Optimal. Leaf size=161 \[ -\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )} \]
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Rubi [A]
time = 0.20, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2815, 2759,
2758, 2761, 8} \begin {gather*} -\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a \sin (e+f x)+a)^7}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3 \sin (e+f x)+a^3\right )}-\frac {63 c^5 x}{2 a^3}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a \sin (e+f x)+a)^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a \sin (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2758
Rule 2759
Rule 2761
Rule 2815
Rubi steps
\begin {align*} \int \frac {(c-c \sin (e+f x))^5}{(a+a \sin (e+f x))^3} \, dx &=\left (a^5 c^5\right ) \int \frac {\cos ^{10}(e+f x)}{(a+a \sin (e+f x))^8} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}-\frac {1}{5} \left (9 a^3 c^5\right ) \int \frac {\cos ^8(e+f x)}{(a+a \sin (e+f x))^6} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}+\frac {1}{5} \left (21 a c^5\right ) \int \frac {\cos ^6(e+f x)}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {\left (21 c^5\right ) \int \frac {\cos ^4(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{a}\\ &=-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int \frac {\cos ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{2 a^2}\\ &=-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {\left (63 c^5\right ) \int 1 \, dx}{2 a^3}\\ &=-\frac {63 c^5 x}{2 a^3}-\frac {63 c^5 \cos (e+f x)}{2 a^3 f}-\frac {2 a^4 c^5 \cos ^9(e+f x)}{5 f (a+a \sin (e+f x))^7}+\frac {6 a^2 c^5 \cos ^7(e+f x)}{5 f (a+a \sin (e+f x))^5}-\frac {42 c^5 \cos ^5(e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {21 c^5 \cos ^3(e+f x)}{2 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.57, size = 303, normalized size = 1.88 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^5 \left (256 \sin \left (\frac {1}{2} (e+f x)\right )-128 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )-896 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+448 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+2304 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-630 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-160 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sin (2 (e+f x))\right )}{20 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10} (a+a \sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 156, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
default | \(\frac {2 c^{5} \left (-\frac {\frac {\left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {\tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}+8}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}-\frac {63 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}-\frac {128}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {64}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {16}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {32}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{f \,a^{3}}\) | \(156\) |
risch | \(-\frac {63 c^{5} x}{2 a^{3}}-\frac {i c^{5} {\mathrm e}^{2 i \left (f x +e \right )}}{8 f \,a^{3}}-\frac {4 c^{5} {\mathrm e}^{i \left (f x +e \right )}}{a^{3} f}-\frac {4 c^{5} {\mathrm e}^{-i \left (f x +e \right )}}{a^{3} f}+\frac {i c^{5} {\mathrm e}^{-2 i \left (f x +e \right )}}{8 f \,a^{3}}-\frac {32 \left (-105 c^{5} {\mathrm e}^{2 i \left (f x +e \right )}+75 i c^{5} {\mathrm e}^{3 i \left (f x +e \right )}+25 c^{5} {\mathrm e}^{4 i \left (f x +e \right )}-65 i c^{5} {\mathrm e}^{i \left (f x +e \right )}+18 c^{5}\right )}{5 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(179\) |
norman | \(\frac {-\frac {8505 c^{5} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x}{2 a}-\frac {2036 c^{5} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2516 c^{5} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {4412 c^{5} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7915 c^{5} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7736 c^{5} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {4095 c^{5} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {8505 c^{5} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {9765 c^{5} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {956 c^{5} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {5477 c^{5} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {19741 c^{5} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}-\frac {945 c^{5} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {309 c^{5} \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {65 c^{5} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {9765 c^{5} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {4095 c^{5} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {945 c^{5} x \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {1179 c^{5} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {6071 c^{5} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {7640 c^{5} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}-\frac {2205 c^{5} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {6363 c^{5} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {431 c^{5} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}-\frac {315 c^{5} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}-\frac {2205 c^{5} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {315 c^{5} x \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {63 c^{5} x \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}-\frac {496 c^{5}}{5 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(658\) |
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. 1628 vs.
\(2 (158) = 316\).
time = 0.57, size = 1628, normalized size = 10.11 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 301, normalized size = 1.87 \begin {gather*} -\frac {5 \, c^{5} \cos \left (f x + e\right )^{5} + 70 \, c^{5} \cos \left (f x + e\right )^{4} - 1260 \, c^{5} f x - 64 \, c^{5} + 7 \, {\left (45 \, c^{5} f x + 113 \, c^{5}\right )} \cos \left (f x + e\right )^{3} + {\left (945 \, c^{5} f x - 502 \, c^{5}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (315 \, c^{5} f x + 646 \, c^{5}\right )} \cos \left (f x + e\right ) - {\left (5 \, c^{5} \cos \left (f x + e\right )^{4} - 65 \, c^{5} \cos \left (f x + e\right )^{3} + 1260 \, c^{5} f x - 64 \, c^{5} - 3 \, {\left (105 \, c^{5} f x - 242 \, c^{5}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (315 \, c^{5} f x + 614 \, c^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{10 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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. 3643 vs.
\(2 (153) = 306\).
time = 29.23, size = 3643, normalized size = 22.63 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 186, normalized size = 1.16 \begin {gather*} -\frac {\frac {315 \, {\left (f x + e\right )} c^{5}}{a^{3}} + \frac {10 \, {\left (c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 16 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 16 \, c^{5}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a^{3}} + \frac {64 \, {\left (10 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 45 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 85 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 55 \, c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, c^{5}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{10 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 11.14, size = 364, normalized size = 2.26 \begin {gather*} \frac {\frac {63\,c^5\,\left (e+f\,x\right )}{2}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+4310\right )}{10}\right )-\frac {c^5\,\left (315\,e+315\,f\,x+992\right )}{10}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (\frac {315\,c^5\,\left (e+f\,x\right )}{2}-\frac {c^5\,\left (1575\,e+1575\,f\,x+650\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+3090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (378\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (3780\,e+3780\,f\,x+8814\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+7610\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (630\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (6300\,e+6300\,f\,x+12230\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+11090\right )}{10}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (819\,c^5\,\left (e+f\,x\right )-\frac {c^5\,\left (8190\,e+8190\,f\,x+14702\right )}{10}\right )}{a^3\,f\,{\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+1\right )}^5\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^2}-\frac {63\,c^5\,x}{2\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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