Optimal. Leaf size=127 \[ -\frac {4 \sec ^5(c+d x)}{5 a^4 d}+\frac {12 \sec ^7(c+d x)}{7 a^4 d}-\frac {8 \sec ^9(c+d x)}{9 a^4 d}+\frac {\tan ^3(c+d x)}{3 a^4 d}+\frac {9 \tan ^5(c+d x)}{5 a^4 d}+\frac {16 \tan ^7(c+d x)}{7 a^4 d}+\frac {8 \tan ^9(c+d x)}{9 a^4 d} \]
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Rubi [A]
time = 0.23, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2790, 2687,
276, 2686, 14} \begin {gather*} \frac {8 \tan ^9(c+d x)}{9 a^4 d}+\frac {16 \tan ^7(c+d x)}{7 a^4 d}+\frac {9 \tan ^5(c+d x)}{5 a^4 d}+\frac {\tan ^3(c+d x)}{3 a^4 d}-\frac {8 \sec ^9(c+d x)}{9 a^4 d}+\frac {12 \sec ^7(c+d x)}{7 a^4 d}-\frac {4 \sec ^5(c+d x)}{5 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 276
Rule 2686
Rule 2687
Rule 2790
Rubi steps
\begin {align*} \int \frac {\tan ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\int \left (a^4 \sec ^8(c+d x) \tan ^2(c+d x)-4 a^4 \sec ^7(c+d x) \tan ^3(c+d x)+6 a^4 \sec ^6(c+d x) \tan ^4(c+d x)-4 a^4 \sec ^5(c+d x) \tan ^5(c+d x)+a^4 \sec ^4(c+d x) \tan ^6(c+d x)\right ) \, dx}{a^8}\\ &=\frac {\int \sec ^8(c+d x) \tan ^2(c+d x) \, dx}{a^4}+\frac {\int \sec ^4(c+d x) \tan ^6(c+d x) \, dx}{a^4}-\frac {4 \int \sec ^7(c+d x) \tan ^3(c+d x) \, dx}{a^4}-\frac {4 \int \sec ^5(c+d x) \tan ^5(c+d x) \, dx}{a^4}+\frac {6 \int \sec ^6(c+d x) \tan ^4(c+d x) \, dx}{a^4}\\ &=\frac {\text {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^3 \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac {6 \text {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,\tan (c+d x)\right )}{a^4 d}\\ &=\frac {\text {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}+\frac {\text {Subst}\left (\int \left (x^2+3 x^4+3 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}-\frac {4 \text {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}+\frac {6 \text {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,\tan (c+d x)\right )}{a^4 d}\\ &=-\frac {4 \sec ^5(c+d x)}{5 a^4 d}+\frac {12 \sec ^7(c+d x)}{7 a^4 d}-\frac {8 \sec ^9(c+d x)}{9 a^4 d}+\frac {\tan ^3(c+d x)}{3 a^4 d}+\frac {9 \tan ^5(c+d x)}{5 a^4 d}+\frac {16 \tan ^7(c+d x)}{7 a^4 d}+\frac {8 \tan ^9(c+d x)}{9 a^4 d}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 124, normalized size = 0.98 \begin {gather*} \frac {\sec (c+d x) (16128+1554 \cos (c+d x)-16896 \cos (2 (c+d x))-999 \cos (3 (c+d x))+2816 \cos (4 (c+d x))+37 \cos (5 (c+d x))+34944 \sin (c+d x)+1776 \sin (2 (c+d x))-9504 \sin (3 (c+d x))-296 \sin (4 (c+d x))+352 \sin (5 (c+d x)))}{80640 a^4 d (1+\sin (c+d x))^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 158, normalized size = 1.24
method | result | size |
risch | \(-\frac {4 i \left (504 i {\mathrm e}^{5 i \left (d x +c \right )}+315 \,{\mathrm e}^{6 i \left (d x +c \right )}-528 i {\mathrm e}^{3 i \left (d x +c \right )}-777 \,{\mathrm e}^{4 i \left (d x +c \right )}+88 i {\mathrm e}^{i \left (d x +c \right )}+297 \,{\mathrm e}^{2 i \left (d x +c \right )}-11\right )}{315 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{9} d \,a^{4}}\) | \(109\) |
derivativedivides | \(\frac {-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {16}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {116}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {62}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {83}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {17}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {29}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128}}{d \,a^{4}}\) | \(158\) |
default | \(\frac {-\frac {1}{16 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {16}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{9}}+\frac {8}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{8}}-\frac {116}{7 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{7}}+\frac {62}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{6}}-\frac {83}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}+\frac {17}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {29}{12 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {8}{128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+128}}{d \,a^{4}}\) | \(158\) |
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. 356 vs.
\(2 (113) = 226\).
time = 0.29, size = 356, normalized size = 2.80 \begin {gather*} \frac {8 \, {\left (\frac {16 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {54 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {201 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {294 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {378 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {210 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {105 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + 2\right )}}{315 \, {\left (a^{4} + \frac {8 \, a^{4} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {27 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, a^{4} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {42 \, a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {42 \, a^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {48 \, a^{4} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {27 \, a^{4} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {8 \, a^{4} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{4} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 129, normalized size = 1.02 \begin {gather*} \frac {88 \, \cos \left (d x + c\right )^{4} - 220 \, \cos \left (d x + c\right )^{2} + {\left (22 \, \cos \left (d x + c\right )^{4} - 165 \, \cos \left (d x + c\right )^{2} + 175\right )} \sin \left (d x + c\right ) + 140}{315 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} - 8 \, a^{4} d \cos \left (d x + c\right )^{3} + 8 \, a^{4} d \cos \left (d x + c\right ) - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{3} - 2 \, a^{4} d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tan ^{2}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 14.26, size = 146, normalized size = 1.15 \begin {gather*} -\frac {\frac {315}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}} - \frac {315 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3150 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 630 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 8064 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 6006 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5274 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 846 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 59}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{9}}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.58, size = 231, normalized size = 1.82 \begin {gather*} \frac {\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{315}+\frac {128\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{315}+\frac {48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{35}+\frac {536\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{105}+\frac {112\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{15}+\frac {48\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}}{a^4\,d\,\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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