Optimal. Leaf size=73 \[ \frac {a (4 A-B) \tan ^3(c+d x)}{15 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)}{5 d} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2855, 3767} \[ \frac {a (4 A-B) \tan ^3(c+d x)}{15 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {(A+B) \sec ^5(c+d x) (a \sin (c+d x)+a)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2855
Rule 3767
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+a \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}+\frac {1}{5} (a (4 A-B)) \int \sec ^4(c+d x) \, dx\\ &=\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}-\frac {(a (4 A-B)) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {(A+B) \sec ^5(c+d x) (a+a \sin (c+d x))}{5 d}+\frac {a (4 A-B) \tan (c+d x)}{5 d}+\frac {a (4 A-B) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [B] time = 1.27, size = 223, normalized size = 3.05 \[ \frac {a \sec (c) (-54 (A+B) \cos (c+d x)+18 A \sin (2 (c+d x))+9 A \sin (4 (c+d x))+128 A \sin (2 c+3 d x)-18 A \cos (3 (c+d x))+128 A \cos (c+2 d x)+64 A \cos (3 c+4 d x)+384 A \sin (d x)+18 B \sin (2 (c+d x))+9 B \sin (4 (c+d x))-32 B \sin (2 c+3 d x)-18 B \cos (3 (c+d x))-32 B \cos (c+2 d x)-16 B \cos (3 c+4 d x)+240 B \cos (c)-96 B \sin (d x))}{960 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5 \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 112, normalized size = 1.53 \[ -\frac {2 \, {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{4} - {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{2} - {\left (A - 4 \, B\right )} a + {\left (2 \, {\left (4 \, A - B\right )} a \cos \left (d x + c\right )^{2} + {\left (4 \, A - B\right )} a\right )} \sin \left (d x + c\right )}{15 \, {\left (d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d \cos \left (d x + c\right )^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 225, normalized size = 3.08 \[ -\frac {\frac {5 \, {\left (15 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13 \, A a - 7 \, B a\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{3}} + \frac {165 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 45 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 60 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 650 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 70 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 400 \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 20 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 113 \, A a + 13 \, B a}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 102, normalized size = 1.40 \[ \frac {\frac {a A}{5 \cos \left (d x +c \right )^{5}}+a B \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )-a A \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+\frac {a B}{5 \cos \left (d x +c \right )^{5}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 86, normalized size = 1.18 \[ \frac {{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a + {\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} B a + \frac {3 \, A a}{\cos \left (d x + c\right )^{5}} + \frac {3 \, B a}{\cos \left (d x + c\right )^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.14, size = 224, normalized size = 3.07 \[ -\frac {a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {A\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{4}-\frac {9\,A\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{4}-A\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )-\frac {15\,B\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}+\frac {3\,B\,\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{2}-B\,\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )+\frac {B\,\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{4}-\frac {73\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {25\,A\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}-\frac {19\,A\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {3\,A\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}+\frac {7\,B\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {5\,B\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{8}+\frac {B\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{8}+\frac {3\,B\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{8}\right )}{120\,d\,{\cos \left (\frac {c}{2}-\frac {\pi }{4}+\frac {d\,x}{2}\right )}^3\,{\cos \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d\,x}{2}\right )}^5} \]
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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