Optimal. Leaf size=136 \[ \frac {49 a^4 \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {49 a^4 \sec (c+d x) \tan (c+d x)}{16 d}+\frac {41 a^4 \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac {a^4 \sec ^5(c+d x) \tan (c+d x)}{6 d}+\frac {4 a^4 \tan ^3(c+d x)}{d}+\frac {4 a^4 \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.14, 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 = {3876, 3853,
3855, 3852} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^4 \, 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 ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (4 a^4\right ) \text {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 = 0.84, size = 211, normalized size = 1.55 \begin {gather*} -\frac {a^4 (1+\cos (c+d x))^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 (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-\sec (c) (-11520 \sin (c)+3750 \sin (d x)+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))\right )}{122880 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 204, normalized size = 1.50
method | result | size |
norman | \(\frac {\frac {207 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {1471 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {1967 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}-\frac {1617 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20 d}+\frac {833 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {49 a^{4} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{6}}-\frac {49 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{16 d}+\frac {49 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{16 d}\) | \(171\) |
risch | \(-\frac {i a^{4} \left (735 \,{\mathrm e}^{11 i \left (d x +c \right )}+3845 \,{\mathrm e}^{9 i \left (d x +c \right )}-1920 \,{\mathrm e}^{8 i \left (d x +c \right )}+3750 \,{\mathrm e}^{7 i \left (d x +c \right )}-11520 \,{\mathrm e}^{6 i \left (d x +c \right )}-3750 \,{\mathrm e}^{5 i \left (d x +c \right )}-15360 \,{\mathrm e}^{4 i \left (d x +c \right )}-3845 \,{\mathrm e}^{3 i \left (d x +c \right )}-6912 \,{\mathrm e}^{2 i \left (d x +c \right )}-735 \,{\mathrm e}^{i \left (d x +c \right )}-1152\right )}{120 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{6}}-\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{16 d}+\frac {49 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{16 d}\) | \(178\) |
derivativedivides | \(\frac {a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+6 a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(204\) |
default | \(\frac {a^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-4 a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+6 a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(204\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 270 vs.
\(2 (126) = 252\).
time = 0.28, size = 270, normalized size = 1.99 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.29, size = 137, normalized size = 1.01 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} a^{4} \left (\int \sec ^{3}{\left (c + d x \right )}\, dx + \int 4 \sec ^{4}{\left (c + d x \right )}\, dx + \int 6 \sec ^{5}{\left (c + d x \right )}\, dx + \int 4 \sec ^{6}{\left (c + d x \right )}\, dx + \int \sec ^{7}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 154, normalized size = 1.13 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.65, size = 199, normalized size = 1.46 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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