Optimal. Leaf size=97 \[ -\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {2 \sin (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {2 \log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.12, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2836, 12, 88} \[ -\frac {\sin ^3(c+d x)}{3 a d}+\frac {\sin ^2(c+d x)}{2 a d}+\frac {2 \sin (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\csc (c+d x)}{a d}-\frac {2 \log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (a-x)^3 (a+x)^2}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a^2+\frac {a^5}{x^3}-\frac {a^4}{x^2}-\frac {2 a^3}{x}+a x-x^2\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\csc (c+d x)}{a d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {2 \log (\sin (c+d x))}{a d}+\frac {2 \sin (c+d x)}{a d}+\frac {\sin ^2(c+d x)}{2 a d}-\frac {\sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 66, normalized size = 0.68 \[ \frac {-2 \sin ^3(c+d x)+3 \sin ^2(c+d x)+12 \sin (c+d x)-3 \csc ^2(c+d x)+6 \csc (c+d x)-12 \log (\sin (c+d x))}{6 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 91, normalized size = 0.94 \[ -\frac {6 \, \cos \left (d x + c\right )^{4} - 9 \, \cos \left (d x + c\right )^{2} + 24 \, {\left (\cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (\cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 3}{12 \, {\left (a d \cos \left (d x + c\right )^{2} - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 94, normalized size = 0.97 \[ -\frac {\frac {12 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{3} - 3 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, a^{2} \sin \left (d x + c\right )}{a^{3}} - \frac {3 \, {\left (6 \, \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 94, normalized size = 0.97 \[ -\frac {\sin ^{3}\left (d x +c \right )}{3 d a}+\frac {\sin ^{2}\left (d x +c \right )}{2 a d}+\frac {2 \sin \left (d x +c \right )}{a d}+\frac {1}{d a \sin \left (d x +c \right )}-\frac {2 \ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {1}{2 a d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 74, normalized size = 0.76 \[ -\frac {\frac {2 \, \sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right )}{a} + \frac {12 \, \log \left (\sin \left (d x + c\right )\right )}{a} - \frac {3 \, {\left (2 \, \sin \left (d x + c\right ) - 1\right )}}{a \sin \left (d x + c\right )^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.23, size = 231, normalized size = 2.38 \[ \frac {18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{2}+\frac {82\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {13\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+22\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\frac {1}{2}}{d\,\left (4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a\,d}+\frac {2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a\,d} \]
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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