Optimal. Leaf size=260 \[ \frac {5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) x}{2 a^2}+\frac {2 d \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}+\frac {d^2 \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \cos (e+f x) \sin (e+f x)}{6 a^2 f}+\frac {d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}-\frac {(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2} \]
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Rubi [A]
time = 0.33, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2844, 3056,
2832, 2813} \begin {gather*} \frac {d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}+\frac {5 d^2 x (2 c-d) \left (2 c^2-3 c d+2 d^2\right )}{2 a^2}+\frac {d^2 \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sin (e+f x) \cos (e+f x)}{6 a^2 f}+\frac {2 d \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}-\frac {(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (\sin (e+f x)+1)}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a \sin (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2813
Rule 2832
Rule 2844
Rule 3056
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^5}{(a+a \sin (e+f x))^2} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x))^3 \left (-a \left (c^2+6 c d-4 d^2\right )+3 a (c-2 d) d \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac {(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac {\int (c+d \sin (e+f x))^2 \left (-3 a^2 (11 c-10 d) d^2+3 a^2 d \left (c^2+10 c d-12 d^2\right ) \sin (e+f x)\right ) \, dx}{3 a^4}\\ &=\frac {d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}-\frac {(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}-\frac {\int (c+d \sin (e+f x)) \left (-3 a^2 d^2 \left (31 c^2-50 c d+24 d^2\right )+3 a^2 d \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \sin (e+f x)\right ) \, dx}{9 a^4}\\ &=\frac {5 (2 c-d) d^2 \left (2 c^2-3 c d+2 d^2\right ) x}{2 a^2}+\frac {2 d \left (c^4+10 c^3 d-44 c^2 d^2+40 c d^3-12 d^4\right ) \cos (e+f x)}{3 a^2 f}+\frac {d^2 \left (2 c^3+20 c^2 d-57 c d^2+30 d^3\right ) \cos (e+f x) \sin (e+f x)}{6 a^2 f}+\frac {d \left (c^2+10 c d-12 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{3 a^2 f}-\frac {(c-d) (c+10 d) \cos (e+f x) (c+d \sin (e+f x))^3}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^4}{3 f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(837\) vs. \(2(260)=520\).
time = 1.12, size = 837, normalized size = 3.22 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (3 d \left (80 c^4+80 c^3 d (-4+3 e+3 f x)-80 c^2 d^2 (-5+6 e+6 f x)+35 c d^3 (-7+12 e+12 f x)-4 d^4 (-13+30 e+30 f x)\right ) \cos \left (\frac {1}{2} (e+f x)\right )-\left (16 c^5+160 c^4 d+80 c^3 d^2 (-10+3 e+3 f x)-40 c^2 d^3 (-41+12 e+12 f x)-6 d^5 (-57+20 e+20 f x)+5 c d^4 (-239+84 e+84 f x)\right ) \cos \left (\frac {3}{2} (e+f x)\right )+120 c^2 d^3 \cos \left (\frac {5}{2} (e+f x)\right )-75 c d^4 \cos \left (\frac {5}{2} (e+f x)\right )+30 d^5 \cos \left (\frac {5}{2} (e+f x)\right )+15 c d^4 \cos \left (\frac {7}{2} (e+f x)\right )-3 d^5 \cos \left (\frac {7}{2} (e+f x)\right )-d^5 \cos \left (\frac {9}{2} (e+f x)\right )+48 c^5 \sin \left (\frac {1}{2} (e+f x)\right )+240 c^4 d \sin \left (\frac {1}{2} (e+f x)\right )-1440 c^3 d^2 \sin \left (\frac {1}{2} (e+f x)\right )+2640 c^2 d^3 \sin \left (\frac {1}{2} (e+f x)\right )-1905 c d^4 \sin \left (\frac {1}{2} (e+f x)\right )+516 d^5 \sin \left (\frac {1}{2} (e+f x)\right )+720 c^3 d^2 e \sin \left (\frac {1}{2} (e+f x)\right )-1440 c^2 d^3 e \sin \left (\frac {1}{2} (e+f x)\right )+1260 c d^4 e \sin \left (\frac {1}{2} (e+f x)\right )-360 d^5 e \sin \left (\frac {1}{2} (e+f x)\right )+720 c^3 d^2 f x \sin \left (\frac {1}{2} (e+f x)\right )-1440 c^2 d^3 f x \sin \left (\frac {1}{2} (e+f x)\right )+1260 c d^4 f x \sin \left (\frac {1}{2} (e+f x)\right )-360 d^5 f x \sin \left (\frac {1}{2} (e+f x)\right )-360 c^2 d^3 \sin \left (\frac {3}{2} (e+f x)\right )+315 c d^4 \sin \left (\frac {3}{2} (e+f x)\right )-118 d^5 \sin \left (\frac {3}{2} (e+f x)\right )+240 c^3 d^2 e \sin \left (\frac {3}{2} (e+f x)\right )-480 c^2 d^3 e \sin \left (\frac {3}{2} (e+f x)\right )+420 c d^4 e \sin \left (\frac {3}{2} (e+f x)\right )-120 d^5 e \sin \left (\frac {3}{2} (e+f x)\right )+240 c^3 d^2 f x \sin \left (\frac {3}{2} (e+f x)\right )-480 c^2 d^3 f x \sin \left (\frac {3}{2} (e+f x)\right )+420 c d^4 f x \sin \left (\frac {3}{2} (e+f x)\right )-120 d^5 f x \sin \left (\frac {3}{2} (e+f x)\right )-120 c^2 d^3 \sin \left (\frac {5}{2} (e+f x)\right )+75 c d^4 \sin \left (\frac {5}{2} (e+f x)\right )-30 d^5 \sin \left (\frac {5}{2} (e+f x)\right )+15 c d^4 \sin \left (\frac {7}{2} (e+f x)\right )-3 d^5 \sin \left (\frac {7}{2} (e+f x)\right )+d^5 \sin \left (\frac {9}{2} (e+f x)\right )\right )}{48 a^2 f (1+\sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.50, size = 342, normalized size = 1.32
method | result | size |
derivativedivides | \(\frac {2 d^{2} \left (\frac {\left (\frac {5}{2} c \,d^{2}-d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-10 c^{2} d +10 c \,d^{2}-3 d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-20 c^{2} d +20 c \,d^{2}-8 d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {5}{2} c \,d^{2}+d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-10 c^{2} d +10 c \,d^{2}-\frac {11 d^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {5 \left (4 c^{3}-8 c^{2} d +7 c \,d^{2}-2 d^{3}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )-\frac {2 \left (c^{5}-10 c^{3} d^{2}+20 c^{2} d^{3}-15 c \,d^{4}+4 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 c^{5}+10 c^{4} d -20 c^{3} d^{2}+20 c^{2} d^{3}-10 c \,d^{4}+2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c^{5}-10 c^{4} d +20 c^{3} d^{2}-20 c^{2} d^{3}+10 c \,d^{4}-2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{2}}\) | \(342\) |
default | \(\frac {2 d^{2} \left (\frac {\left (\frac {5}{2} c \,d^{2}-d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-10 c^{2} d +10 c \,d^{2}-3 d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-20 c^{2} d +20 c \,d^{2}-8 d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-\frac {5}{2} c \,d^{2}+d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-10 c^{2} d +10 c \,d^{2}-\frac {11 d^{3}}{3}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {5 \left (4 c^{3}-8 c^{2} d +7 c \,d^{2}-2 d^{3}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )-\frac {2 \left (c^{5}-10 c^{3} d^{2}+20 c^{2} d^{3}-15 c \,d^{4}+4 d^{5}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 c^{5}+10 c^{4} d -20 c^{3} d^{2}+20 c^{2} d^{3}-10 c \,d^{4}+2 d^{5}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c^{5}-10 c^{4} d +20 c^{3} d^{2}-20 c^{2} d^{3}+10 c \,d^{4}-2 d^{5}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}}{f \,a^{2}}\) | \(342\) |
risch | \(\frac {10 d^{2} x \,c^{3}}{a^{2}}-\frac {20 d^{3} x \,c^{2}}{a^{2}}+\frac {35 d^{4} x c}{2 a^{2}}-\frac {5 d^{5} x}{a^{2}}+\frac {5 i d^{4} {\mathrm e}^{2 i \left (f x +e \right )} c}{8 f \,a^{2}}-\frac {2 i \left (-50 i c^{3} d^{2}+i c^{5}+10 i c^{4} d +80 i c^{2} d^{3}+60 i c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-15 i c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+14 i d^{5}-55 i c \,d^{4}-90 i c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+60 i {\mathrm e}^{2 i \left (f x +e \right )} c \,d^{4}-15 i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}+3 c^{5} {\mathrm e}^{i \left (f x +e \right )}+15 c^{4} d \,{\mathrm e}^{i \left (f x +e \right )}-90 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}+150 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}-105 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}+27 d^{5} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}-\frac {5 d^{3} {\mathrm e}^{i \left (f x +e \right )} c^{2}}{a^{2} f}+\frac {5 d^{4} {\mathrm e}^{i \left (f x +e \right )} c}{a^{2} f}-\frac {15 d^{5} {\mathrm e}^{i \left (f x +e \right )}}{8 a^{2} f}-\frac {5 d^{3} {\mathrm e}^{-i \left (f x +e \right )} c^{2}}{a^{2} f}+\frac {5 d^{4} {\mathrm e}^{-i \left (f x +e \right )} c}{a^{2} f}-\frac {15 d^{5} {\mathrm e}^{-i \left (f x +e \right )}}{8 a^{2} f}-\frac {5 i d^{4} {\mathrm e}^{-2 i \left (f x +e \right )} c}{8 f \,a^{2}}+\frac {i d^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{4 f \,a^{2}}-\frac {i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{4 f \,a^{2}}+\frac {d^{5} \cos \left (3 f x +3 e \right )}{12 a^{2} f}\) | \(522\) |
norman | \(\text {Expression too large to display}\) | \(1355\) |
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. 1426 vs.
\(2 (259) = 518\).
time = 0.52, size = 1426, normalized size = 5.48 \begin {gather*} \text {Too large to display} \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. 592 vs.
\(2 (259) = 518\).
time = 0.35, size = 592, normalized size = 2.28 \begin {gather*} \frac {2 \, d^{5} \cos \left (f x + e\right )^{5} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} - {\left (15 \, c d^{4} - 4 \, d^{5}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (30 \, c^{2} d^{3} - 15 \, c d^{4} + 8 \, d^{5}\right )} \cos \left (f x + e\right )^{3} - 30 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x + {\left (2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 220 \, c^{2} d^{3} - 155 \, c d^{4} + 46 \, d^{5} + 15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )^{2} + {\left (4 \, c^{5} + 10 \, c^{4} d - 80 \, c^{3} d^{2} + 260 \, c^{2} d^{3} - 190 \, c d^{4} + 62 \, d^{5} - 15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (2 \, d^{5} \cos \left (f x + e\right )^{4} + 2 \, c^{5} - 10 \, c^{4} d + 20 \, c^{3} d^{2} - 20 \, c^{2} d^{3} + 10 \, c d^{4} - 2 \, d^{5} + {\left (15 \, c d^{4} - 2 \, d^{5}\right )} \cos \left (f x + e\right )^{3} + 30 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x - 3 \, {\left (20 \, c^{2} d^{3} - 15 \, c d^{4} + 6 \, d^{5}\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, c^{5} + 20 \, c^{4} d - 100 \, c^{3} d^{2} + 280 \, c^{2} d^{3} - 200 \, c d^{4} + 64 \, d^{5} - 15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 17899 vs.
\(2 (250) = 500\).
time = 17.40, size = 17899, normalized size = 68.84 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 977 vs.
\(2 (259) = 518\).
time = 0.45, size = 977, normalized size = 3.76 \begin {gather*} \frac {\frac {15 \, {\left (4 \, c^{3} d^{2} - 8 \, c^{2} d^{3} + 7 \, c d^{4} - 2 \, d^{5}\right )} {\left (f x + e\right )}}{a^{2}} - \frac {2 \, {\left (6 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 60 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 120 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 105 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 30 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} + 6 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 30 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 180 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 360 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 315 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 90 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 22 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 10 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 260 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 680 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 595 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 170 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 18 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 90 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 540 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1200 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 945 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 270 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 30 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 420 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1200 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 975 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 306 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 90 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 540 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1320 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1005 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 310 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 18 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 30 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 300 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 840 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 645 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 198 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 180 \, c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 480 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 375 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 114 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, c^{5} + 10 \, c^{4} d - 80 \, c^{3} d^{2} + 200 \, c^{2} d^{3} - 160 \, c d^{4} + 48 \, d^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3} a^{2}}}{6 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.43, size = 692, normalized size = 2.66 \begin {gather*} \frac {5\,d^2\,\mathrm {atan}\left (\frac {5\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-d\right )\,\left (2\,c^2-3\,c\,d+2\,d^2\right )}{20\,c^3\,d^2-40\,c^2\,d^3+35\,c\,d^4-10\,d^5}\right )\,\left (2\,c-d\right )\,\left (2\,c^2-3\,c\,d+2\,d^2\right )}{a^2\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,c^5+10\,c^4\,d-60\,c^3\,d^2+120\,c^2\,d^3-105\,c\,d^4+30\,d^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,c^5+10\,c^4\,d-100\,c^3\,d^2+280\,c^2\,d^3-215\,c\,d^4+66\,d^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (10\,c^5+10\,c^4\,d-140\,c^3\,d^2+400\,c^2\,d^3-325\,c\,d^4+102\,d^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (6\,c^5+30\,c^4\,d-180\,c^3\,d^2+400\,c^2\,d^3-315\,c\,d^4+90\,d^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,c^5+30\,c^4\,d-180\,c^3\,d^2+440\,c^2\,d^3-335\,c\,d^4+\frac {310\,d^5}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {22\,c^5}{3}+\frac {10\,c^4\,d}{3}-\frac {260\,c^3\,d^2}{3}+\frac {680\,c^2\,d^3}{3}-\frac {595\,c\,d^4}{3}+\frac {170\,d^5}{3}\right )-\frac {160\,c\,d^4}{3}+\frac {10\,c^4\,d}{3}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (2\,c^5-20\,c^3\,d^2+40\,c^2\,d^3-35\,c\,d^4+10\,d^5\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^5+10\,c^4\,d-60\,c^3\,d^2+160\,c^2\,d^3-125\,c\,d^4+38\,d^5\right )+\frac {4\,c^5}{3}+16\,d^5+\frac {200\,c^2\,d^3}{3}-\frac {80\,c^3\,d^2}{3}}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+6\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+10\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+12\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+12\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+6\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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