Optimal. Leaf size=195 \[ -\frac {\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}+\frac {a^2}{48 d (a+a \sin (c+d x))^6}-\frac {7 a}{80 d (a+a \sin (c+d x))^5}+\frac {1}{8 d (a+a \sin (c+d x))^4}-\frac {5}{96 a d (a+a \sin (c+d x))^3}+\frac {1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {5}{256 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac {1}{256 d \left (a^4+a^4 \sin (c+d x)\right )} \]
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Rubi [A]
time = 0.10, antiderivative size = 195, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2786, 90, 212}
\begin {gather*} -\frac {3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac {1}{256 d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}+\frac {a^2}{48 d (a \sin (c+d x)+a)^6}+\frac {1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {5}{256 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {7 a}{80 d (a \sin (c+d x)+a)^5}+\frac {1}{8 d (a \sin (c+d x)+a)^4}-\frac {5}{96 a d (a \sin (c+d x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 212
Rule 2786
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\text {Subst}\left (\int \frac {x^5}{(a-x)^3 (a+x)^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{128 a^2 (a-x)^3}-\frac {3}{256 a^3 (a-x)^2}-\frac {a^2}{8 (a+x)^7}+\frac {7 a}{16 (a+x)^6}-\frac {1}{2 (a+x)^5}+\frac {5}{32 a (a+x)^4}+\frac {5}{128 a^2 (a+x)^3}+\frac {1}{256 a^3 (a+x)^2}-\frac {1}{128 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2}{48 d (a+a \sin (c+d x))^6}-\frac {7 a}{80 d (a+a \sin (c+d x))^5}+\frac {1}{8 d (a+a \sin (c+d x))^4}-\frac {5}{96 a d (a+a \sin (c+d x))^3}+\frac {1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {5}{256 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac {1}{256 d \left (a^4+a^4 \sin (c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{128 a^3 d}\\ &=-\frac {\tanh ^{-1}(\sin (c+d x))}{128 a^4 d}+\frac {a^2}{48 d (a+a \sin (c+d x))^6}-\frac {7 a}{80 d (a+a \sin (c+d x))^5}+\frac {1}{8 d (a+a \sin (c+d x))^4}-\frac {5}{96 a d (a+a \sin (c+d x))^3}+\frac {1}{256 d \left (a^2-a^2 \sin (c+d x)\right )^2}-\frac {5}{256 d \left (a^2+a^2 \sin (c+d x)\right )^2}-\frac {3}{256 d \left (a^4-a^4 \sin (c+d x)\right )}-\frac {1}{256 d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 112, normalized size = 0.57 \begin {gather*} -\frac {30 \tanh ^{-1}(\sin (c+d x))-\frac {2 \left (-48-177 \sin (c+d x)-132 \sin ^2(c+d x)+257 \sin ^3(c+d x)+440 \sin ^4(c+d x)+65 \sin ^5(c+d x)+60 \sin ^6(c+d x)+15 \sin ^7(c+d x)\right )}{(-1+\sin (c+d x))^2 (1+\sin (c+d x))^6}}{3840 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 127, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {7}{80 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{96 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d \,a^{4}}\) | \(127\) |
default | \(\frac {\frac {1}{256 \left (\sin \left (d x +c \right )-1\right )^{2}}+\frac {3}{256 \left (\sin \left (d x +c \right )-1\right )}+\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{256}+\frac {1}{48 \left (1+\sin \left (d x +c \right )\right )^{6}}-\frac {7}{80 \left (1+\sin \left (d x +c \right )\right )^{5}}+\frac {1}{8 \left (1+\sin \left (d x +c \right )\right )^{4}}-\frac {5}{96 \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {5}{256 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {1}{256 \left (1+\sin \left (d x +c \right )\right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{256}}{d \,a^{4}}\) | \(127\) |
risch | \(\frac {i \left (4133 \,{\mathrm e}^{7 i \left (d x +c \right )}+5727 \,{\mathrm e}^{11 i \left (d x +c \right )}+365 \,{\mathrm e}^{3 i \left (d x +c \right )}-15 \,{\mathrm e}^{i \left (d x +c \right )}-5727 \,{\mathrm e}^{5 i \left (d x +c \right )}-4240 i {\mathrm e}^{4 i \left (d x +c \right )}+11656 i {\mathrm e}^{6 i \left (d x +c \right )}-8928 i {\mathrm e}^{8 i \left (d x +c \right )}+11656 i {\mathrm e}^{10 i \left (d x +c \right )}+120 i {\mathrm e}^{14 i \left (d x +c \right )}-365 \,{\mathrm e}^{13 i \left (d x +c \right )}+120 i {\mathrm e}^{2 i \left (d x +c \right )}-4240 i {\mathrm e}^{12 i \left (d x +c \right )}+15 \,{\mathrm e}^{15 i \left (d x +c \right )}-4133 \,{\mathrm e}^{9 i \left (d x +c \right )}\right )}{960 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{12} d \,a^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{128 d \,a^{4}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{128 d \,a^{4}}\) | \(254\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 213, normalized size = 1.09 \begin {gather*} \frac {\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{7} + 60 \, \sin \left (d x + c\right )^{6} + 65 \, \sin \left (d x + c\right )^{5} + 440 \, \sin \left (d x + c\right )^{4} + 257 \, \sin \left (d x + c\right )^{3} - 132 \, \sin \left (d x + c\right )^{2} - 177 \, \sin \left (d x + c\right ) - 48\right )}}{a^{4} \sin \left (d x + c\right )^{8} + 4 \, a^{4} \sin \left (d x + c\right )^{7} + 4 \, a^{4} \sin \left (d x + c\right )^{6} - 4 \, a^{4} \sin \left (d x + c\right )^{5} - 10 \, a^{4} \sin \left (d x + c\right )^{4} - 4 \, a^{4} \sin \left (d x + c\right )^{3} + 4 \, a^{4} \sin \left (d x + c\right )^{2} + 4 \, a^{4} \sin \left (d x + c\right ) + a^{4}} - \frac {15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4}}}{3840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 290, normalized size = 1.49 \begin {gather*} -\frac {120 \, \cos \left (d x + c\right )^{6} - 1240 \, \cos \left (d x + c\right )^{4} + 1856 \, \cos \left (d x + c\right )^{2} + 15 \, {\left (\cos \left (d x + c\right )^{8} - 8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{4} - 4 \, {\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (\cos \left (d x + c\right )^{8} - 8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{4} - 4 \, {\left (\cos \left (d x + c\right )^{6} - 2 \, \cos \left (d x + c\right )^{4}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (15 \, \cos \left (d x + c\right )^{6} - 110 \, \cos \left (d x + c\right )^{4} + 432 \, \cos \left (d x + c\right )^{2} - 160\right )} \sin \left (d x + c\right ) - 640}{3840 \, {\left (a^{4} d \cos \left (d x + c\right )^{8} - 8 \, a^{4} d \cos \left (d x + c\right )^{6} + 8 \, a^{4} d \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{4} d \cos \left (d x + c\right )^{6} - 2 \, a^{4} d \cos \left (d x + c\right )^{4}\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 ^{5}{\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 = 24.28, size = 146, normalized size = 0.75 \begin {gather*} -\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4}} + \frac {30 \, {\left (3 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) + 7\right )}}{a^{4} {\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac {147 \, \sin \left (d x + c\right )^{6} + 822 \, \sin \left (d x + c\right )^{5} + 1605 \, \sin \left (d x + c\right )^{4} + 340 \, \sin \left (d x + c\right )^{3} - 675 \, \sin \left (d x + c\right )^{2} - 522 \, \sin \left (d x + c\right ) - 117}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{6}}}{15360 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.66, size = 476, normalized size = 2.44 \begin {gather*} \frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{64}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{8}+\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{192}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{12}-\frac {139\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{320}+\frac {1073\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{120}+\frac {10277\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{960}+\frac {237\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{10}+\frac {10277\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{960}+\frac {1073\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{120}-\frac {139\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{320}+\frac {5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{12}+\frac {73\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{192}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}}{d\,\left (a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}-36\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-120\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-88\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+88\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+198\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+88\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-88\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-120\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-36\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a^4\right )}-\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,a^4\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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