Optimal. Leaf size=202 \[ -\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac {5 b^2 x}{2} \]
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Rubi [A] time = 0.17, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2722, 2591, 288, 302, 203, 2592, 321, 206, 3473, 8} \[ -\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot (c+d x)}{d}-a^2 x+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}+\frac {5 b^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 203
Rule 206
Rule 288
Rule 302
Rule 321
Rule 2591
Rule 2592
Rule 2722
Rule 3473
Rubi steps
\begin {align*} \int \cot ^6(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \left (b^2 \cos ^2(c+d x) \cot ^4(c+d x)+2 a b \cos (c+d x) \cot ^5(c+d x)+a^2 \cot ^6(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^6(c+d x) \, dx+(2 a b) \int \cos (c+d x) \cot ^5(c+d x) \, dx+b^2 \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx\\ &=-\frac {a^2 \cot ^5(c+d x)}{5 d}-a^2 \int \cot ^4(c+d x) \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+a^2 \int \cot ^2(c+d x) \, dx+\frac {(5 a b) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-\frac {a^2 \cot (c+d x)}{d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-a^2 \int 1 \, dx-\frac {(15 a b) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {\left (5 b^2\right ) \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=-a^2 x+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {(15 a b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {\left (5 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 {5 b^2 x}{2}-\frac {15 a b \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac {15 a b \cos (c+d x)}{4 d}-\frac {a^2 \cot (c+d x)}{d}+\frac {5 b^2 \cot (c+d x)}{2 d}+\frac {5 a b \cos (c+d x) \cot ^2(c+d x)}{4 d}+\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {5 b^2 \cot ^3(c+d x)}{6 d}+\frac {b^2 \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a b \cos (c+d x) \cot ^4(c+d x)}{2 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 1.12, size = 351, normalized size = 1.74 \[ \frac {\left (560 b^2-368 a^2\right ) \cot \left (\frac {1}{2} (c+d x)\right )+368 a^2 \tan \left (\frac {1}{2} (c+d x)\right )-\frac {3}{2} a^2 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+96 a^2 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)+\frac {41}{2} a^2 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-328 a^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)-480 a^2 c-480 a^2 d x+960 a b \cos (c+d x)-15 a b \csc ^4\left (\frac {1}{2} (c+d x)\right )+270 a b \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 a b \sec ^4\left (\frac {1}{2} (c+d x)\right )-270 a b \sec ^2\left (\frac {1}{2} (c+d x)\right )+1800 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1800 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+120 b^2 \sin (2 (c+d x))-560 b^2 \tan \left (\frac {1}{2} (c+d x)\right )-10 b^2 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+160 b^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+1200 b^2 c+1200 b^2 d x}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 306, normalized size = 1.51 \[ -\frac {60 \, b^{2} \cos \left (d x + c\right )^{7} + 92 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 140 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 225 \, {\left (a b \cos \left (d x + c\right )^{4} - 2 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 60 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} \cos \left (d x + c\right ) + 30 \, {\left (2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{4} - 8 \, a b \cos \left (d x + c\right )^{5} - 4 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x \cos \left (d x + c\right )^{2} + 25 \, a b \cos \left (d x + c\right )^{3} + 2 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} d x - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 337, normalized size = 1.67 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1800 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, {\left (2 \, a^{2} - 5 \, b^{2}\right )} {\left (d x + c\right )} - \frac {480 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {4110 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 330 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 20 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 302, normalized size = 1.50 \[ -\frac {a^{2} \left (\cot ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a^{2} \cot \left (d x +c \right )}{d}-a^{2} x -\frac {a^{2} c}{d}-\frac {a b \left (\cos ^{7}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{4}}+\frac {3 a b \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{2}}+\frac {3 a b \left (\cos ^{5}\left (d x +c \right )\right )}{4 d}+\frac {5 a b \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {15 a b \cos \left (d x +c \right )}{4 d}+\frac {15 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{4 d}-\frac {b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {4 b^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}+\frac {4 b^{2} \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{3 d}+\frac {5 b^{2} \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {5 b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {5 b^{2} x}{2}+\frac {5 b^{2} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.12, size = 183, normalized size = 0.91 \[ -\frac {8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} - 5 \, \tan \left (d x + c\right )^{2} + 3}{\tan \left (d x + c\right )^{5}}\right )} a^{2} - 20 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} b^{2} + 15 \, a b {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.28, size = 888, normalized size = 4.40 \[ \text {result too large to display} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot ^{6}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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