Optimal. Leaf size=169 \[ \frac {a \left (4 a^2+15 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan (c+d x)}{5 d}+\frac {b \left (9 a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \left (9 a^2+4 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d} \]
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Rubi [A] time = 0.22, antiderivative size = 169, 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 = {2792, 3021, 2748, 3767, 3768, 3770} \[ \frac {a \left (4 a^2+15 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan (c+d x)}{5 d}+\frac {b \left (9 a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {b \left (9 a^2+4 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {11 a^2 b \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac {a^2 \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))}{5 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2792
Rule 3021
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \sec ^6(c+d x) \, dx &=\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{5} \int \left (11 a^2 b+a \left (4 a^2+15 b^2\right ) \cos (c+d x)+b \left (3 a^2+5 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {11 a^2 b \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{20} \int \left (4 a \left (4 a^2+15 b^2\right )+5 b \left (9 a^2+4 b^2\right ) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {11 a^2 b \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{4} \left (b \left (9 a^2+4 b^2\right )\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{5} \left (a \left (4 a^2+15 b^2\right )\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac {b \left (9 a^2+4 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {11 a^2 b \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {1}{8} \left (b \left (9 a^2+4 b^2\right )\right ) \int \sec (c+d x) \, dx-\frac {\left (a \left (4 a^2+15 b^2\right )\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {b \left (9 a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan (c+d x)}{5 d}+\frac {b \left (9 a^2+4 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {11 a^2 b \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {a^2 (a+b \cos (c+d x)) \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac {a \left (4 a^2+15 b^2\right ) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 0.90, size = 120, normalized size = 0.71 \[ \frac {15 b \left (9 a^2+4 b^2\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (8 a \left (5 \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)+15 \left (a^2+3 b^2\right )+3 a^2 \tan ^4(c+d x)\right )+15 b \left (9 a^2+4 b^2\right ) \sec (c+d x)+90 a^2 b \sec ^3(c+d x)\right )}{120 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 170, normalized size = 1.01 \[ \frac {15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + 90 \, a^{2} b \cos \left (d x + c\right ) + 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \cos \left (d x + c\right )^{3} + 24 \, a^{3} + 8 \, {\left (4 \, a^{3} + 15 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 367, normalized size = 2.17 \[ \frac {15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (9 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 225 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 90 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 960 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1200 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 160 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 90 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 960 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 225 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 206, normalized size = 1.22 \[ \frac {8 a^{3} \tan \left (d x +c \right )}{15 d}+\frac {a^{3} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 a^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d}+\frac {3 a^{2} b \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {9 a^{2} b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {9 a^{2} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {2 b^{2} a \tan \left (d x +c \right )}{d}+\frac {b^{2} a \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {b^{3} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 181, normalized size = 1.07 \[ \frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} + 240 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{2} - 45 \, a^{2} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 260, normalized size = 1.54 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {9\,a^2\,b}{4}+b^3\right )}{d}-\frac {\left (2\,a^3-\frac {15\,a^2\,b}{4}+6\,a\,b^2-b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {8\,a^3}{3}+\frac {3\,a^2\,b}{2}-16\,a\,b^2+2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {116\,a^3}{15}+20\,a\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {8\,a^3}{3}-\frac {3\,a^2\,b}{2}-16\,a\,b^2-2\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^3+\frac {15\,a^2\,b}{4}+6\,a\,b^2+b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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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