Optimal. Leaf size=128 \[ -\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {6 a^3 \cot (c+d x)}{d}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]
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Rubi [A] time = 0.21, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2872, 3770, 3767, 8, 3768, 2650, 2648} \[ -\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {6 a^3 \cot (c+d x)}{d}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2650
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^4(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^4 \int \left (\frac {7 \csc (c+d x)}{a}+\frac {5 \csc ^2(c+d x)}{a}+\frac {3 \csc ^3(c+d x)}{a}+\frac {\csc ^4(c+d x)}{a}+\frac {2}{a (-1+\sin (c+d x))^2}-\frac {7}{a (-1+\sin (c+d x))}\right ) \, dx\\ &=a^3 \int \csc ^4(c+d x) \, dx+\left (2 a^3\right ) \int \frac {1}{(-1+\sin (c+d x))^2} \, dx+\left (3 a^3\right ) \int \csc ^3(c+d x) \, dx+\left (5 a^3\right ) \int \csc ^2(c+d x) \, dx+\left (7 a^3\right ) \int \csc (c+d x) \, dx-\left (7 a^3\right ) \int \frac {1}{-1+\sin (c+d x)} \, dx\\ &=-\frac {7 a^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {7 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 {1}{2} \left (3 a^3\right ) \int \csc (c+d x) \, dx-\frac {a^3 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (5 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{d}\\ &=-\frac {17 a^3 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {6 a^3 \cot (c+d x)}{d}-\frac {a^3 \cot ^3(c+d x)}{3 d}-\frac {3 a^3 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {2 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))^2}+\frac {23 a^3 \cos (c+d x)}{3 d (1-\sin (c+d x))}\\ \end {align*}
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Mathematica [B] time = 6.18, size = 287, normalized size = 2.24 \[ a^3 \left (\frac {17 \tan \left (\frac {1}{2} (c+d x)\right )}{6 d}-\frac {17 \cot \left (\frac {1}{2} (c+d x)\right )}{6 d}-\frac {3 \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {3 \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {17 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {17 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {46 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.50, size = 528, normalized size = 4.12 \[ \frac {160 \, a^{3} \cos \left (d x + c\right )^{5} - 58 \, a^{3} \cos \left (d x + c\right )^{4} - 356 \, a^{3} \cos \left (d x + c\right )^{3} + 70 \, a^{3} \cos \left (d x + c\right )^{2} + 200 \, a^{3} \cos \left (d x + c\right ) - 8 \, a^{3} - 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, 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 )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 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 ) + 51 \, {\left (a^{3} \cos \left (d x + c\right )^{5} + 2 \, a^{3} \cos \left (d x + c\right )^{4} - 2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, 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 )^{4} - a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 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 (80 \, a^{3} \cos \left (d x + c\right )^{4} + 109 \, a^{3} \cos \left (d x + c\right )^{3} - 69 \, a^{3} \cos \left (d x + c\right )^{2} - 104 \, a^{3} \cos \left (d x + c\right ) - 4 \, a^{3}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{5} + 2 \, d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{3} - 4 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 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 = 0.31, size = 194, normalized size = 1.52 \[ \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 204 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 69 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {187 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 60 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 405 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 394 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.70, size = 214, normalized size = 1.67 \[ \frac {a^{3}}{3 d \cos \left (d x +c \right )^{3}}+\frac {17 a^{3}}{2 d \cos \left (d x +c \right )}+\frac {17 a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}+\frac {a^{3}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {20 a^{3}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {40 a^{3} \cot \left (d x +c \right )}{3 d}+\frac {a^{3}}{d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5 a^{3}}{2 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {a^{3}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3}}-\frac {2 a^{3}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 205, normalized size = 1.60 \[ \frac {12 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{3} + 4 \, {\left (\tan \left (d x + c\right )^{3} - \frac {9 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} + 9 \, \tan \left (d x + c\right )\right )} a^{3} + 3 \, a^{3} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.41, size = 239, normalized size = 1.87 \[ \frac {a^3\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-581\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+897\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-303\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-181\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-612\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+204\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+1\right )}{24\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}^3} \]
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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