Optimal. Leaf size=118 \[ \frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{35 a c^4 f} \]
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Rubi [A]
time = 0.15, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2751,
3852, 8} \begin {gather*} \frac {8 \tan (e+f x)}{35 a c^4 f}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2751
Rule 2815
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx &=\frac {\int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a c}\\ &=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{7 a c^2}\\ &=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {12 \int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{35 a c^3}\\ &=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {8 \int \sec ^2(e+f x) \, dx}{35 a c^4}\\ &=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}-\frac {8 \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{35 a c^4 f}\\ &=\frac {\sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {4 \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {4 \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {8 \tan (e+f x)}{35 a c^4 f}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 131, normalized size = 1.11 \begin {gather*} \frac {-406+896 \cos (e+f x)-232 \cos (2 (e+f x))+832 \cos (3 (e+f x))+174 \cos (4 (e+f x))-64 \cos (5 (e+f x))+406 \sin (e+f x)+512 \sin (2 (e+f x))+377 \sin (3 (e+f x))-384 \sin (4 (e+f x))-29 \sin (5 (e+f x))}{4480 a c^4 f (-1+\sin (e+f x))^4 (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 133, normalized size = 1.13
method | result | size |
risch | \(-\frac {16 \left (-6 \,{\mathrm e}^{i \left (f x +e \right )}+i+14 \,{\mathrm e}^{3 i \left (f x +e \right )}-14 i {\mathrm e}^{2 i \left (f x +e \right )}\right )}{35 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a f \,c^{4}}\) | \(77\) |
derivativedivides | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {17}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a f \,c^{4}}\) | \(133\) |
default | \(\frac {-\frac {1}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {8}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {38}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {9}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {15}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {17}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {15}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}}{a f \,c^{4}}\) | \(133\) |
norman | \(\frac {\frac {6 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {26}{35 a c f}-\frac {2 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {6 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {10 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {2 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}-\frac {22 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}+\frac {86 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 a c f}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(195\) |
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. 347 vs.
\(2 (120) = 240\).
time = 0.34, size = 347, normalized size = 2.94 \begin {gather*} -\frac {2 \, {\left (\frac {43 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {77 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {175 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {35 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - 13\right )}}{35 \, {\left (a c^{4} - \frac {6 \, a c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {6 \, a c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 120, normalized size = 1.02 \begin {gather*} \frac {8 \, \cos \left (f x + e\right )^{4} - 36 \, \cos \left (f x + e\right )^{2} + 4 \, {\left (6 \, \cos \left (f x + e\right )^{2} - 5\right )} \sin \left (f x + e\right ) + 15}{35 \, {\left (3 \, a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right ) - {\left (a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right )\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. 1307 vs.
\(2 (97) = 194\).
time = 6.24, size = 1307, normalized size = 11.08 \begin {gather*} \begin {cases} - \frac {70 \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} + \frac {210 \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} - \frac {350 \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} + \frac {210 \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} + \frac {14 \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} - \frac {154 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} + \frac {86 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} - \frac {26}{35 a c^{4} f \tan ^{8}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 210 a c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 490 a c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 490 a c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 210 a c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 35 a c^{4} f} & \text {for}\: f \neq 0 \\\frac {x}{\left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 133, normalized size = 1.13 \begin {gather*} -\frac {\frac {35}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {525 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1960 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 4025 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4480 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3143 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1176 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 243}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{280 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.27, size = 96, normalized size = 0.81 \begin {gather*} \frac {\frac {2\,\sin \left (e+f\,x\right )}{5}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{5}-\frac {\cos \left (4\,e+4\,f\,x\right )}{35}-\frac {6\,\sin \left (3\,e+3\,f\,x\right )}{35}}{a\,c^4\,f\,\left (\frac {7\,\cos \left (e+f\,x\right )}{4}-\frac {3\,\cos \left (3\,e+3\,f\,x\right )}{4}-\frac {7\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {\sin \left (4\,e+4\,f\,x\right )}{8}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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