Optimal. Leaf size=125 \[ \frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}-\frac {16 a^2 \cot (c+d x)}{3 d}-\frac {7 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))} \]
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Rubi [A] time = 0.31, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2869, 2766, 2978, 2748, 3768, 3770, 3767, 8} \[ -\frac {16 a^2 \cot (c+d x)}{3 d}-\frac {7 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2766
Rule 2869
Rule 2978
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x))^2 \, dx &=a^4 \int \frac {\csc ^3(c+d x)}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} a^2 \int \frac {\csc ^3(c+d x) (5 a+3 a \sin (c+d x))}{a-a \sin (c+d x)} \, dx\\ &=\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} \int \csc ^3(c+d x) \left (21 a^2+16 a^2 \sin (c+d x)\right ) \, dx\\ &=\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{3} \left (16 a^2\right ) \int \csc ^2(c+d x) \, dx+\left (7 a^2\right ) \int \csc ^3(c+d x) \, dx\\ &=-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}+\frac {1}{2} \left (7 a^2\right ) \int \csc (c+d x) \, dx-\frac {\left (16 a^2\right ) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=-\frac {7 a^2 \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {16 a^2 \cot (c+d x)}{3 d}-\frac {7 a^2 \cot (c+d x) \csc (c+d x)}{2 d}+\frac {8 a^2 \cot (c+d x) \csc (c+d x)}{3 d (1-\sin (c+d x))}+\frac {a^4 \cot (c+d x) \csc (c+d x)}{3 d (a-a \sin (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 1.99, size = 190, normalized size = 1.52 \[ \frac {a^2 \left (24 \tan \left (\frac {1}{2} (c+d x)\right )-24 \cot \left (\frac {1}{2} (c+d x)\right )-3 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+84 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-84 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\frac {160 \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {16 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {8}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 428, normalized size = 3.42 \[ -\frac {64 \, a^{2} \cos \left (d x + c\right )^{4} + 86 \, a^{2} \cos \left (d x + c\right )^{3} - 54 \, a^{2} \cos \left (d x + c\right )^{2} - 80 \, a^{2} \cos \left (d x + c\right ) - 4 \, a^{2} + 21 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 21 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right ) + 2 \, a^{2} + {\left (a^{2} \cos \left (d x + c\right )^{3} + 2 \, a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - 2 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (32 \, a^{2} \cos \left (d x + c\right )^{3} - 11 \, a^{2} \cos \left (d x + c\right )^{2} - 38 \, a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - 2 \, d\right )} \sin \left (d x + c\right ) + 2 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 150, normalized size = 1.20 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 84 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {3 \, {\left (42 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {16 \, {\left (12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 21 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 11 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 168, normalized size = 1.34 \[ \frac {a^{2}}{3 d \cos \left (d x +c \right )^{3}}+\frac {7 a^{2}}{2 d \cos \left (d x +c \right )}+\frac {7 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}+\frac {2 a^{2}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {8 a^{2}}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {16 a^{2} \cot \left (d x +c \right )}{3 d}+\frac {a^{2}}{3 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5 a^{2}}{6 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 160, normalized size = 1.28 \[ \frac {8 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a^{2} + a^{2} {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )}}{\cos \left (d x + c\right )^{3}} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.07, size = 182, normalized size = 1.46 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {7\,a^2\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d}-\frac {-36\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {135\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {239\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6}+\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+\frac {a^2}{2}}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d} \]
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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