Optimal. Leaf size=127 \[ \frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}-\frac {4 a b \sin (c+d x)}{d}-\frac {2 a b \csc (c+d x)}{d}+\frac {b^2 \sin ^4(c+d x)}{4 d} \]
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Rubi [A] time = 0.16, antiderivative size = 127, 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 = {2837, 12, 948} \[ \frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}-\frac {4 a b \sin (c+d x)}{d}-\frac {2 a b \csc (c+d x)}{d}+\frac {b^2 \sin ^4(c+d x)}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 948
Rule 2837
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^3 (a+x)^2 \left (b^2-x^2\right )^2}{x^3} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^2 \left (b^2-x^2\right )^2}{x^3} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-4 a b^2+\frac {a^2 b^4}{x^3}+\frac {2 a b^4}{x^2}+\frac {-2 a^2 b^2+b^4}{x}+\left (a^2-2 b^2\right ) x+2 a x^2+x^3\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=-\frac {2 a b \csc (c+d x)}{d}-\frac {a^2 \csc ^2(c+d x)}{2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{d}-\frac {4 a b \sin (c+d x)}{d}+\frac {\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 d}+\frac {2 a b \sin ^3(c+d x)}{3 d}+\frac {b^2 \sin ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 103, normalized size = 0.81 \[ \frac {6 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+12 \left (b^2-2 a^2\right ) \log (\sin (c+d x))-6 a^2 \csc ^2(c+d x)+8 a b \sin ^3(c+d x)-48 a b \sin (c+d x)-24 a b \csc (c+d x)+3 b^2 \sin ^4(c+d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 160, normalized size = 1.26 \[ \frac {24 \, b^{2} \cos \left (d x + c\right )^{6} - 24 \, {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 9 \, {\left (8 \, a^{2} - 9 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + 24 \, a^{2} + 33 \, b^{2} - 96 \, {\left ({\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} + b^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 64 \, {\left (a b \cos \left (d x + c\right )^{4} + 4 \, a b \cos \left (d x + c\right )^{2} - 8 \, a b\right )} \sin \left (d x + c\right )}{96 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 140, normalized size = 1.10 \[ \frac {3 \, b^{2} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right )^{3} + 6 \, a^{2} \sin \left (d x + c\right )^{2} - 12 \, b^{2} \sin \left (d x + c\right )^{2} - 48 \, a b \sin \left (d x + c\right ) - 12 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + \frac {6 \, {\left (6 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} - 4 \, a b \sin \left (d x + c\right ) - a^{2}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 197, normalized size = 1.55 \[ -\frac {a^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{2 d}-\frac {a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{d}-\frac {2 a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {2 a b \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {16 a b \sin \left (d x +c \right )}{3 d}-\frac {2 a b \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{d}-\frac {8 a b \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{3 d}+\frac {b^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {b^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 104, normalized size = 0.82 \[ \frac {3 \, b^{2} \sin \left (d x + c\right )^{4} + 8 \, a b \sin \left (d x + c\right )^{3} - 48 \, a b \sin \left (d x + c\right ) + 6 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{2} - 12 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) - \frac {6 \, {\left (4 \, a b \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )^{2}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.68, size = 331, normalized size = 2.61 \[ \frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (2\,a^2-b^2\right )}{d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (14\,a^2-16\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {15\,a^2}{2}-16\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (5\,a^2-16\,b^2\right )+\frac {a^2}{2}+48\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {296\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {272\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+36\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-b^2\right )}{d}-\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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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