Optimal. Leaf size=72 \[ \frac {5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2872, 3770, 2650, 2648} \[ \frac {5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2872
Rule 3770
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^4 \int \left (\frac {\csc (c+d x)}{a}+\frac {2}{a (-1+\sin (c+d x))^2}-\frac {1}{a (-1+\sin (c+d x))}\right ) \, dx\\ &=a^3 \int \csc (c+d x) \, dx-a^3 \int \frac {1}{-1+\sin (c+d x)} \, dx+\left (2 a^3\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx\\ &=-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {a^3 \cos (c+d x)}{d (1-\sin (c+d x))}-\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx\\ &=-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {5 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 144, normalized size = 2.00 \[ \frac {a^3 (\sin (c+d x)+1)^3 \left (3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right ) (5 \sin (c+d x)-7)}{\left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 231, normalized size = 3.21 \[ -\frac {10 \, a^{3} \cos \left (d x + c\right )^{2} + 14 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3} + 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 3 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3} \cos \left (d x + c\right ) - 2 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (5 \, a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right ) + 2 \, d\right )} \sin \left (d x + c\right ) - 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 73, normalized size = 1.01 \[ \frac {3 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 7 \, a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 115, normalized size = 1.60 \[ \frac {a^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}+\frac {4 a^{3}}{3 d \cos \left (d x +c \right )^{3}}+\frac {2 a^{3} \tan \left (d x +c \right )}{d}+\frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {a^{3}}{d \cos \left (d x +c \right )}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 103, normalized size = 1.43 \[ \frac {2 \, a^{3} \tan \left (d x + c\right )^{3} + 6 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} + a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {6 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.23, size = 98, normalized size = 1.36 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-8\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {14\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )} \]
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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