Optimal. Leaf size=131 \[ \frac {6}{d \left (a^4 \sin (c+d x)+a^4\right )}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {4 \csc (c+d x)}{a^4 d}+\frac {10 \log (\sin (c+d x))}{a^4 d}-\frac {10 \log (\sin (c+d x)+1)}{a^4 d}+\frac {3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
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Rubi [A] time = 0.12, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2833, 12, 44} \[ \frac {6}{d \left (a^4 \sin (c+d x)+a^4\right )}+\frac {3}{2 d \left (a^2 \sin (c+d x)+a^2\right )^2}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {4 \csc (c+d x)}{a^4 d}+\frac {10 \log (\sin (c+d x))}{a^4 d}-\frac {10 \log (\sin (c+d x)+1)}{a^4 d}+\frac {1}{3 a d (a \sin (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2833
Rubi steps
\begin {align*} \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3}{x^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (\frac {1}{a^4 x^3}-\frac {4}{a^5 x^2}+\frac {10}{a^6 x}-\frac {1}{a^3 (a+x)^4}-\frac {3}{a^4 (a+x)^3}-\frac {6}{a^5 (a+x)^2}-\frac {10}{a^6 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {4 \csc (c+d x)}{a^4 d}-\frac {\csc ^2(c+d x)}{2 a^4 d}+\frac {10 \log (\sin (c+d x))}{a^4 d}-\frac {10 \log (1+\sin (c+d x))}{a^4 d}+\frac {1}{3 a d (a+a \sin (c+d x))^3}+\frac {3}{2 d \left (a^2+a^2 \sin (c+d x)\right )^2}+\frac {6}{d \left (a^4+a^4 \sin (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 3.42, size = 85, normalized size = 0.65 \[ \frac {\frac {36}{\sin (c+d x)+1}+\frac {9}{(\sin (c+d x)+1)^2}+\frac {2}{(\sin (c+d x)+1)^3}-3 \csc ^2(c+d x)+24 \csc (c+d x)+60 \log (\sin (c+d x))-60 \log (\sin (c+d x)+1)}{6 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 242, normalized size = 1.85 \[ \frac {60 \, \cos \left (d x + c\right )^{4} - 230 \, \cos \left (d x + c\right )^{2} + 60 \, {\left (3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 60 \, {\left (3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{4} - 5 \, \cos \left (d x + c\right )^{2} + 4\right )} \sin \left (d x + c\right ) + 4\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (10 \, \cos \left (d x + c\right )^{2} - 11\right )} \sin \left (d x + c\right ) + 167}{6 \, {\left (3 \, a^{4} d \cos \left (d x + c\right )^{4} - 7 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d + {\left (a^{4} d \cos \left (d x + c\right )^{4} - 5 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 97, normalized size = 0.74 \[ -\frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4}} - \frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {60 \, \sin \left (d x + c\right )^{4} + 150 \, \sin \left (d x + c\right )^{3} + 110 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 3}{a^{4} {\left (\sin \left (d x + c\right ) + 1\right )}^{3} \sin \left (d x + c\right )^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 120, normalized size = 0.92 \[ -\frac {1}{2 a^{4} d \sin \left (d x +c \right )^{2}}+\frac {4}{a^{4} d \sin \left (d x +c \right )}+\frac {10 \ln \left (\sin \left (d x +c \right )\right )}{a^{4} d}+\frac {1}{3 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{3}}+\frac {3}{2 d \,a^{4} \left (1+\sin \left (d x +c \right )\right )^{2}}+\frac {6}{d \,a^{4} \left (1+\sin \left (d x +c \right )\right )}-\frac {10 \ln \left (1+\sin \left (d x +c \right )\right )}{a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 126, normalized size = 0.96 \[ \frac {\frac {60 \, \sin \left (d x + c\right )^{4} + 150 \, \sin \left (d x + c\right )^{3} + 110 \, \sin \left (d x + c\right )^{2} + 15 \, \sin \left (d x + c\right ) - 3}{a^{4} \sin \left (d x + c\right )^{5} + 3 \, a^{4} \sin \left (d x + c\right )^{4} + 3 \, a^{4} \sin \left (d x + c\right )^{3} + a^{4} \sin \left (d x + c\right )^{2}} - \frac {60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} + \frac {60 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.69, size = 286, normalized size = 2.18 \[ \frac {10\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^4\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^4\,d}-\frac {72\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {465\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {881\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {255\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-30\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {81\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {1}{2}}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+60\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+80\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+60\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {20\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{a^4\,d}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d} \]
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 \[ \frac {\int \frac {\cos {\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\sin ^{4}{\left (c + d x \right )} + 4 \sin ^{3}{\left (c + d x \right )} + 6 \sin ^{2}{\left (c + d x \right )} + 4 \sin {\left (c + d x \right )} + 1}\, dx}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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