Optimal. Leaf size=126 \[ \frac {\left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 d}+\frac {\left (a^2-2 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \csc ^4(c+d x)}{4 d}+\frac {2 a b \sin (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{3 d}+\frac {4 a b \csc (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 126, 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 = {2721, 948} \[ \frac {\left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 d}+\frac {\left (a^2-2 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \csc ^4(c+d x)}{4 d}+\frac {2 a b \sin (c+d x)}{d}-\frac {2 a b \csc ^3(c+d x)}{3 d}+\frac {4 a b \csc (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 948
Rule 2721
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^5} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a+\frac {a^2 b^4}{x^5}+\frac {2 a b^4}{x^4}+\frac {-2 a^2 b^2+b^4}{x^3}-\frac {4 a b^2}{x^2}+\frac {a^2-2 b^2}{x}+x\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {4 a b \csc (c+d x)}{d}+\frac {\left (2 a^2-b^2\right ) \csc ^2(c+d x)}{2 d}-\frac {2 a b \csc ^3(c+d x)}{3 d}-\frac {a^2 \csc ^4(c+d x)}{4 d}+\frac {\left (a^2-2 b^2\right ) \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.73, size = 107, normalized size = 0.85 \[ \frac {6 \left (2 a^2-b^2\right ) \csc ^2(c+d x)+6 \left (2 \left (a^2-2 b^2\right ) \log (\sin (c+d x))+4 a b \sin (c+d x)+b^2 \sin ^2(c+d x)\right )-3 a^2 \csc ^4(c+d x)-8 a b \csc ^3(c+d x)+48 a b \csc (c+d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 177, normalized size = 1.40 \[ -\frac {6 \, b^{2} \cos \left (d x + c\right )^{6} - 15 \, b^{2} \cos \left (d x + c\right )^{4} + 6 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 9 \, a^{2} + 3 \, b^{2} - 12 \, {\left ({\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} - 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 8 \, {\left (3 \, a b \cos \left (d x + c\right )^{4} - 12 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.87, size = 138, normalized size = 1.10 \[ \frac {6 \, b^{2} \sin \left (d x + c\right )^{2} + 24 \, a b \sin \left (d x + c\right ) + 12 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {25 \, a^{2} \sin \left (d x + c\right )^{4} - 50 \, b^{2} \sin \left (d x + c\right )^{4} - 48 \, a b \sin \left (d x + c\right )^{3} - 12 \, a^{2} \sin \left (d x + c\right )^{2} + 6 \, b^{2} \sin \left (d x + c\right )^{2} + 8 \, a b \sin \left (d x + c\right ) + 3 \, a^{2}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 220, normalized size = 1.75 \[ -\frac {a^{2} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {2 a b \left (\cos ^{6}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {2 a b \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}+\frac {16 a b \sin \left (d x +c \right )}{3 d}+\frac {2 a b \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{d}+\frac {8 a b \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}-\frac {b^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{2 d}-\frac {b^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{d}-\frac {2 b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 105, normalized size = 0.83 \[ \frac {6 \, b^{2} \sin \left (d x + c\right )^{2} + 24 \, a b \sin \left (d x + c\right ) + 12 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + \frac {48 \, a b \sin \left (d x + c\right )^{3} - 8 \, a b \sin \left (d x + c\right ) + 6 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - 3 \, a^{2}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.44, size = 310, normalized size = 2.46 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^2}{2}-2\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {23\,a^2}{4}-4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (3\,a^2+30\,b^2\right )-\frac {a^2}{4}+\frac {76\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {356\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+92\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (a^2-2\,b^2\right )}{d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2-2\,b^2\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^2}{16}-\frac {b^2}{8}\right )}{d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,d}+\frac {7\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,d} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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