Optimal. Leaf size=110 \[ \frac {3 a^3 \tan (c+d x)}{d}+\frac {11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.23, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3872, 2872, 2650, 2648, 3770, 3767, 8, 3768} \[ \frac {3 a^3 \tan (c+d x)}{d}+\frac {11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}+\frac {a^3 \tan (c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2648
Rule 2650
Rule 2872
Rule 3767
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\int (-a-a \cos (c+d x))^3 \csc ^4(c+d x) \sec ^3(c+d x) \, dx\\ &=a^4 \int \left (\frac {2}{a (1-\cos (c+d x))^2}+\frac {5}{a (1-\cos (c+d x))}+\frac {5 \sec (c+d x)}{a}+\frac {3 \sec ^2(c+d x)}{a}+\frac {\sec ^3(c+d x)}{a}\right ) \, dx\\ &=a^3 \int \sec ^3(c+d x) \, dx+\left (2 a^3\right ) \int \frac {1}{(1-\cos (c+d x))^2} \, dx+\left (3 a^3\right ) \int \sec ^2(c+d x) \, dx+\left (5 a^3\right ) \int \frac {1}{1-\cos (c+d x)} \, dx+\left (5 a^3\right ) \int \sec (c+d x) \, dx\\ &=\frac {5 a^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}-\frac {5 a^3 \sin (c+d x)}{d (1-\cos (c+d x))}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} a^3 \int \sec (c+d x) \, dx+\frac {1}{3} \left (2 a^3\right ) \int \frac {1}{1-\cos (c+d x)} \, dx-\frac {\left (3 a^3\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {11 a^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac {2 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))^2}-\frac {17 a^3 \sin (c+d x)}{3 d (1-\cos (c+d x))}+\frac {3 a^3 \tan (c+d x)}{d}+\frac {a^3 \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [B] time = 6.25, size = 678, normalized size = 6.16 \[ \frac {3 \sin \left (\frac {d x}{2}\right ) \cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{8 d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {3 \sin \left (\frac {d x}{2}\right ) \cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{8 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {\cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{32 d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {\cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{32 d \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {11 \cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3 \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{16 d}+\frac {11 \cos ^3(c+d x) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3 \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{16 d}-\frac {\cot \left (\frac {c}{2}\right ) \cos ^3(c+d x) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{24 d}+\frac {\csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^3(c+d x) \csc ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{24 d}+\frac {17 \csc \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \cos ^3(c+d x) \csc \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \sec (c+d x)+a)^3}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 178, normalized size = 1.62 \[ -\frac {104 \, a^{3} \cos \left (d x + c\right )^{4} - 38 \, a^{3} \cos \left (d x + c\right )^{3} - 118 \, a^{3} \cos \left (d x + c\right )^{2} + 30 \, a^{3} \cos \left (d x + c\right ) + 6 \, a^{3} - 33 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 33 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - a^{3} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 123, normalized size = 1.12 \[ \frac {33 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 33 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {2 \, {\left (18 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{3}\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.03, size = 188, normalized size = 1.71 \[ -\frac {26 a^{3} \cot \left (d x +c \right )}{3 d}-\frac {a^{3} \cot \left (d x +c \right ) \left (\csc ^{2}\left (d x +c \right )\right )}{3 d}-\frac {a^{3}}{d \sin \left (d x +c \right )^{3}}-\frac {11 a^{3}}{2 d \sin \left (d x +c \right )}+\frac {11 a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}-\frac {a^{3}}{d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4 a^{3}}{d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {a^{3}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{2}}+\frac {5 a^{3}}{6 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 188, normalized size = 1.71 \[ -\frac {a^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{4} - 10 \, \sin \left (d x + c\right )^{2} - 2\right )}}{\sin \left (d x + c\right )^{5} - \sin \left (d x + c\right )^{3}} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, a^{3} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )} + \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{3}}{\tan \left (d x + c\right )^{3}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.35, size = 116, normalized size = 1.05 \[ \frac {11\,a^3\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {56\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}+\frac {16\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )} \]
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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