Optimal. Leaf size=121 \[ \frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos (c+d x)}{d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}-\frac {15 a x}{8} \]
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Rubi [A] time = 0.13, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2838, 2591, 288, 321, 203, 2592, 302, 206} \[ \frac {a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos (c+d x)}{d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}-\frac {15 a x}{8} \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 288
Rule 302
Rule 321
Rule 2591
Rule 2592
Rule 2838
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^5(c+d x) \cot (c+d x) \, dx+a \int \cos ^4(c+d x) \cot ^2(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {x^6}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}-\frac {a \operatorname {Subst}\left (\int \left (-1-x^2-x^4+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 d}\\ &=\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^5(c+d x)}{5 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}-\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {(15 a) \operatorname {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^5(c+d x)}{5 d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}+\frac {(15 a) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=-\frac {15 a x}{8}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \cos (c+d x)}{d}+\frac {a \cos ^3(c+d x)}{3 d}+\frac {a \cos ^5(c+d x)}{5 d}-\frac {15 a \cot (c+d x)}{8 d}+\frac {5 a \cos ^2(c+d x) \cot (c+d x)}{8 d}+\frac {a \cos ^4(c+d x) \cot (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 98, normalized size = 0.81 \[ -\frac {a \left (240 \sin (2 (c+d x))+15 \sin (4 (c+d x))-660 \cos (c+d x)-70 \cos (3 (c+d x))-6 \cos (5 (c+d x))+480 \cot (c+d x)-480 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+900 c+900 d x\right )}{480 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 129, normalized size = 1.07 \[ \frac {30 \, a \cos \left (d x + c\right )^{5} + 75 \, a \cos \left (d x + c\right )^{3} - 60 \, a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 60 \, a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 225 \, a \cos \left (d x + c\right ) + {\left (24 \, a \cos \left (d x + c\right )^{5} + 40 \, a \cos \left (d x + c\right )^{3} - 225 \, a d x + 120 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 198, normalized size = 1.64 \[ -\frac {225 \, {\left (d x + c\right )} a - 120 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 60 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {60 \, {\left (2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (135 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 720 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 1120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 150 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 560 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 135 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 184 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 153, normalized size = 1.26 \[ \frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}+\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \cos \left (d x +c \right )}{d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}-\frac {15 a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}-\frac {15 a x}{8}-\frac {15 c a}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 121, normalized size = 1.00 \[ \frac {4 \, {\left (6 \, \cos \left (d x + c\right )^{5} + 10 \, \cos \left (d x + c\right )^{3} + 30 \, \cos \left (d x + c\right ) - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} a - 15 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 25 \, \tan \left (d x + c\right )^{2} + 8}{\tan \left (d x + c\right )^{5} + 2 \, \tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.92, size = 313, normalized size = 2.59 \[ \frac {\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+12\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-10\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {112\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}-15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {56\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {92\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{15}-a}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}+\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {15\,a\,\mathrm {atan}\left (\frac {225\,a^2}{16\,\left (\frac {15\,a^2}{2}+\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}-\frac {15\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {15\,a^2}{2}+\frac {225\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d} \]
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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