Optimal. Leaf size=91 \[ \frac {a \tan (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {2 a \cot (c+d x)}{d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
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Rubi [A] time = 0.13, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2838, 2620, 270, 2622, 288, 321, 207} \[ \frac {a \tan (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {2 a \cot (c+d x)}{d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 288
Rule 321
Rule 2620
Rule 2622
Rule 2838
Rubi steps
\begin {align*} \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+a \int \csc ^4(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \operatorname {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \tan (c+d x)}{d}+\frac {(3 a) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 4.72, size = 205, normalized size = 2.25 \[ \frac {a \tan (c+d x)}{d}-\frac {5 a \cot (c+d x)}{3 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 308, normalized size = 3.38 \[ -\frac {32 \, a \cos \left (d x + c\right )^{4} + 14 \, a \cos \left (d x + c\right )^{3} - 48 \, a \cos \left (d x + c\right )^{2} - 18 \, a \cos \left (d x + c\right ) + 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (16 \, a \cos \left (d x + c\right )^{3} + 9 \, a \cos \left (d x + c\right )^{2} - 15 \, a \cos \left (d x + c\right ) - 6 \, a\right )} \sin \left (d x + c\right ) + 12 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 130, normalized size = 1.43 \[ \frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} - \frac {66 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.44, size = 116, normalized size = 1.27 \[ -\frac {a}{2 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3 a}{2 d \cos \left (d x +c \right )}+\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {a}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4 a}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 a \cot \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 98, normalized size = 1.08 \[ \frac {3 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, a {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.99, size = 144, normalized size = 1.58 \[ \frac {7\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]
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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