Optimal. Leaf size=111 \[ \frac {7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {7 a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {8 a^4 \tan ^3(c+d x)}{3 d}+\frac {a^4 \tan ^5(c+d x)}{5 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3876, 3852, 8,
3853, 3855} \begin {gather*} \frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {8 a^4 \tan ^3(c+d x)}{3 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {a^4 \tan (c+d x) \sec ^3(c+d x)}{d}+\frac {7 a^4 \tan (c+d x) \sec (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3853
Rule 3855
Rule 3876
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \, dx &=\int \left (a^4 \sec ^2(c+d x)+4 a^4 \sec ^3(c+d x)+6 a^4 \sec ^4(c+d x)+4 a^4 \sec ^5(c+d x)+a^4 \sec ^6(c+d x)\right ) \, dx\\ &=a^4 \int \sec ^2(c+d x) \, dx+a^4 \int \sec ^6(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^3(c+d x) \, dx+\left (4 a^4\right ) \int \sec ^5(c+d x) \, dx+\left (6 a^4\right ) \int \sec ^4(c+d x) \, dx\\ &=\frac {2 a^4 \sec (c+d x) \tan (c+d x)}{d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{d}+\left (2 a^4\right ) \int \sec (c+d x) \, dx+\left (3 a^4\right ) \int \sec ^3(c+d x) \, dx-\frac {a^4 \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}-\frac {a^4 \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}-\frac {\left (6 a^4\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {2 a^4 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {7 a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {8 a^4 \tan ^3(c+d x)}{3 d}+\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {1}{2} \left (3 a^4\right ) \int \sec (c+d x) \, dx\\ &=\frac {7 a^4 \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac {8 a^4 \tan (c+d x)}{d}+\frac {7 a^4 \sec (c+d x) \tan (c+d x)}{2 d}+\frac {a^4 \sec ^3(c+d x) \tan (c+d x)}{d}+\frac {8 a^4 \tan ^3(c+d x)}{3 d}+\frac {a^4 \tan ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(498\) vs. \(2(111)=222\).
time = 1.54, size = 498, normalized size = 4.49 \begin {gather*} -\frac {a^4 \sec (c) \sec ^5(c+d x) \left (525 \cos (2 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+525 \cos (4 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (4 c+5 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos (6 c+5 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+1050 \cos (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 )+1050 \cos (2 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 )-525 \cos (2 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-525 \cos (4 c+3 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (4 c+5 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 \cos (6 c+5 d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2360 \sin (d x)+960 \sin (2 c+d x)-660 \sin (c+2 d x)-660 \sin (3 c+2 d x)-1600 \sin (2 c+3 d x)+60 \sin (4 c+3 d x)-210 \sin (3 c+4 d x)-210 \sin (5 c+4 d x)-332 \sin (4 c+5 d x)\right )}{960 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 157, normalized size = 1.41
method | result | size |
norman | \(\frac {-\frac {25 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {158 a^{4} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {896 a^{4} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {98 a^{4} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {7 a^{4} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {7 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {7 a^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(152\) |
risch | \(-\frac {i a^{4} \left (105 \,{\mathrm e}^{9 i \left (d x +c \right )}-30 \,{\mathrm e}^{8 i \left (d x +c \right )}+330 \,{\mathrm e}^{7 i \left (d x +c \right )}-480 \,{\mathrm e}^{6 i \left (d x +c \right )}-1180 \,{\mathrm e}^{4 i \left (d x +c \right )}-330 \,{\mathrm e}^{3 i \left (d x +c \right )}-800 \,{\mathrm e}^{2 i \left (d x +c \right )}-105 \,{\mathrm e}^{i \left (d x +c \right )}-166\right )}{15 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 d}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 d}\) | \(156\) |
derivativedivides | \(\frac {-a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} \tan \left (d x +c \right )}{d}\) | \(157\) |
default | \(\frac {-a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+a^{4} \tan \left (d x +c \right )}{d}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 190, normalized size = 1.71 \begin {gather*} \frac {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 )} a^{4} + 120 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4} - 15 \, 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 )} - 60 \, 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 )} + 60 \, a^{4} \tan \left (d x + c\right )}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.42, size = 124, normalized size = 1.12 \begin {gather*} \frac {105 \, a^{4} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, a^{4} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (166 \, a^{4} \cos \left (d x + c\right )^{4} + 105 \, a^{4} \cos \left (d x + c\right )^{3} + 68 \, a^{4} \cos \left (d x + c\right )^{2} + 30 \, a^{4} \cos \left (d x + c\right ) + 6 \, a^{4}\right )} \sin \left (d x + c\right )}{60 \, d \cos \left (d x + c\right )^{5}} \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 ^{2}{\left (c + d x \right )}\, dx + \int 4 \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 \sec ^{5}{\left (c + d x \right )}\, dx + \int \sec ^{6}{\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 = 0.49, size = 138, normalized size = 1.24 \begin {gather*} \frac {105 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 490 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 896 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 790 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 375 \, 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 )}^{5}}}{30 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.47, size = 170, normalized size = 1.53 \begin {gather*} \frac {7\,a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {7\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-\frac {98\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{3}+\frac {896\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}-\frac {158\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+25\,a^4\,\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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