Optimal. Leaf size=133 \[ -\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {3 a b \cos (c+d x)}{d}-\frac {a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 b^2 x}{2} \]
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Rubi [A] time = 0.16, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2722, 2591, 288, 321, 203, 2592, 206, 3473, 8} \[ -\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}+a^2 x-\frac {3 a b \cos (c+d x)}{d}-\frac {a b \cos (c+d x) \cot ^2(c+d x)}{d}+\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 b^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 203
Rule 206
Rule 288
Rule 321
Rule 2591
Rule 2592
Rule 2722
Rule 3473
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \left (b^2 \cos ^2(c+d x) \cot ^2(c+d x)+2 a b \cos (c+d x) \cot ^3(c+d x)+a^2 \cot ^4(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \, dx+(2 a b) \int \cos (c+d x) \cot ^3(c+d x) \, dx+b^2 \int \cos ^2(c+d x) \cot ^2(c+d x) \, dx\\ &=-\frac {a^2 \cot ^3(c+d x)}{3 d}-a^2 \int \cot ^2(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot (c+d x)}{d}+\frac {b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+a^2 \int 1 \, dx+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=a^2 x-\frac {3 a b \cos (c+d x)}{d}+\frac {a^2 \cot (c+d x)}{d}-\frac {3 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=a^2 x-\frac {3 b^2 x}{2}+\frac {3 a b \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a b \cos (c+d x)}{d}+\frac {a^2 \cot (c+d x)}{d}-\frac {3 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^2(c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.19, size = 293, normalized size = 2.20 \[ \frac {\left (2 a^2-3 b^2\right ) (c+d x)}{2 d}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (4 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-3 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{6 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (3 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-4 a^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 d}-\frac {a^2 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}+\frac {a^2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}-\frac {2 a b \cos (c+d x)}{d}-\frac {a b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{4 d}+\frac {a b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{4 d}-\frac {3 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {3 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {b^2 \sin (2 (c+d x))}{4 d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 218, normalized size = 1.64 \[ \frac {3 \, b^{2} \cos \left (d x + c\right )^{5} + 4 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 9 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 9 \, {\left (a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 3 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \cos \left (d x + c\right ) + 3 \, {\left ({\left (2 \, a^{2} - 3 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} - 4 \, a b \cos \left (d x + c\right )^{3} - {\left (2 \, a^{2} - 3 \, b^{2}\right )} d x + 6 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 241, normalized size = 1.81 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} {\left (d x + c\right )} + \frac {24 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} + \frac {132 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 199, normalized size = 1.50 \[ -\frac {a^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} \cot \left (d x +c \right )}{d}+a^{2} x +\frac {a^{2} c}{d}-\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )^{2}}-\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{d}-\frac {3 a b \cos \left (d x +c \right )}{d}-\frac {3 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}-\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {3 b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}-\frac {3 b^{2} x}{2}-\frac {3 b^{2} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 138, normalized size = 1.04 \[ \frac {2 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} - 3 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} b^{2} + 3 \, a b {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.75, size = 584, normalized size = 4.39 \[ -\frac {\frac {5\,b^2\,\cos \left (c+d\,x\right )}{16}+\frac {a^2\,\cos \left (3\,c+3\,d\,x\right )}{3}-\frac {11\,b^2\,\cos \left (3\,c+3\,d\,x\right )}{32}+\frac {b^2\,\cos \left (5\,c+5\,d\,x\right )}{32}+\frac {a^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (3\,c+3\,d\,x\right )}{2}-\frac {3\,b^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (3\,c+3\,d\,x\right )}{4}+\frac {3\,a\,b\,\sin \left (c+d\,x\right )}{2}-\frac {3\,a^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (c+d\,x\right )}{2}+\frac {9\,b^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+6\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\sin \left (c+d\,x\right )}{4}+a\,b\,\sin \left (2\,c+2\,d\,x\right )-\frac {a\,b\,\sin \left (3\,c+3\,d\,x\right )}{2}-\frac {a\,b\,\sin \left (4\,c+4\,d\,x\right )}{4}+\frac {9\,a\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {3\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{4}}{d\,{\sin \left (c+d\,x\right )}^3} \]
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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