Optimal. Leaf size=129 \[ \frac {a^6}{d (a-a \sin (c+d x))^2}-\frac {11 a^5}{d (a-a \sin (c+d x))}-\frac {a^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {9 a^4 \sin ^2(c+d x)}{2 d}-\frac {16 a^4 \sin (c+d x)}{d}-\frac {25 a^4 \log (1-\sin (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2707, 77} \[ -\frac {a^4 \sin ^4(c+d x)}{4 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {9 a^4 \sin ^2(c+d x)}{2 d}+\frac {a^6}{d (a-a \sin (c+d x))^2}-\frac {11 a^5}{d (a-a \sin (c+d x))}-\frac {16 a^4 \sin (c+d x)}{d}-\frac {25 a^4 \log (1-\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 77
Rule 2707
Rubi steps
\begin {align*} \int (a+a \sin (c+d x))^4 \tan ^5(c+d x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5 (a+x)}{(a-x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-16 a^3+\frac {2 a^6}{(a-x)^3}-\frac {11 a^5}{(a-x)^2}+\frac {25 a^4}{a-x}-9 a^2 x-4 a x^2-x^3\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {25 a^4 \log (1-\sin (c+d x))}{d}-\frac {16 a^4 \sin (c+d x)}{d}-\frac {9 a^4 \sin ^2(c+d x)}{2 d}-\frac {4 a^4 \sin ^3(c+d x)}{3 d}-\frac {a^4 \sin ^4(c+d x)}{4 d}+\frac {a^6}{d (a-a \sin (c+d x))^2}-\frac {11 a^5}{d (a-a \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 83, normalized size = 0.64 \[ -\frac {a^4 \left (3 \sin ^4(c+d x)+16 \sin ^3(c+d x)+54 \sin ^2(c+d x)+192 \sin (c+d x)+\frac {120-132 \sin (c+d x)}{(\sin (c+d x)-1)^2}+300 \log (1-\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 154, normalized size = 1.19 \[ -\frac {24 \, a^{4} \cos \left (d x + c\right )^{6} - 272 \, a^{4} \cos \left (d x + c\right )^{4} - 2393 \, a^{4} \cos \left (d x + c\right )^{2} + 1906 \, a^{4} + 2400 \, {\left (a^{4} \cos \left (d x + c\right )^{2} + 2 \, a^{4} \sin \left (d x + c\right ) - 2 \, a^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (8 \, a^{4} \cos \left (d x + c\right )^{4} - 96 \, a^{4} \cos \left (d x + c\right )^{2} + 181 \, a^{4}\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{2} + 2 \, d \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.24, size = 387, normalized size = 3.00 \[ -\frac {4 a^{4} \left (\sin ^{6}\left (d x +c \right )\right )}{d}-\frac {5 a^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{d}-\frac {3 a^{4} \left (\sin ^{8}\left (d x +c \right )\right )}{4 d}-\frac {5 a^{4} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d}-\frac {25 a^{4} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {25 a^{4} \sin \left (d x +c \right )}{d}+\frac {25 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a^{4} \left (\sin ^{10}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{4} \left (\sin ^{10}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{2}}+\frac {a^{4} \left (\sin ^{9}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}-\frac {5 a^{4} \left (\sin ^{9}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {3 a^{4} \left (\sin ^{8}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{4}}-\frac {3 a^{4} \left (\sin ^{8}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {a^{4} \left (\sin ^{7}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{4}}-\frac {3 a^{4} \left (\sin ^{7}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}-\frac {6 a^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{d}-\frac {12 a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{d}-\frac {25 a^{4} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{4} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{4} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 109, normalized size = 0.84 \[ -\frac {3 \, a^{4} \sin \left (d x + c\right )^{4} + 16 \, a^{4} \sin \left (d x + c\right )^{3} + 54 \, a^{4} \sin \left (d x + c\right )^{2} + 300 \, a^{4} \log \left (\sin \left (d x + c\right ) - 1\right ) + 192 \, a^{4} \sin \left (d x + c\right ) - \frac {12 \, {\left (11 \, a^{4} \sin \left (d x + c\right ) - 10 \, a^{4}\right )}}{\sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.88, size = 379, normalized size = 2.94 \[ \frac {25\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {50\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d}-\frac {50\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-150\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {950\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}-\frac {1700\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}+\frac {2180\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}-\frac {2452\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {2180\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-\frac {1700\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {950\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-150\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+50\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+44\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+31\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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