Optimal. Leaf size=123 \[ \frac {b \left (11 a^2+4 b^2\right ) \sin (c+d x)}{4 d}+\frac {3 a \left (a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a x \left (a^2+4 b^2\right )-\frac {3 a^2 b \sin ^3(c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))}{4 d} \]
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Rubi [A] time = 0.18, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3841, 4047, 2635, 8, 4044, 3013} \[ \frac {b \left (11 a^2+4 b^2\right ) \sin (c+d x)}{4 d}+\frac {3 a \left (a^2+4 b^2\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3}{8} a x \left (a^2+4 b^2\right )-\frac {3 a^2 b \sin ^3(c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3013
Rule 3841
Rule 4044
Rule 4047
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \, dx &=\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) \left (9 a^2 b+3 a \left (a^2+4 b^2\right ) \sec (c+d x)+2 b \left (a^2+2 b^2\right ) \sec ^2(c+d x)\right ) \, dx\\ &=\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos ^3(c+d x) \left (9 a^2 b+2 b \left (a^2+2 b^2\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{4} \left (3 a \left (a^2+4 b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac {3 a \left (a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{4 d}+\frac {1}{4} \int \cos (c+d x) \left (2 b \left (a^2+2 b^2\right )+9 a^2 b \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (3 a \left (a^2+4 b^2\right )\right ) \int 1 \, dx\\ &=\frac {3}{8} a \left (a^2+4 b^2\right ) x+\frac {3 a \left (a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \left (9 a^2 b+2 b \left (a^2+2 b^2\right )-9 a^2 b x^2\right ) \, dx,x,-\sin (c+d x)\right )}{4 d}\\ &=\frac {3}{8} a \left (a^2+4 b^2\right ) x+\frac {b \left (11 a^2+4 b^2\right ) \sin (c+d x)}{4 d}+\frac {3 a \left (a^2+4 b^2\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) (a+b \sec (c+d x)) \sin (c+d x)}{4 d}-\frac {3 a^2 b \sin ^3(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 100, normalized size = 0.81 \[ \frac {8 b \left (9 a^2+4 b^2\right ) \sin (c+d x)+a \left (8 \left (a^2+3 b^2\right ) \sin (2 (c+d x))+a^2 \sin (4 (c+d x))+12 a^2 c+12 a^2 d x+8 a b \sin (3 (c+d x))+48 b^2 c+48 b^2 d x\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 84, normalized size = 0.68 \[ \frac {3 \, {\left (a^{3} + 4 \, a b^{2}\right )} d x + {\left (2 \, a^{3} \cos \left (d x + c\right )^{3} + 8 \, a^{2} b \cos \left (d x + c\right )^{2} + 16 \, a^{2} b + 8 \, b^{3} + 3 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 297, normalized size = 2.41 \[ \frac {3 \, {\left (a^{3} + 4 \, a b^{2}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 8 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 12 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.00, size = 102, normalized size = 0.83 \[ \frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{2} b \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 b^{2} a \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b^{3} \sin \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 95, normalized size = 0.77 \[ \frac {{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a^{2} b + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{2} + 32 \, b^{3} \sin \left (d x + c\right )}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.83, size = 250, normalized size = 2.03 \[ \frac {\left (-\frac {5\,a^3}{4}+6\,a^2\,b-3\,a\,b^2+2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,a^3}{4}+10\,a^2\,b-3\,a\,b^2+6\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {3\,a^3}{4}+10\,a^2\,b+3\,a\,b^2+6\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,a^3}{4}+6\,a^2\,b+3\,a\,b^2+2\,b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {3\,a\,\mathrm {atan}\left (\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^2+4\,b^2\right )}{4\,\left (\frac {3\,a^3}{4}+3\,a\,b^2\right )}\right )\,\left (a^2+4\,b^2\right )}{4\,d} \]
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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