Optimal. Leaf size=89 \[ -\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8} \]
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Rubi [A] time = 0.11, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3872, 2838, 2592, 302, 206, 2635, 8} \[ -\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin (c+d x)}{d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \sin ^3(c+d x) \cos (c+d x)}{4 d}-\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 302
Rule 2592
Rule 2635
Rule 2838
Rule 3872
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \sin ^4(c+d x) \, dx &=-\int (-a-a \cos (c+d x)) \sin ^3(c+d x) \tan (c+d x) \, dx\\ &=a \int \sin ^4(c+d x) \, dx+a \int \sin ^3(c+d x) \tan (c+d x) \, dx\\ &=-\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{4} (3 a) \int \sin ^2(c+d x) \, dx+\frac {a \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=-\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {1}{8} (3 a) \int 1 \, dx+\frac {a \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {3 a x}{8}-\frac {a \sin (c+d x)}{d}-\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac {3 a x}{8}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \sin (c+d x)}{d}-\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \cos (c+d x) \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 86, normalized size = 0.97 \[ \frac {3 a (c+d x)}{8 d}-\frac {a \sin ^3(c+d x)}{3 d}-\frac {a \sin (c+d x)}{d}-\frac {a \sin (2 (c+d x))}{4 d}+\frac {a \sin (4 (c+d x))}{32 d}+\frac {a \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 79, normalized size = 0.89 \[ \frac {9 \, a d x + 12 \, a \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, a \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, a \cos \left (d x + c\right )^{3} + 8 \, a \cos \left (d x + c\right )^{2} - 15 \, a \cos \left (d x + c\right ) - 32 \, a\right )} \sin \left (d x + c\right )}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 118, normalized size = 1.33 \[ \frac {9 \, {\left (d x + c\right )} a + 24 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 24 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 71 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 137 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 33 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.61, size = 96, normalized size = 1.08 \[ -\frac {a \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 a x}{8}+\frac {3 c a}{8 d}-\frac {a \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}-\frac {a \sin \left (d x +c \right )}{d}+\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 81, normalized size = 0.91 \[ -\frac {16 \, {\left (2 \, \sin \left (d x + c\right )^{3} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 6 \, \sin \left (d x + c\right )\right )} a - 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.03, size = 90, normalized size = 1.01 \[ \frac {3\,a\,x}{8}+\frac {2\,a\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {a\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {a\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}-\frac {5\,a\,\sin \left (c+d\,x\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \sin ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \sin ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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