Optimal. Leaf size=134 \[ \frac {15 a \cos (c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {5 a x}{2} \]
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Rubi [A] time = 0.13, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2838, 2592, 288, 321, 206, 2591, 302, 203} \[ \frac {15 a \cos (c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {5 a x}{2} \]
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 (c+d x) \cot ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cos ^2(c+d x) \cot ^4(c+d x) \, dx+a \int \cos (c+d x) \cot ^5(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^3} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{4 d}-\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac {(15 a) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}-\frac {(5 a) \operatorname {Subst}\left (\int \left (-1+x^2+\frac {1}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac {15 a \cos (c+d x)}{8 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}-\frac {(15 a) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{8 d}-\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}\\ &=\frac {5 a x}{2}-\frac {15 a \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {15 a \cos (c+d x)}{8 d}+\frac {5 a \cot (c+d x)}{2 d}+\frac {5 a \cos (c+d x) \cot ^2(c+d x)}{8 d}-\frac {5 a \cot ^3(c+d x)}{6 d}+\frac {a \cos ^2(c+d x) \cot ^3(c+d x)}{2 d}-\frac {a \cos (c+d x) \cot ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 1.43, size = 138, normalized size = 1.03 \[ \frac {a \left (192 \cos (c+d x)-64 \cot (c+d x) \left (\csc ^2(c+d x)-7\right )+3 \left (16 \sin (2 (c+d x))-\csc ^4\left (\frac {1}{2} (c+d x)\right )+18 \csc ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )-18 \sec ^2\left (\frac {1}{2} (c+d x)\right )+40 \left (3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+4 c+4 d x\right )\right )\right )}{192 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 202, normalized size = 1.51 \[ \frac {120 \, a d x \cos \left (d x + c\right )^{4} + 48 \, a \cos \left (d x + c\right )^{5} - 240 \, a d x \cos \left (d x + c\right )^{2} - 150 \, a \cos \left (d x + c\right )^{3} + 120 \, a d x + 90 \, a \cos \left (d x + c\right ) - 45 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 45 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 8 \, {\left (3 \, a \cos \left (d x + c\right )^{5} - 20 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 213, normalized size = 1.59 \[ \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, {\left (d x + c\right )} a + 360 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 216 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {192 \, {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {750 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 216 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 221, normalized size = 1.65 \[ -\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}+\frac {4 a \left (\cos ^{7}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )}+\frac {4 a \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {5 a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {5 a x}{2}+\frac {5 c a}{2 d}-\frac {a \left (\cos ^{7}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {3 a \left (\cos ^{7}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {3 a \left (\cos ^{5}\left (d x +c \right )\right )}{8 d}+\frac {5 a \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {15 a \cos \left (d x +c \right )}{8 d}+\frac {15 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 136, normalized size = 1.01 \[ \frac {8 \, {\left (15 \, d x + 15 \, c + \frac {15 \, \tan \left (d x + c\right )^{4} + 10 \, \tan \left (d x + c\right )^{2} - 2}{\tan \left (d x + c\right )^{5} + \tan \left (d x + c\right )^{3}}\right )} a - 3 \, a {\left (\frac {2 \, {\left (9 \, \cos \left (d x + c\right )^{3} - 7 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - 16 \, \cos \left (d x + c\right ) + 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.82, size = 300, normalized size = 2.24 \[ \frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {9\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}+\frac {15\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+36\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {154\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+\frac {159\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{4}+\frac {50\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}-\frac {a}{4}}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {5\,a\,\mathrm {atan}\left (\frac {25\,a^2}{\frac {75\,a^2}{4}-25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {75\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,\left (\frac {75\,a^2}{4}-25\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}\right )}{d} \]
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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