Optimal. Leaf size=120 \[ \frac {4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {12 \csc (c+d x)}{a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (\sin (c+d x)+1)}{a^4 d} \]
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Rubi [A] time = 0.08, antiderivative size = 120, 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, 88} \[ \frac {4}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {12 \csc (c+d x)}{a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (\sin (c+d x)+1)}{a^4 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 2707
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2}{x^5 (a+x)^2} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {1}{x^5}-\frac {4}{a x^4}+\frac {8}{a^2 x^3}-\frac {12}{a^3 x^2}+\frac {16}{a^4 x}-\frac {4}{a^3 (a+x)^2}-\frac {16}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {12 \csc (c+d x)}{a^4 d}-\frac {4 \csc ^2(c+d x)}{a^4 d}+\frac {4 \csc ^3(c+d x)}{3 a^4 d}-\frac {\csc ^4(c+d x)}{4 a^4 d}+\frac {16 \log (\sin (c+d x))}{a^4 d}-\frac {16 \log (1+\sin (c+d x))}{a^4 d}+\frac {4}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 81, normalized size = 0.68 \[ \frac {\frac {48}{\sin (c+d x)+1}-3 \csc ^4(c+d x)+16 \csc ^3(c+d x)-48 \csc ^2(c+d x)+144 \csc (c+d x)+192 \log (\sin (c+d x))-192 \log (\sin (c+d x)+1)}{12 a^4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 235, normalized size = 1.96 \[ \frac {192 \, \cos \left (d x + c\right )^{4} - 352 \, \cos \left (d x + c\right )^{2} + 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 192 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (96 \, \cos \left (d x + c\right )^{2} - 109\right )} \sin \left (d x + c\right ) + 157}{12 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{4} - 2 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 218, normalized size = 1.82 \[ -\frac {\frac {6144 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac {3072 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {1536 \, {\left (6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{2}} + \frac {6400 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 1248 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 204 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 32 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}} + \frac {3 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 32 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 204 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1248 \, a^{12} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{16}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.68, size = 116, normalized size = 0.97 \[ -\frac {1}{4 d \,a^{4} \sin \left (d x +c \right )^{4}}+\frac {4}{3 d \,a^{4} \sin \left (d x +c \right )^{3}}-\frac {4}{d \,a^{4} \sin \left (d x +c \right )^{2}}+\frac {12}{d \,a^{4} \sin \left (d x +c \right )}+\frac {16 \ln \left (\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {4}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )}-\frac {16 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 100, normalized size = 0.83 \[ \frac {\frac {192 \, \sin \left (d x + c\right )^{4} + 96 \, \sin \left (d x + c\right )^{3} - 32 \, \sin \left (d x + c\right )^{2} + 13 \, \sin \left (d x + c\right ) - 3}{a^{4} \sin \left (d x + c\right )^{5} + a^{4} \sin \left (d x + c\right )^{4}} - \frac {192 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {192 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.89, size = 233, normalized size = 1.94 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6\,a^4\,d}-\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a^4\,d}+\frac {16\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {32\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {-24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+191\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {218\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {143\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{12}+\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{6}-\frac {1}{4}}{d\,\left (16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+32\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {13\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^4\,d} \]
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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