Optimal. Leaf size=76 \[ a^3 A x-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {a^3 A \cos ^3(c+d x)}{3 d}+\frac {a^3 A \cos (c+d x) \sin (c+d x)}{d} \]
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Rubi [A]
time = 0.07, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3045, 3855,
2715, 8, 2713} \begin {gather*} -\frac {a^3 A \cos ^3(c+d x)}{3 d}+\frac {a^3 A \cos (c+d x)}{d}+\frac {a^3 A \sin (c+d x) \cos (c+d x)}{d}-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+a^3 A x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2713
Rule 2715
Rule 3045
Rule 3855
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx &=\int \left (2 a^3 A+a^3 A \csc (c+d x)-2 a^3 A \sin ^2(c+d x)-a^3 A \sin ^3(c+d x)\right ) \, dx\\ &=2 a^3 A x+\left (a^3 A\right ) \int \csc (c+d x) \, dx-\left (a^3 A\right ) \int \sin ^3(c+d x) \, dx-\left (2 a^3 A\right ) \int \sin ^2(c+d x) \, dx\\ &=2 a^3 A x-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 A \cos (c+d x) \sin (c+d x)}{d}-\left (a^3 A\right ) \int 1 \, dx+\frac {\left (a^3 A\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^3 A x-\frac {a^3 A \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 A \cos (c+d x)}{d}-\frac {a^3 A \cos ^3(c+d x)}{3 d}+\frac {a^3 A \cos (c+d x) \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 74, normalized size = 0.97 \begin {gather*} \frac {a^3 A \left (9 \cos (c+d x)-\cos (3 (c+d x))+6 \left (-2 c+2 d x-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sin (2 (c+d x))\right )\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 88, normalized size = 1.16
method | result | size |
derivativedivides | \(\frac {2 a^{3} A \left (d x +c \right )+a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )-2 a^{3} A \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} A \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(88\) |
default | \(\frac {2 a^{3} A \left (d x +c \right )+a^{3} A \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )-2 a^{3} A \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{3} A \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(88\) |
risch | \(a^{3} x A +\frac {3 a^{3} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 a^{3} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}-\frac {a^{3} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}-\frac {a^{3} \cos \left (3 d x +3 c \right ) A}{12 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{2 d}\) | \(121\) |
norman | \(\frac {a^{3} x A +a^{3} x A \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 a^{3} A}{3 d}+\frac {4 a^{3} A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {16 a^{3} A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 a^{3} A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{3} A \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{3} A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {2 a^{3} A \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+4 a^{3} x A \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 a^{3} x A \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 a^{3} x A \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {a^{3} A \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}\) | \(241\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 85, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a^{3} + 3 \, {\left (2 \, d x + 2 \, c - \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 12 \, {\left (d x + c\right )} A a^{3} + 6 \, A a^{3} \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 92, normalized size = 1.21 \begin {gather*} -\frac {2 \, A a^{3} \cos \left (d x + c\right )^{3} - 6 \, A a^{3} d x - 6 \, A a^{3} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 6 \, A a^{3} \cos \left (d x + c\right ) + 3 \, A a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, A a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{6 \, d} \end {gather*}
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 \begin {gather*} - A a^{3} \left (\int \left (- 2 \sin {\left (c + d x \right )} \csc {\left (c + d x \right )}\right )\, dx + \int 2 \sin ^{3}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \left (- \csc {\left (c + d x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 107, normalized size = 1.41 \begin {gather*} \frac {3 \, {\left (d x + c\right )} A a^{3} + 3 \, A a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 3 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.23, size = 212, normalized size = 2.79 \begin {gather*} \frac {-2\,A\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+4\,A\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,A\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,A\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {2\,A\,a^3\,\mathrm {atan}\left (\frac {4\,A^2\,a^6}{4\,A^2\,a^6-4\,A^2\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {4\,A^2\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,A^2\,a^6-4\,A^2\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {A\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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