Optimal. Leaf size=167 \[ \frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f} \]
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Rubi [A]
time = 0.16, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2815, 2751,
3852} \begin {gather*} \frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2751
Rule 2815
Rule 3852
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \, dx &=\frac {\int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^3} \, dx}{a^3 c^3}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \int \frac {\sec ^6(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{11 a^3 c^4}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {56 \int \frac {\sec ^6(e+f x)}{c-c \sin (e+f x)} \, dx}{99 a^3 c^5}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \int \sec ^6(e+f x) \, dx}{33 a^3 c^6}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}-\frac {16 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (e+f x)\right )}{33 a^3 c^6 f}\\ &=\frac {\sec ^5(e+f x)}{11 a^3 f \left (c^2-c^2 \sin (e+f x)\right )^3}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^3-c^3 \sin (e+f x)\right )^2}+\frac {8 \sec ^5(e+f x)}{99 a^3 f \left (c^6-c^6 \sin (e+f x)\right )}+\frac {16 \tan (e+f x)}{33 a^3 c^6 f}+\frac {32 \tan ^3(e+f x)}{99 a^3 c^6 f}+\frac {16 \tan ^5(e+f x)}{165 a^3 c^6 f}\\ \end {align*}
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Mathematica [A]
time = 1.05, size = 233, normalized size = 1.40 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-411950 \cos (e+f x)+1081344 \cos (2 (e+f x))-127330 \cos (3 (e+f x))+819200 \cos (4 (e+f x))+37450 \cos (5 (e+f x))+163840 \cos (6 (e+f x))+22470 \cos (7 (e+f x))-16384 \cos (8 (e+f x))+1802240 \sin (e+f x)+247170 \sin (2 (e+f x))+557056 \sin (3 (e+f x))+187250 \sin (4 (e+f x))-163840 \sin (5 (e+f x))+37450 \sin (6 (e+f x))-98304 \sin (7 (e+f x))-3745 \sin (8 (e+f x)))}{8110080 f (a+a \sin (e+f x))^3 (c-c \sin (e+f x))^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.59, size = 253, normalized size = 1.51
method | result | size |
risch | \(-\frac {256 \left (-6 \,{\mathrm e}^{i \left (f x +e \right )}+i-50 i {\mathrm e}^{4 i \left (f x +e \right )}-10 i {\mathrm e}^{2 i \left (f x +e \right )}+110 \,{\mathrm e}^{7 i \left (f x +e \right )}-10 \,{\mathrm e}^{3 i \left (f x +e \right )}+34 \,{\mathrm e}^{5 i \left (f x +e \right )}-66 i {\mathrm e}^{6 i \left (f x +e \right )}\right )}{495 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{11} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5} f \,c^{6} a^{3}}\) | \(123\) |
derivativedivides | \(\frac {-\frac {8}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {106}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {23}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {33}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {217}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {623}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {169}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {365}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {303}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {219}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{40 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {7}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {37}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{6} a^{3}}\) | \(253\) |
default | \(\frac {-\frac {8}{11 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}-\frac {4}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{10}}-\frac {106}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {23}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {33}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {217}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {623}{20 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {169}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {365}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {303}{64 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {219}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {1}{40 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}+\frac {1}{16 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {7}{48 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+\frac {5}{32 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {37}{128 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}}{f \,c^{6} a^{3}}\) | \(253\) |
norman | \(\frac {\frac {106 \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {50}{99 a c f}-\frac {2 \left (\tan ^{15}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {10 \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {6 \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}-\frac {22 \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}+\frac {34 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{33 a c f}-\frac {142 \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {166 \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a c f}+\frac {298 \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {94 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}+\frac {2 \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a c f}+\frac {74 \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{9 a c f}+\frac {1510 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}-\frac {1226 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{99 a c f}+\frac {1334 \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{495 a c f}}{a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5} c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{11}}\) | \(374\) |
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. 763 vs.
\(2 (167) = 334\).
time = 0.35, size = 763, normalized size = 4.57 \begin {gather*} -\frac {2 \, {\left (\frac {255 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {235 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {3065 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3775 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {667 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {8217 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {2035 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {8745 \, \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}} - \frac {11715 \, \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} + \frac {33 \, \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} + \frac {4917 \, \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} - \frac {2475 \, \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - \frac {1815 \, \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} + \frac {1485 \, \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}} - \frac {495 \, \sin \left (f x + e\right )^{15}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{15}} - 125\right )}}{495 \, {\left (a^{3} c^{6} - \frac {6 \, a^{3} c^{6} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} c^{6} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} c^{6} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {50 \, a^{3} c^{6} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {34 \, a^{3} c^{6} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {66 \, a^{3} c^{6} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {110 \, a^{3} c^{6} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} + \frac {110 \, a^{3} c^{6} \sin \left (f x + e\right )^{9}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{9}} - \frac {66 \, a^{3} c^{6} \sin \left (f x + e\right )^{10}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{10}} - \frac {34 \, a^{3} c^{6} \sin \left (f x + e\right )^{11}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{11}} + \frac {50 \, a^{3} c^{6} \sin \left (f x + e\right )^{12}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{12}} - \frac {10 \, a^{3} c^{6} \sin \left (f x + e\right )^{13}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{13}} - \frac {10 \, a^{3} c^{6} \sin \left (f x + e\right )^{14}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{14}} + \frac {6 \, a^{3} c^{6} \sin \left (f x + e\right )^{15}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{15}} - \frac {a^{3} c^{6} \sin \left (f x + e\right )^{16}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{16}}\right )} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 176, normalized size = 1.05 \begin {gather*} \frac {128 \, \cos \left (f x + e\right )^{8} - 576 \, \cos \left (f x + e\right )^{6} + 240 \, \cos \left (f x + e\right )^{4} + 56 \, \cos \left (f x + e\right )^{2} + 8 \, {\left (48 \, \cos \left (f x + e\right )^{6} - 40 \, \cos \left (f x + e\right )^{4} - 14 \, \cos \left (f x + e\right )^{2} - 9\right )} \sin \left (f x + e\right ) + 27}{495 \, {\left (3 \, a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5} - {\left (a^{3} c^{6} f \cos \left (f x + e\right )^{7} - 4 \, a^{3} c^{6} f \cos \left (f x + e\right )^{5}\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. 5661 vs.
\(2 (151) = 302\).
time = 90.01, size = 5661, normalized size = 33.90 \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 = 0.47, size = 245, normalized size = 1.47 \begin {gather*} -\frac {\frac {33 \, {\left (555 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 1920 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2710 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1760 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 463\right )}}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} + \frac {108405 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} - 784080 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 2901195 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 6652800 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 10407474 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 11435424 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 8949270 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4899840 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1816265 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 411664 \, \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 47279}{a^{3} c^{6} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{11}}}{63360 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.62, size = 185, normalized size = 1.11 \begin {gather*} -\frac {\frac {2\,\sin \left (e+f\,x\right )}{9}+\frac {2\,\cos \left (2\,e+2\,f\,x\right )}{15}+\frac {10\,\cos \left (4\,e+4\,f\,x\right )}{99}+\frac {2\,\cos \left (6\,e+6\,f\,x\right )}{99}-\frac {\cos \left (8\,e+8\,f\,x\right )}{495}+\frac {34\,\sin \left (3\,e+3\,f\,x\right )}{495}-\frac {2\,\sin \left (5\,e+5\,f\,x\right )}{99}-\frac {2\,\sin \left (7\,e+7\,f\,x\right )}{165}}{a^3\,c^6\,f\,\left (\frac {5\,\cos \left (5\,e+5\,f\,x\right )}{64}-\frac {17\,\cos \left (3\,e+3\,f\,x\right )}{64}-\frac {55\,\cos \left (e+f\,x\right )}{64}+\frac {3\,\cos \left (7\,e+7\,f\,x\right )}{64}+\frac {33\,\sin \left (2\,e+2\,f\,x\right )}{64}+\frac {25\,\sin \left (4\,e+4\,f\,x\right )}{64}+\frac {5\,\sin \left (6\,e+6\,f\,x\right )}{64}-\frac {\sin \left (8\,e+8\,f\,x\right )}{128}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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