Optimal. Leaf size=222 \[ \frac {7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac {a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d}+\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \tan (c+d x) \sec (c+d x)}{16 d} \]
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Rubi [A] time = 0.38, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2792, 3031, 3021, 2748, 3767, 3768, 3770} \[ \frac {4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac {\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^2 \left (5 a^2+32 b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {\left (36 a^2 b^2+5 a^4+8 b^4\right ) \tan (c+d x) \sec (c+d x)}{16 d}+\frac {7 a^3 b \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac {a^2 \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2792
Rule 3021
Rule 3031
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \sec ^7(c+d x) \, dx &=\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{6} \int (a+b \cos (c+d x)) \left (14 a^2 b+a \left (5 a^2+18 b^2\right ) \cos (c+d x)+3 b \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx\\ &=\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{30} \int \left (-5 a^2 \left (5 a^2+32 b^2\right )-24 a b \left (4 a^2+5 b^2\right ) \cos (c+d x)-15 b^2 \left (a^2+2 b^2\right ) \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx\\ &=\frac {a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{120} \int \left (-96 a b \left (4 a^2+5 b^2\right )-15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{5} \left (4 a b \left (4 a^2+5 b^2\right )\right ) \int \sec ^4(c+d x) \, dx-\frac {1}{8} \left (-5 a^4-36 a^2 b^2-8 b^4\right ) \int \sec ^3(c+d x) \, dx\\ &=\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}-\frac {1}{16} \left (-5 a^4-36 a^2 b^2-8 b^4\right ) \int \sec (c+d x) \, dx-\frac {\left (4 a b \left (4 a^2+5 b^2\right )\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan (c+d x)}{5 d}+\frac {\left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x) \tan (c+d x)}{16 d}+\frac {a^2 \left (5 a^2+32 b^2\right ) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {7 a^3 b \sec ^4(c+d x) \tan (c+d x)}{15 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a b \left (4 a^2+5 b^2\right ) \tan ^3(c+d x)}{15 d}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 154, normalized size = 0.69 \[ \frac {15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (40 a^4 \sec ^5(c+d x)+64 a b \left (5 \left (2 a^2+b^2\right ) \tan ^2(c+d x)+15 \left (a^2+b^2\right )+3 a^2 \tan ^4(c+d x)\right )+10 a^2 \left (5 a^2+36 b^2\right ) \sec ^3(c+d x)+15 \left (5 a^4+36 a^2 b^2+8 b^4\right ) \sec (c+d x)\right )}{240 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 217, normalized size = 0.98 \[ \frac {15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (128 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} + 192 \, a^{3} b \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 40 \, a^{4} + 64 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 10 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.78, size = 592, normalized size = 2.67 \[ \frac {15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (5 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (165 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 960 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 900 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 960 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 120 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 25 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 1260 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 3520 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 4992 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5760 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 450 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4992 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5760 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 240 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2240 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1260 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3520 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 165 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 960 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 900 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 960 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, b^{4} \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 )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 302, normalized size = 1.36 \[ \frac {a^{4} \left (\sec ^{5}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{6 d}+\frac {5 a^{4} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{24 d}+\frac {5 a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {5 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {32 a^{3} b \tan \left (d x +c \right )}{15 d}+\frac {4 a^{3} b \left (\sec ^{4}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{5 d}+\frac {16 a^{3} b \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{15 d}+\frac {3 a^{2} b^{2} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{2 d}+\frac {9 a^{2} b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{4 d}+\frac {9 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{4 d}+\frac {8 a \,b^{3} \tan \left (d x +c \right )}{3 d}+\frac {4 a \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {b^{4} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{2 d}+\frac {b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 275, normalized size = 1.24 \[ \frac {128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{3} b + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a b^{3} - 5 \, a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, a^{2} b^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.31, size = 370, normalized size = 1.67 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {5\,a^4}{8}+\frac {9\,a^2\,b^2}{2}+b^4\right )}{d}+\frac {\left (\frac {11\,a^4}{8}-8\,a^3\,b+\frac {15\,a^2\,b^2}{2}-8\,a\,b^3+b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,a^4}{24}+\frac {56\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}+\frac {88\,a\,b^3}{3}-3\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {15\,a^4}{4}-\frac {208\,a^3\,b}{5}+3\,a^2\,b^2-48\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,a^4}{4}+\frac {208\,a^3\,b}{5}+3\,a^2\,b^2+48\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5\,a^4}{24}-\frac {56\,a^3\,b}{3}-\frac {21\,a^2\,b^2}{2}-\frac {88\,a\,b^3}{3}-3\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,a^4}{8}+8\,a^3\,b+\frac {15\,a^2\,b^2}{2}+8\,a\,b^3+b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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