Optimal. Leaf size=154 \[ \frac {5 a^3 b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac {4 a b \left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d}+\frac {\left (3 a^4+24 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d} \]
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Rubi [A] time = 0.34, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, 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, 8, 3770} \[ \frac {4 a b \left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {\left (24 a^2 b^2+3 a^4+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac {5 a^3 b \tan (c+d x) \sec ^2(c+d x)}{6 d}+\frac {a^2 \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2792
Rule 3021
Rule 3031
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \sec ^5(c+d x) \, dx &=\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x)) \left (10 a^2 b+3 a \left (a^2+4 b^2\right ) \cos (c+d x)+b \left (a^2+4 b^2\right ) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {5 a^3 b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{12} \int \left (-3 a^2 \left (3 a^2+22 b^2\right )-16 a b \left (2 a^2+3 b^2\right ) \cos (c+d x)-3 b^2 \left (a^2+4 b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {a^2 \left (3 a^2+22 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a^3 b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-32 a b \left (2 a^2+3 b^2\right )-3 \left (3 a^4+24 a^2 b^2+8 b^4\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a^2 \left (3 a^2+22 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a^3 b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{3} \left (4 a b \left (2 a^2+3 b^2\right )\right ) \int \sec ^2(c+d x) \, dx-\frac {1}{8} \left (-3 a^4-24 a^2 b^2-8 b^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {\left (3 a^4+24 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a^3 b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {\left (4 a b \left (2 a^2+3 b^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {\left (3 a^4+24 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {4 a b \left (2 a^2+3 b^2\right ) \tan (c+d x)}{3 d}+\frac {a^2 \left (3 a^2+22 b^2\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {5 a^3 b \sec ^2(c+d x) \tan (c+d x)}{6 d}+\frac {a^2 (a+b \cos (c+d x))^2 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.49, size = 101, normalized size = 0.66 \[ \frac {3 \left (3 a^4+24 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) \left (6 a^3 \sec ^3(c+d x)+32 b \left (3 \left (a^2+b^2\right )+a^2 \tan ^2(c+d x)\right )+9 a \left (a^2+8 b^2\right ) \sec (c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 163, normalized size = 1.06 \[ \frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (32 \, a^{3} b \cos \left (d x + c\right ) + 6 \, a^{4} + 32 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 9 \, {\left (a^{4} + 8 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.70, size = 360, normalized size = 2.34 \[ \frac {3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, a^{4} + 24 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 188, normalized size = 1.22 \[ \frac {a^{4} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{4 d}+\frac {3 a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {8 a^{3} b \tan \left (d x +c \right )}{3 d}+\frac {4 a^{3} b \left (\sec ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{3 d}+\frac {3 a^{2} b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{d}+\frac {3 a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a \,b^{3} \tan \left (d x +c \right )}{d}+\frac {b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 187, normalized size = 1.21 \[ \frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} b - 3 \, a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 192 \, a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.29, size = 245, normalized size = 1.59 \[ \frac {\left (\frac {5\,a^4}{4}-8\,a^3\,b+6\,a^2\,b^2-8\,a\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,a^4}{4}+\frac {40\,a^3\,b}{3}-6\,a^2\,b^2+24\,a\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,a^4}{4}-\frac {40\,a^3\,b}{3}-6\,a^2\,b^2-24\,a\,b^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,a^4}{4}+8\,a^3\,b+6\,a^2\,b^2+8\,a\,b^3\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,a^4}{4}+6\,a^2\,b^2+2\,b^4\right )}{d} \]
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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