Optimal. Leaf size=179 \[ \frac {a b \left (4 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\left (3 a^2+4 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \tan (c+d x) \sec (c+d x)}{30 d}+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \tan (c+d x)}{15 d}+\frac {\tan (c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {a \tan (c+d x) (a+b \sec (c+d x))^3}{5 d} \]
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Rubi [A] time = 0.30, antiderivative size = 179, 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 = {3835, 4002, 3997, 3787, 3770, 3767, 8} \[ \frac {2 \left (28 a^2 b^2+3 a^4+4 b^4\right ) \tan (c+d x)}{15 d}+\frac {a b \left (4 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {\left (3 a^2+4 b^2\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \tan (c+d x) \sec (c+d x)}{30 d}+\frac {\tan (c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {a \tan (c+d x) (a+b \sec (c+d x))^3}{5 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3787
Rule 3835
Rule 3997
Rule 4002
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \, dx &=\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {4}{5} \int \sec (c+d x) (b+a \sec (c+d x)) (a+b \sec (c+d x))^3 \, dx\\ &=\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{5} \int \sec (c+d x) (a+b \sec (c+d x))^2 \left (7 a b+\left (3 a^2+4 b^2\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {\left (3 a^2+4 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{15} \int \sec (c+d x) (a+b \sec (c+d x)) \left (b \left (27 a^2+8 b^2\right )+a \left (6 a^2+29 b^2\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {a b \left (6 a^2+29 b^2\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{30} \int \sec (c+d x) \left (15 a b \left (4 a^2+3 b^2\right )+4 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \sec (c+d x)\right ) \, dx\\ &=\frac {a b \left (6 a^2+29 b^2\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}+\frac {1}{2} \left (a b \left (4 a^2+3 b^2\right )\right ) \int \sec (c+d x) \, dx+\frac {1}{15} \left (2 \left (3 a^4+28 a^2 b^2+4 b^4\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {a b \left (4 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a b \left (6 a^2+29 b^2\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}-\frac {\left (2 \left (3 a^4+28 a^2 b^2+4 b^4\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 d}\\ &=\frac {a b \left (4 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {2 \left (3 a^4+28 a^2 b^2+4 b^4\right ) \tan (c+d x)}{15 d}+\frac {a b \left (6 a^2+29 b^2\right ) \sec (c+d x) \tan (c+d x)}{30 d}+\frac {\left (3 a^2+4 b^2\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{15 d}+\frac {a (a+b \sec (c+d x))^3 \tan (c+d x)}{5 d}+\frac {(a+b \sec (c+d x))^4 \tan (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.79, size = 125, normalized size = 0.70 \[ \frac {15 a b \left (4 a^2+3 b^2\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (20 b^2 \left (3 a^2+b^2\right ) \tan ^2(c+d x)+15 a b \left (4 a^2+3 b^2\right ) \sec (c+d x)+30 \left (a^4+6 a^2 b^2+b^4\right )+30 a b^3 \sec ^3(c+d x)+6 b^4 \tan ^4(c+d x)\right )}{30 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 182, normalized size = 1.02 \[ \frac {15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (30 \, a b^{3} \cos \left (d x + c\right ) + 2 \, {\left (15 \, a^{4} + 60 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 6 \, b^{4} + 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (15 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 461, normalized size = 2.58 \[ \frac {15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 15 \, {\left (4 \, a^{3} b + 3 \, a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 30 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 480 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 30 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 40 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 180 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 116 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 30 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 30 \, 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 )}^{5}}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.34, size = 225, normalized size = 1.26 \[ \frac {a^{4} \tan \left (d x +c \right )}{d}+\frac {2 a^{3} b \sec \left (d x +c \right ) \tan \left (d x +c \right )}{d}+\frac {2 a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a^{2} b^{2} \tan \left (d x +c \right )}{d}+\frac {2 a^{2} b^{2} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{d}+\frac {a \,b^{3} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{d}+\frac {3 a \,b^{3} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {3 a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {8 b^{4} \tan \left (d x +c \right )}{15 d}+\frac {b^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {4 b^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{15 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.39, size = 195, normalized size = 1.09 \[ \frac {120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} b^{2} + 4 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} b^{4} - 15 \, a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 60 \, a^{4} \tan \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.97, size = 304, normalized size = 1.70 \[ \frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (4\,a^3\,b+3\,a\,b^3\right )}{d}-\frac {\left (2\,a^4-4\,a^3\,b+12\,a^2\,b^2-5\,a\,b^3+2\,b^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-8\,a^4+8\,a^3\,b-32\,a^2\,b^2+2\,a\,b^3-\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (12\,a^4+40\,a^2\,b^2+\frac {116\,b^4}{15}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-8\,a^4-8\,a^3\,b-32\,a^2\,b^2-2\,a\,b^3-\frac {8\,b^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a^4+4\,a^3\,b+12\,a^2\,b^2+5\,a\,b^3+2\,b^4\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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