Optimal. Leaf size=110 \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {2 a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec ^3(c+d x)}{6 d}+\frac {5 a \sec (c+d x)}{2 d}-\frac {5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]
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Rubi [A] time = 0.14, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2838, 2622, 288, 302, 207, 2620, 270} \[ \frac {a \tan ^3(c+d x)}{3 d}+\frac {2 a \tan (c+d x)}{d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec ^3(c+d x)}{6 d}+\frac {5 a \sec (c+d x)}{2 d}-\frac {5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 207
Rule 270
Rule 288
Rule 302
Rule 2620
Rule 2622
Rule 2838
Rubi steps
\begin {align*} \int \csc ^3(c+d x) \sec ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \csc ^2(c+d x) \sec ^4(c+d x) \, dx+a \int \csc ^3(c+d x) \sec ^4(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^6}{\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^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {a \operatorname {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {x^4}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {(5 a) \operatorname {Subst}\left (\int \left (1+x^2+\frac {1}{-1+x^2}\right ) \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec (c+d x)}{2 d}+\frac {5 a \sec ^3(c+d x)}{6 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d}\\ &=-\frac {5 a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x)}{d}+\frac {5 a \sec (c+d x)}{2 d}+\frac {5 a \sec ^3(c+d x)}{6 d}-\frac {a \csc ^2(c+d x) \sec ^3(c+d x)}{2 d}+\frac {2 a \tan (c+d x)}{d}+\frac {a \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [B] time = 6.10, size = 359, normalized size = 3.26 \[ \frac {5 a \tan (c+d x)}{3 d}-\frac {a \cot (c+d x)}{d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {5 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {5 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {13 a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {13 a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a}{12 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {a}{12 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{6 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {a \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 222, normalized size = 2.02 \[ \frac {32 \, a \cos \left (d x + c\right )^{4} - 18 \, a \cos \left (d x + c\right )^{2} - 15 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{3} - a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (a \cos \left (d x + c\right )^{2} + 2 \, a\right )} \sin \left (d x + c\right ) - 8 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right ) - {\left (d \cos \left (d x + c\right )^{3} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 148, normalized size = 1.35 \[ \frac {3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 60 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1} - \frac {3 \, {\left (30 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}} - \frac {4 \, {\left (27 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 25 \, a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 138, normalized size = 1.25 \[ \frac {a}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{3}}+\frac {4 a}{3 d \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 a \cot \left (d x +c \right )}{3 d}+\frac {a}{3 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}-\frac {5 a}{6 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {5 a}{2 d \cos \left (d x +c \right )}+\frac {5 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 106, normalized size = 0.96 \[ \frac {4 \, {\left (\tan \left (d x + c\right )^{3} - \frac {3}{\tan \left (d x + c\right )} + 6 \, \tan \left (d x + c\right )\right )} a + a {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{5} - \cos \left (d x + c\right )^{3}} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.98, size = 180, normalized size = 1.64 \[ \frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d}-\frac {-18\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+\frac {67\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {68\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a}{2}}{d\,\left (-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {5\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]
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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