Optimal. Leaf size=48 \[ \frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x) \]
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Rubi [A] time = 0.10, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2872, 3770, 2648} \[ \frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+a^3 (-x) \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2872
Rule 3770
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^2 \int \left (-a+a \csc (c+d x)-\frac {4 a}{-1+\sin (c+d x)}\right ) \, dx\\ &=-a^3 x+a^3 \int \csc (c+d x) \, dx-\left (4 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx\\ &=-a^3 x-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {4 a^3 \cos (c+d x)}{d (1-\sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 74, normalized size = 1.54 \[ -\frac {a^3 \left (-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {8 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+c+d x\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 151, normalized size = 3.15 \[ -\frac {2 \, a^{3} d x - 8 \, a^{3} + 2 \, {\left (a^{3} d x - 4 \, a^{3}\right )} \cos \left (d x + c\right ) + {\left (a^{3} \cos \left (d x + c\right ) - a^{3} \sin \left (d x + c\right ) + a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (a^{3} \cos \left (d x + c\right ) - a^{3} \sin \left (d x + c\right ) + a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (a^{3} d x + 4 \, a^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (d \cos \left (d x + c\right ) - d \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 49, normalized size = 1.02 \[ -\frac {{\left (d x + c\right )} a^{3} - a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {8 \, a^{3}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 70, normalized size = 1.46 \[ -a^{3} x +\frac {4 a^{3} \tan \left (d x +c \right )}{d}-\frac {a^{3} c}{d}+\frac {4 a^{3}}{d \cos \left (d x +c \right )}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 84, normalized size = 1.75 \[ -\frac {2 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{3} - a^{3} {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 6 \, a^{3} \tan \left (d x + c\right ) - \frac {6 \, a^{3}}{\cos \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.93, size = 112, normalized size = 2.33 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {2\,a^3\,\mathrm {atan}\left (\frac {4\,a^6}{4\,a^6+4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,a^6+4\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {8\,a^3}{d\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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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