{5} d^{5} + 370 \, c^{4} d^{6} + 240 \, c^{3} d^{7} + 100 \, c^{2} d^{8} + 25 \, c d^{9} + 3 \, d^{10}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{10} + 5 \, c^{9} d + 20 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 110 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 100 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 25 \, c^{2} d^{8} + 5 \, c d^{9}\right )} f \cos \left (f x + e\right ) - {\left (c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f - {\left ({\left (c^{5} d^{5} + 5 \, c^{4} d^{6} + 10 \, c^{3} d^{7} + 10 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{5} + {\left (5 \, c^{6} d^{4} + 26 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 35 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{4} - 2 \, {\left (5 \, c^{7} d^{3} + 25 \, c^{6} d^{4} + 51 \, c^{5} d^{5} + 55 \, c^{4} d^{6} + 35 \, c^{3} d^{7} + 15 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, c^{8} d^{2} + 30 \, c^{7} d^{3} + 80 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 130 \, c^{4} d^{6} + 90 \, c^{3} d^{7} + 40 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right )^{2} + {\left (5 \, c^{9} d + 25 \, c^{8} d^{2} + 60 \, c^{7} d^{3} + 100 \, c^{6} d^{4} + 126 \, c^{5} d^{5} + 110 \, c^{4} d^{6} + 60 \, c^{3} d^{7} + 20 \, c^{2} d^{8} + 5 \, c d^{9} + d^{10}\right )} f \cos \left (f x + e\right ) + {\left (c^{10} + 10 \, c^{9} d + 45 \, c^{8} d^{2} + 120 \, c^{7} d^{3} + 210 \, c^{6} d^{4} + 252 \, c^{5} d^{5} + 210 \, c^{4} d^{6} + 120 \, c^{3} d^{7} + 45 \, c^{2} d^{8} + 10 \, c d^{9} + d^{10}\right )} f\right )} \sin \left (f x + e\right )\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 = 26.14, size = 1155, normalized size = 3.64 \begin {gather*} -\frac {\sqrt {c+d\,\sin \left (e+f\,x\right )}\,\left (\frac {32\,a^2\,{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,1{}\mathrm {i}+c\,d\,10{}\mathrm {i}+d^2\,73{}\mathrm {i}\right )}{315\,d^3\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2+10\,c\,d+73\,d^2\right )}{315\,d^3\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (25\,c^4-25\,c^3\,d+57\,c^2\,d^2-15\,c\,d^3+6\,d^4\right )}{5\,d^5\,f\,{\left (c+d\right )}^5}+\frac {32\,a^2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,25{}\mathrm {i}-c^3\,d\,25{}\mathrm {i}+c^2\,d^2\,57{}\mathrm {i}-c\,d^3\,15{}\mathrm {i}+d^4\,6{}\mathrm {i}\right )}{5\,d^5\,f\,{\left (c+d\right )}^5}-\frac {16\,a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (25\,c^4+318\,c^3\,d-20\,c^2\,d^2+194\,c\,d^3-5\,d^4\right )}{15\,d^5\,f\,{\left (c+d\right )}^5}+\frac {16\,a^2\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,25{}\mathrm {i}+c^3\,d\,318{}\mathrm {i}-c^2\,d^2\,20{}\mathrm {i}+c\,d^3\,194{}\mathrm {i}-d^4\,5{}\mathrm {i}\right )}{15\,d^5\,f\,{\left (c+d\right )}^5}+\frac {16\,a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (7\,c^4+70\,c^3\,d+512\,c^2\,d^2+10\,c\,d^3+73\,d^4\right )}{35\,d^5\,f\,{\left (c+d\right )}^5}-\frac {16\,a^2\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^4\,7{}\mathrm {i}+c^3\,d\,70{}\mathrm {i}+c^2\,d^2\,512{}\mathrm {i}+c\,d^3\,10{}\mathrm {i}+d^4\,73{}\mathrm {i}\right )}{35\,d^5\,f\,{\left (c+d\right )}^5}+\frac {32\,a^2\,c\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2+10\,c\,d+73\,d^2\right )}{35\,d^4\,f\,{\left (c+d\right )}^5}-\frac {32\,a^2\,c\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c^2\,1{}\mathrm {i}+c\,d\,10{}\mathrm {i}+d^2\,73{}\mathrm {i}\right )}{35\,d^4\,f\,{\left (c+d\right )}^5}\right )}{{\mathrm {e}}^{e\,11{}\mathrm {i}+f\,x\,11{}\mathrm {i}}-\frac {{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5}{{\left (c+d\right )}^5}+\frac {10\,{\mathrm {e}}^{e\,7{}\mathrm {i}+f\,x\,7{}\mathrm {i}}\,\left (8\,c^4+8\,c^3\,d+12\,c^2\,d^2+4\,c\,d^3+d^4\right )}{d^4}+\frac {5\,{\mathrm {e}}^{e\,3{}\mathrm {i}+f\,x\,3{}\mathrm {i}}\,\left (16\,c^3+8\,c^2\,d+8\,c\,d^2+d^3\right )}{d^3}-\frac {5\,{\mathrm {e}}^{e\,9{}\mathrm {i}+f\,x\,9{}\mathrm {i}}\,\left (8\,c^2+2\,c\,d+d^2\right )}{d^2}-\frac {2\,{\mathrm {e}}^{e\,5{}\mathrm {i}+f\,x\,5{}\mathrm {i}}\,\left (16\,c^5+40\,c^4\,d+80\,c^3\,d^2+60\,c^2\,d^3+30\,c\,d^4+5\,d^5\right )}{d^5}-\frac {{\mathrm {e}}^{e\,1{}\mathrm {i}+f\,x\,1{}\mathrm {i}}\,\left (10\,c+d\right )}{d}-\frac {5\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (16\,c^3+8\,c^2\,d+8\,c\,d^2+d^3\right )}{d^3\,{\left (c+d\right )}^5}+\frac {5\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (8\,c^2+2\,c\,d+d^2\right )}{d^2\,{\left (c+d\right )}^5}+\frac {2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (16\,c^5+40\,c^4\,d+80\,c^3\,d^2+60\,c^2\,d^3+30\,c\,d^4+5\,d^5\right )}{d^5\,{\left (c+d\right )}^5}+\frac {{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (10\,c+d\right )\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5}{d\,{\left (c+d\right )}^5}-\frac {10\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,{\left (c\,1{}\mathrm {i}+d\,1{}\mathrm {i}\right )}^5\,\left (8\,c^4+8\,c^3\,d+12\,c^2\,d^2+4\,c\,d^3+d^4\right )}{d^4\,{\left (c+d\right )}^5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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