Optimal. Leaf size=136 \[ \frac {4 a^4 \tan ^5(c+d x)}{5 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {49 a^4 \tan (c+d x) \sec (c+d x)}{16 d} \]
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Rubi [A] time = 0.18, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2757, 3768, 3770, 3767} \[ \frac {4 a^4 \tan ^5(c+d x)}{5 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {a^4 \tan (c+d x) \sec ^5(c+d x)}{6 d}+\frac {41 a^4 \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac {49 a^4 \tan (c+d x) \sec (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^4 \sec ^7(c+d x) \, dx &=\int \left (a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+6 a^4 \sec ^5(c+d x)+4 a^4 \sec ^6(c+d x)+a^4 \sec ^7(c+d x)\right ) \, dx\\ &=a^4 \int \sec ^3(c+d x) \, dx+a^4 \int \sec ^7(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^4(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^6(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^5(c+d x) \, dx\\ &=\frac {a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {3 a^4 \sec ^3(c+d x) \tan (c+d x)}{2 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {1}{2} a^4 \int \sec (c+d x) \, dx+\frac {1}{6} \left (5 a^4\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{2} \left (9 a^4\right ) \int \sec ^3(c+d x) \, dx-\frac {\left (4 a^4\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (4 a^4\right ) \operatorname {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {11 a^4 \sec (c+d x) \tan (c+d x)}{4 d}+\frac {41 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {4 a^4 \tan ^5(c+d x)}{5 d}+\frac {1}{8} \left (5 a^4\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{4} \left (9 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {11 a^4 \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {49 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {41 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {4 a^4 \tan ^5(c+d x)}{5 d}+\frac {1}{16} \left (5 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {49 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {41 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {4 a^4 \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 211, normalized size = 1.55 \[ -\frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \sec ^6(c+d x) \left (23520 \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) (3750 \sin (2 c+d x)+15360 \sin (c+2 d x)-1920 \sin (3 c+2 d x)+3845 \sin (2 c+3 d x)+3845 \sin (4 c+3 d x)+6912 \sin (3 c+4 d x)+735 \sin (4 c+5 d x)+735 \sin (6 c+5 d x)+1152 \sin (5 c+6 d x)-11520 \sin (c)+3750 \sin (d x))\right )}{122880 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 137, normalized size = 1.01 \[ \frac {735 \, a^{4} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 735 \, a^{4} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (1152 \, a^{4} \cos \left (d x + c\right )^{5} + 735 \, a^{4} \cos \left (d x + c\right )^{4} + 576 \, a^{4} \cos \left (d x + c\right )^{3} + 410 \, a^{4} \cos \left (d x + c\right )^{2} + 192 \, a^{4} \cos \left (d x + c\right ) + 40 \, a^{4}\right )} \sin \left (d x + c\right )}{480 \, d \cos \left (d x + c\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.71, size = 154, normalized size = 1.13 \[ \frac {735 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 735 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (735 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 4165 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 11802 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{6}}}{240 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 146, normalized size = 1.07 \[ \frac {49 a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{16 d}+\frac {49 a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16 d}+\frac {24 a^{4} \tan \left (d x +c \right )}{5 d}+\frac {12 a^{4} \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{5 d}+\frac {41 a^{4} \left (\sec ^{3}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{24 d}+\frac {4 a^{4} \tan \left (d x +c \right ) \left (\sec ^{4}\left (d x +c \right )\right )}{5 d}+\frac {a^{4} \left (\sec ^{5}\left (d x +c \right )\right ) \tan \left (d x +c \right )}{6 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 270, normalized size = 1.99 \[ \frac {128 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{4} + 640 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} - 5 \, a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 180 \, a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 120 \, a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 199, normalized size = 1.46 \[ \frac {49\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d}-\frac {\frac {49\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}-\frac {833\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {1617\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{20}-\frac {1967\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{20}+\frac {1471\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}-\frac {207\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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