Optimal. Leaf size=92 \[ \frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3} \]
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Rubi [A]
time = 0.12, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2815, 2751,
2750} \begin {gather*} \frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
Rule 2815
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^5} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {1}{7} (2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^4} \, dx\\ &=\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {(2 a) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{35 c}\\ &=\frac {a c \cos ^3(e+f x)}{7 f (c-c \sin (e+f x))^5}+\frac {2 a \cos ^3(e+f x)}{35 f (c-c \sin (e+f x))^4}+\frac {2 a \cos ^3(e+f x)}{105 c f (c-c \sin (e+f x))^3}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 109, normalized size = 1.18 \begin {gather*} \frac {a \left (70 \cos \left (e+\frac {f x}{2}\right )-21 \cos \left (e+\frac {3 f x}{2}\right )+\cos \left (3 e+\frac {7 f x}{2}\right )+35 \sin \left (\frac {f x}{2}\right )+7 \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{210 c^4 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 116, normalized size = 1.26
method | result | size |
risch | \(\frac {\frac {4 a}{105}+\frac {4 i a \,{\mathrm e}^{3 i \left (f x +e \right )}}{3}+\frac {8 a \,{\mathrm e}^{4 i \left (f x +e \right )}}{3}+\frac {4 i a \,{\mathrm e}^{i \left (f x +e \right )}}{15}-\frac {4 a \,{\mathrm e}^{2 i \left (f x +e \right )}}{5}}{\left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\) | \(75\) |
derivativedivides | \(\frac {2 a \left (-\frac {28}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {68}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {16}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(116\) |
default | \(\frac {2 a \left (-\frac {28}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {68}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {8}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {16}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {14}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}\right )}{f \,c^{4}}\) | \(116\) |
norman | \(\frac {-\frac {46 a}{105 c f}-\frac {2 a \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {4 a \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {32 a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}-\frac {32 a \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {16 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{15 c f}-\frac {208 a \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}+\frac {116 a \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {592 a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\) | \(201\) |
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. 611 vs.
\(2 (95) = 190\).
time = 0.30, size = 611, normalized size = 6.64 \begin {gather*} \frac {2 \, {\left (\frac {a {\left (\frac {91 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {168 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {280 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {175 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - 13\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}} - \frac {3 \, a {\left (\frac {49 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {147 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {210 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {210 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {105 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 12\right )}}{c^{4} - \frac {7 \, c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {35 \, c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {35 \, c^{4} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {21 \, c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {7 \, c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}}}\right )}}{105 \, 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. 222 vs.
\(2 (95) = 190\).
time = 0.32, size = 222, normalized size = 2.41 \begin {gather*} \frac {2 \, a \cos \left (f x + e\right )^{4} + 8 \, a \cos \left (f x + e\right )^{3} - 9 \, a \cos \left (f x + e\right )^{2} + 15 \, a \cos \left (f x + e\right ) - {\left (2 \, a \cos \left (f x + e\right )^{3} - 6 \, a \cos \left (f x + e\right )^{2} - 15 \, a \cos \left (f x + e\right ) - 30 \, a\right )} \sin \left (f x + e\right ) + 30 \, a}{105 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} 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. 1061 vs.
\(2 (82) = 164\).
time = 6.51, size = 1061, normalized size = 11.53 \begin {gather*} \begin {cases} - \frac {210 a \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} + \frac {420 a \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} - \frac {910 a \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} + \frac {700 a \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} - \frac {546 a \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} + \frac {112 a \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 c^{4} f} - \frac {46 a}{105 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 735 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 2205 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3675 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3675 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 2205 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 735 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 105 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 = 0.51, size = 114, normalized size = 1.24 \begin {gather*} -\frac {2 \, {\left (105 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 210 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 455 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 350 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 273 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 56 \, a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 23 \, a\right )}}{105 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.35, size = 97, normalized size = 1.05 \begin {gather*} \frac {\sqrt {2}\,a\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {25\,\cos \left (3\,e+3\,f\,x\right )}{8}-\frac {595\,\sin \left (e+f\,x\right )}{8}-\frac {43\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {353\,\cos \left (e+f\,x\right )}{8}+\frac {77\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {21\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {171}{2}\right )}{840\,c^4\,f\,{\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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