Optimal. Leaf size=152 \[ -\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \cos ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 a^3 d}+\frac {b \cos ^4(c+d x)}{4 a^2 d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {b \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^6 d} \]
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Rubi [A]
time = 0.14, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2916, 12,
786} \begin {gather*} \frac {b \cos ^4(c+d x)}{4 a^2 d}+\frac {b \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)}{a^6 d}-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \cos ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 a^3 d}-\frac {\cos ^5(c+d x)}{5 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 786
Rule 2916
Rule 3957
Rubi steps
\begin {align*} \int \frac {\sin ^5(c+d x)}{a+b \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) \sin ^5(c+d x)}{-b-a \cos (c+d x)} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^2}{a (-b+x)} \, dx,x,-a \cos (c+d x)\right )}{a^5 d}\\ &=\frac {\text {Subst}\left (\int \frac {x \left (a^2-x^2\right )^2}{-b+x} \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=\frac {\text {Subst}\left (\int \left (\left (a^2-b^2\right )^2-\frac {b \left (-a^2+b^2\right )^2}{b-x}+b \left (-2 a^2+b^2\right ) x-\left (2 a^2-b^2\right ) x^2+b x^3+x^4\right ) \, dx,x,-a \cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\left (a^2-b^2\right )^2 \cos (c+d x)}{a^5 d}-\frac {b \left (2 a^2-b^2\right ) \cos ^2(c+d x)}{2 a^4 d}+\frac {\left (2 a^2-b^2\right ) \cos ^3(c+d x)}{3 a^3 d}+\frac {b \cos ^4(c+d x)}{4 a^2 d}-\frac {\cos ^5(c+d x)}{5 a d}+\frac {b \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^6 d}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 172, normalized size = 1.13 \begin {gather*} \frac {-60 a \left (5 a^4-14 a^2 b^2+8 b^4\right ) \cos (c+d x)-60 \left (3 a^4 b-2 a^2 b^3\right ) \cos (2 (c+d x))+50 a^5 \cos (3 (c+d x))-40 a^3 b^2 \cos (3 (c+d x))+15 a^4 b \cos (4 (c+d x))-6 a^5 \cos (5 (c+d x))+480 a^4 b \log (b+a \cos (c+d x))-960 a^2 b^3 \log (b+a \cos (c+d x))+480 b^5 \log (b+a \cos (c+d x))}{480 a^6 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 160, normalized size = 1.05
method | result | size |
derivativedivides | \(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\cos ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+a^{4} \cos \left (d x +c \right )-2 a^{2} b^{2} \cos \left (d x +c \right )+b^{4} \cos \left (d x +c \right )}{a^{5}}+\frac {b \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{6}}}{d}\) | \(160\) |
default | \(\frac {-\frac {\frac {\left (\cos ^{5}\left (d x +c \right )\right ) a^{4}}{5}-\frac {b \left (\cos ^{4}\left (d x +c \right )\right ) a^{3}}{4}-\frac {2 a^{4} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+\frac {a^{2} b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{3}+a^{3} b \left (\cos ^{2}\left (d x +c \right )\right )-\frac {a \,b^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{2}+a^{4} \cos \left (d x +c \right )-2 a^{2} b^{2} \cos \left (d x +c \right )+b^{4} \cos \left (d x +c \right )}{a^{5}}+\frac {b \left (a^{4}-2 b^{2} a^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{6}}}{d}\) | \(160\) |
norman | \(\frac {\frac {\left (2 b \,a^{3}+2 b^{2} a^{2}-2 b^{3} a -2 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{5} d}+\frac {-16 a^{4}+50 b^{2} a^{2}-30 b^{4}}{15 d \,a^{5}}+\frac {2 \left (5 b \,a^{3}+6 b^{2} a^{2}-3 b^{3} a -4 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5}}+\frac {\left (-16 a^{4}+6 b \,a^{3}+44 b^{2} a^{2}-6 b^{3} a -24 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a^{5} d}+\frac {2 \left (-16 a^{4}+15 b \,a^{3}+32 b^{2} a^{2}-9 b^{3} a -18 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d \,a^{5}}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}+\frac {\left (a +b \right ) b \left (a^{3}-b \,a^{2}-b^{2} a +b^{3}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{a^{6} d}-\frac {\left (a +b \right ) b \left (a^{3}-b \,a^{2}-b^{2} a +b^{3}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{6} d}\) | \(345\) |
risch | \(-\frac {i b x}{a^{2}}-\frac {3 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{16 a^{2} d}+\frac {b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{8 a^{3} d}-\frac {{\mathrm e}^{i \left (d x +c \right )} b^{4}}{2 a^{5} d}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{8 a^{3} d}-\frac {{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{2 a^{5} d}-\frac {3 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{16 a^{2} d}+\frac {b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 a^{4} d}+\frac {4 i b^{3} c}{a^{4} d}-\frac {5 \,{\mathrm e}^{-i \left (d x +c \right )}}{16 a d}+\frac {b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{2} d}-\frac {2 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{4} d}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{6} d}+\frac {b \cos \left (4 d x +4 c \right )}{32 a^{2} d}-\frac {\cos \left (3 d x +3 c \right ) b^{2}}{12 d \,a^{3}}-\frac {\cos \left (5 d x +5 c \right )}{80 a d}-\frac {5 \,{\mathrm e}^{i \left (d x +c \right )}}{16 a d}-\frac {2 i b^{5} c}{a^{6} d}-\frac {2 i b c}{a^{2} d}+\frac {5 \cos \left (3 d x +3 c \right )}{48 a d}+\frac {2 i b^{3} x}{a^{4}}-\frac {i b^{5} x}{a^{6}}\) | \(439\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 141, normalized size = 0.93 \begin {gather*} -\frac {\frac {12 \, a^{4} \cos \left (d x + c\right )^{5} - 15 \, a^{3} b \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (2 \, a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )}{a^{5}} - \frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.58, size = 140, normalized size = 0.92 \begin {gather*} -\frac {12 \, a^{5} \cos \left (d x + c\right )^{5} - 15 \, a^{4} b \cos \left (d x + c\right )^{4} - 20 \, {\left (2 \, a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )^{3} + 30 \, {\left (2 \, a^{4} b - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} + 60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right ) - 60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{60 \, a^{6} 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. 867 vs.
\(2 (144) = 288\).
time = 0.52, size = 867, normalized size = 5.70 \begin {gather*} \frac {\frac {60 \, {\left (a^{5} b - a^{4} b^{2} - 2 \, a^{3} b^{3} + 2 \, a^{2} b^{4} + a b^{5} - b^{6}\right )} \log \left ({\left | a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{7} - a^{6} b} - \frac {60 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{6}} + \frac {64 \, a^{5} - 137 \, a^{4} b - 200 \, a^{3} b^{2} + 274 \, a^{2} b^{3} + 120 \, a b^{4} - 137 \, b^{5} - \frac {320 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {805 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {880 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {1490 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {480 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {685 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {640 \, a^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1970 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1280 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3100 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {720 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1370 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1970 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {720 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3100 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {480 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1370 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {805 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {120 \, a^{3} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1490 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {120 \, a b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {685 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {137 \, a^{4} b {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {274 \, a^{2} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {137 \, b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{6} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{5}}}{60 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 151, normalized size = 0.99 \begin {gather*} -\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a}-\frac {b^2\,\left (\frac {2}{a}-\frac {b^2}{a^3}\right )}{a^2}\right )-{\cos \left (c+d\,x\right )}^3\,\left (\frac {2}{3\,a}-\frac {b^2}{3\,a^3}\right )+\frac {{\cos \left (c+d\,x\right )}^5}{5\,a}-\frac {b\,{\cos \left (c+d\,x\right )}^4}{4\,a^2}-\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (a^4\,b-2\,a^2\,b^3+b^5\right )}{a^6}+\frac {b\,{\cos \left (c+d\,x\right )}^2\,\left (\frac {2}{a}-\frac {b^2}{a^3}\right )}{2\,a}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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