Optimal. Leaf size=119 \[ -\frac {a \cos (c+d x)}{d}+\frac {3 b \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 b \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {b \log (\cos (c+d x))}{d} \]
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Rubi [A]
time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3957, 2916, 12,
780} \begin {gather*} \frac {a \cos ^7(c+d x)}{7 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {a \cos (c+d x)}{d}+\frac {b \cos ^6(c+d x)}{6 d}-\frac {3 b \cos ^4(c+d x)}{4 d}+\frac {3 b \cos ^2(c+d x)}{2 d}-\frac {b \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 780
Rule 2916
Rule 3957
Rubi steps
\begin {align*} \int (a+b \sec (c+d x)) \sin ^7(c+d x) \, dx &=-\int (-b-a \cos (c+d x)) \sin ^6(c+d x) \tan (c+d x) \, dx\\ &=\frac {\text {Subst}\left (\int \frac {a (-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \frac {(-b+x) \left (a^2-x^2\right )^3}{x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {\text {Subst}\left (\int \left (a^6-\frac {a^6 b}{x}+3 a^4 b x-3 a^4 x^2-3 a^2 b x^3+3 a^2 x^4+b x^5-x^6\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=-\frac {a \cos (c+d x)}{d}+\frac {3 b \cos ^2(c+d x)}{2 d}+\frac {a \cos ^3(c+d x)}{d}-\frac {3 b \cos ^4(c+d x)}{4 d}-\frac {3 a \cos ^5(c+d x)}{5 d}+\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \cos ^7(c+d x)}{7 d}-\frac {b \log (\cos (c+d x))}{d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 115, normalized size = 0.97 \begin {gather*} -\frac {35 a \cos (c+d x)}{64 d}+\frac {7 a \cos (3 (c+d x))}{64 d}-\frac {7 a \cos (5 (c+d x))}{320 d}+\frac {a \cos (7 (c+d x))}{448 d}-\frac {b \left (-3 \cos ^2(c+d x)+\frac {3}{2} \cos ^4(c+d x)-\frac {1}{3} \cos ^6(c+d x)+2 \log (\cos (c+d x))\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 87, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {b \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(87\) |
default | \(\frac {b \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right )}{6}-\frac {\left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(87\) |
risch | \(i b x +\frac {2 i b c}{d}+\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {35 a \cos \left (d x +c \right )}{64 d}+\frac {a \cos \left (7 d x +7 c \right )}{448 d}+\frac {b \cos \left (6 d x +6 c \right )}{192 d}-\frac {7 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {b \cos \left (4 d x +4 c \right )}{16 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\) | \(150\) |
norman | \(\frac {-\frac {32 a}{35 d}-\frac {128 b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {2 b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {14 b \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (96 a +70 b \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (96 a +128 b \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {\left (32 a +10 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {b \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(200\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 0.76 \begin {gather*} \frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (\cos \left (d x + c\right )\right )}{420 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.23, size = 93, normalized size = 0.78 \begin {gather*} \frac {60 \, a \cos \left (d x + c\right )^{7} + 70 \, b \cos \left (d x + c\right )^{6} - 252 \, a \cos \left (d x + c\right )^{5} - 315 \, b \cos \left (d x + c\right )^{4} + 420 \, a \cos \left (d x + c\right )^{3} + 630 \, b \cos \left (d x + c\right )^{2} - 420 \, a \cos \left (d x + c\right ) - 420 \, b \log \left (-\cos \left (d x + c\right )\right )}{420 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 317 vs.
\(2 (109) = 218\).
time = 0.46, size = 317, normalized size = 2.66 \begin {gather*} \frac {420 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {384 \, a + 1089 \, b - \frac {2688 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8463 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8064 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {28749 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {13440 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {56035 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {28749 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1089 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}}{420 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.93, size = 89, normalized size = 0.75 \begin {gather*} -\frac {a\,\cos \left (c+d\,x\right )-a\,{\cos \left (c+d\,x\right )}^3+\frac {3\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {a\,{\cos \left (c+d\,x\right )}^7}{7}-\frac {3\,b\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {3\,b\,{\cos \left (c+d\,x\right )}^4}{4}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{6}+b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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