Optimal. Leaf size=117 \[ \frac {a^4 \cos ^5(c+d x)}{5 d}-\frac {7 a^4 \cos ^3(c+d x)}{3 d}+\frac {a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac {5 a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {5 a^4 x}{2} \]
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Rubi [A] time = 0.20, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2873, 2635, 8, 2592, 321, 206, 2565, 30, 2568, 14} \[ \frac {a^4 \cos ^5(c+d x)}{5 d}-\frac {7 a^4 \cos ^3(c+d x)}{3 d}+\frac {a^4 \cos (c+d x)}{d}-\frac {a^4 \sin (c+d x) \cos ^3(c+d x)}{d}+\frac {5 a^4 \sin (c+d x) \cos (c+d x)}{2 d}-\frac {a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {5 a^4 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 14
Rule 30
Rule 206
Rule 321
Rule 2565
Rule 2568
Rule 2592
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^4 \, dx &=\int \left (4 a^4 \cos ^2(c+d x)+a^4 \cos (c+d x) \cot (c+d x)+6 a^4 \cos ^2(c+d x) \sin (c+d x)+4 a^4 \cos ^2(c+d x) \sin ^2(c+d x)+a^4 \cos ^2(c+d x) \sin ^3(c+d x)\right ) \, dx\\ &=a^4 \int \cos (c+d x) \cot (c+d x) \, dx+a^4 \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^2(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^2(c+d x) \sin (c+d x) \, dx\\ &=\frac {2 a^4 \cos (c+d x) \sin (c+d x)}{d}-\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}+a^4 \int \cos ^2(c+d x) \, dx+\left (2 a^4\right ) \int 1 \, dx-\frac {a^4 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^4 \operatorname {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (6 a^4\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=2 a^4 x+\frac {a^4 \cos (c+d x)}{d}-\frac {2 a^4 \cos ^3(c+d x)}{d}+\frac {5 a^4 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}+\frac {1}{2} a^4 \int 1 \, dx-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^4 \operatorname {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {5 a^4 x}{2}-\frac {a^4 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^4 \cos (c+d x)}{d}-\frac {7 a^4 \cos ^3(c+d x)}{3 d}+\frac {a^4 \cos ^5(c+d x)}{5 d}+\frac {5 a^4 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a^4 \cos ^3(c+d x) \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 95, normalized size = 0.81 \[ \frac {a^4 \left (-150 \cos (c+d x)-125 \cos (3 (c+d x))+3 \cos (5 (c+d x))+30 \left (8 \sin (2 (c+d x))-\sin (4 (c+d x))+8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+20 c+20 d x\right )\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 115, normalized size = 0.98 \[ \frac {6 \, a^{4} \cos \left (d x + c\right )^{5} - 70 \, a^{4} \cos \left (d x + c\right )^{3} + 75 \, a^{4} d x + 30 \, a^{4} \cos \left (d x + c\right ) - 15 \, a^{4} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, a^{4} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 15 \, {\left (2 \, a^{4} \cos \left (d x + c\right )^{3} - 5 \, a^{4} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 181, normalized size = 1.55 \[ \frac {75 \, {\left (d x + c\right )} a^{4} + 30 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 150 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 210 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 300 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 40 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 210 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 20 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 34 \, a^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 135, normalized size = 1.15 \[ -\frac {a^{4} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5 d}-\frac {32 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d}-\frac {a^{4} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}+\frac {5 a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {5 a^{4} x}{2}+\frac {5 a^{4} c}{2 d}+\frac {a^{4} \cos \left (d x +c \right )}{d}+\frac {a^{4} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 125, normalized size = 1.07 \[ -\frac {240 \, a^{4} \cos \left (d x + c\right )^{3} - 8 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{4} - 15 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{4} - 60 \, a^{4} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.42, size = 295, normalized size = 2.52 \[ \frac {a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {5\,a^4\,\mathrm {atan}\left (\frac {25\,a^8}{10\,a^8-25\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {10\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{10\,a^8-25\,a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {3\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+10\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+14\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+20\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {8\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-14\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-3\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {34\,a^4}{15}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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