Optimal. Leaf size=133 \[ -\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 90}
\begin {gather*} \frac {a^3 \sin ^5(c+d x)}{5 d}+\frac {3 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}-\frac {5 a^3 \sin (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int \frac {a^3 (a-x)^2 (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {(a-x)^2 (a+x)^5}{x^3} \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=\frac {\text {Subst}\left (\int \left (-5 a^4+\frac {a^7}{x^3}+\frac {3 a^6}{x^2}+\frac {a^5}{x}-5 a^3 x+a^2 x^2+3 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^2 d}\\ &=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}+\frac {a^3 \log (\sin (c+d x))}{d}-\frac {5 a^3 \sin (c+d x)}{d}-\frac {5 a^3 \sin ^2(c+d x)}{2 d}+\frac {a^3 \sin ^3(c+d x)}{3 d}+\frac {3 a^3 \sin ^4(c+d x)}{4 d}+\frac {a^3 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 86, normalized size = 0.65 \begin {gather*} -\frac {a^3 \left (180 \csc (c+d x)+30 \csc ^2(c+d x)-60 \log (\sin (c+d x))+300 \sin (c+d x)+150 \sin ^2(c+d x)-20 \sin ^3(c+d x)-45 \sin ^4(c+d x)-12 \sin ^5(c+d x)\right )}{60 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 174, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+\frac {a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(174\) |
default | \(\frac {a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a^{3} \left (\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cos ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+\frac {a^{3} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(174\) |
risch | \(-i a^{3} x +\frac {7 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{96 d}+\frac {7 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}+\frac {37 i a^{3} {\mathrm e}^{i \left (d x +c \right )}}{16 d}-\frac {37 i a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {7 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 d}-\frac {7 i a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{96 d}-\frac {2 i a^{3} c}{d}-\frac {2 i a^{3} \left (i {\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}+\frac {a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} \cos \left (4 d x +4 c \right )}{32 d}\) | \(235\) |
norman | \(\frac {-\frac {a^{3}}{8 d}-\frac {3 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {19 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {359 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {1174 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {359 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {19 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {3 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {33 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {33 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {109 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {109 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {a^{3} \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(303\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 106, normalized size = 0.80 \begin {gather*} \frac {12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac {30 \, {\left (6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 145, normalized size = 1.09 \begin {gather*} \frac {360 \, a^{3} \cos \left (d x + c\right )^{6} + 120 \, a^{3} \cos \left (d x + c\right )^{4} - 855 \, a^{3} \cos \left (d x + c\right )^{2} + 615 \, a^{3} + 480 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 32 \, {\left (3 \, a^{3} \cos \left (d x + c\right )^{6} - 14 \, a^{3} \cos \left (d x + c\right )^{4} - 56 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.54, size = 120, normalized size = 0.90 \begin {gather*} \frac {12 \, a^{3} \sin \left (d x + c\right )^{5} + 45 \, a^{3} \sin \left (d x + c\right )^{4} + 20 \, a^{3} \sin \left (d x + c\right )^{3} - 150 \, a^{3} \sin \left (d x + c\right )^{2} + 60 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 300 \, a^{3} \sin \left (d x + c\right ) - \frac {30 \, {\left (3 \, a^{3} \sin \left (d x + c\right )^{2} + 6 \, a^{3} \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.01, size = 342, normalized size = 2.57 \begin {gather*} \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {46\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\frac {81\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{2}+\frac {538\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3}+\frac {149\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{2}+\frac {3796\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}+77\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {628\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{3}+45\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+70\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+6\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+40\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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