Optimal. Leaf size=130 \[ \frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}-\frac {\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}+\frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin ^5(c+d x)}{5 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^6(c+d x)}{6 d} \]
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Rubi [A] time = 0.13, antiderivative size = 130, 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 = {2837, 12, 948} \[ \frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}-\frac {\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}+\frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin ^5(c+d x)}{5 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {2 a b \sin (c+d x)}{d}+\frac {b^2 \sin ^6(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 948
Rule 2837
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \cot (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b (a+x)^2 \left (b^2-x^2\right )^2}{x} \, 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} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a b^4+\frac {a^2 b^4}{x}-b^2 \left (2 a^2-b^2\right ) x-4 a b^2 x^2+\left (a^2-2 b^2\right ) x^3+2 a x^4+x^5\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a b \sin (c+d x)}{d}-\frac {\left (2 a^2-b^2\right ) \sin ^2(c+d x)}{2 d}-\frac {4 a b \sin ^3(c+d x)}{3 d}+\frac {\left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 d}+\frac {2 a b \sin ^5(c+d x)}{5 d}+\frac {b^2 \sin ^6(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 105, normalized size = 0.81 \[ \frac {15 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+30 \left (b^2-2 a^2\right ) \sin ^2(c+d x)+60 a^2 \log (\sin (c+d x))+24 a b \sin ^5(c+d x)-80 a b \sin ^3(c+d x)+120 a b \sin (c+d x)+10 b^2 \sin ^6(c+d x)}{60 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 96, normalized size = 0.74 \[ -\frac {10 \, b^{2} \cos \left (d x + c\right )^{6} - 15 \, a^{2} \cos \left (d x + c\right )^{4} - 30 \, a^{2} \cos \left (d x + c\right )^{2} - 60 \, a^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 8 \, {\left (3 \, a b \cos \left (d x + c\right )^{4} + 4 \, a b \cos \left (d x + c\right )^{2} + 8 \, a b\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 118, normalized size = 0.91 \[ \frac {10 \, b^{2} \sin \left (d x + c\right )^{6} + 24 \, a b \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} - 30 \, b^{2} \sin \left (d x + c\right )^{4} - 80 \, a b \sin \left (d x + c\right )^{3} - 60 \, a^{2} \sin \left (d x + c\right )^{2} + 30 \, b^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 120 \, a b \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 119, normalized size = 0.92 \[ \frac {a^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {16 a b \sin \left (d x +c \right )}{15 d}+\frac {2 a b \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {8 a b \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) b^{2}}{6 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 105, normalized size = 0.81 \[ \frac {10 \, b^{2} \sin \left (d x + c\right )^{6} + 24 \, a b \sin \left (d x + c\right )^{5} - 80 \, a b \sin \left (d x + c\right )^{3} + 15 \, {\left (a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )^{4} + 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 120 \, a b \sin \left (d x + c\right ) - 30 \, {\left (2 \, a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.74, size = 153, normalized size = 1.18 \[ \frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a^2\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^2}{2\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^4}{4\,d}-\frac {b^2\,{\cos \left (c+d\,x\right )}^6}{6\,d}+\frac {16\,a\,b\,\sin \left (c+d\,x\right )}{15\,d}+\frac {8\,a\,b\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{15\,d}+\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{5\,d} \]
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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