Optimal. Leaf size=170 \[ -\frac {\sin ^4(c+d x) \left (a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)+b \left (5 a^4-10 a^2 b^2+b^4\right )\right )}{4 d}+\frac {\sin ^2(c+d x) \left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right )}{8 d}+\frac {1}{8} a x \left (3 a^4+10 a^2 b^2+15 b^4\right )-\frac {b^5 \log (\sin (c+d x))}{d}+\frac {b^5 \log (\tan (c+d x))}{d} \]
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Rubi [A] time = 0.22, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3088, 1805, 801, 635, 203, 260} \[ -\frac {\sin ^4(c+d x) \left (a \left (-10 a^2 b^2+a^4+5 b^4\right ) \cot (c+d x)+b \left (-10 a^2 b^2+5 a^4+b^4\right )\right )}{4 d}+\frac {\sin ^2(c+d x) \left (5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)+4 b \left (5 a^4-b^4\right )\right )}{8 d}+\frac {1}{8} a x \left (10 a^2 b^2+3 a^4+15 b^4\right )-\frac {b^5 \log (\sin (c+d x))}{d}+\frac {b^5 \log (\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 801
Rule 1805
Rule 3088
Rubi steps
\begin {align*} \int \sec (c+d x) (a \cos (c+d x)+b \sin (c+d x))^5 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(b+a x)^5}{x \left (1+x^2\right )^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {-4 b^5+a \left (a^4-10 a^2 b^2-15 b^4\right ) x-20 a^4 b x^2-4 a^5 x^3}{x \left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{4 d}\\ &=\frac {\left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{8 d}-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {8 b^5+a \left (3 a^4+10 a^2 b^2+15 b^4\right ) x}{x \left (1+x^2\right )} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=\frac {\left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{8 d}-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \left (\frac {8 b^5}{x}+\frac {3 a^5+10 a^3 b^2+15 a b^4-8 b^5 x}{1+x^2}\right ) \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=\frac {b^5 \log (\tan (c+d x))}{d}+\frac {\left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{8 d}-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}-\frac {\operatorname {Subst}\left (\int \frac {3 a^5+10 a^3 b^2+15 a b^4-8 b^5 x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=\frac {b^5 \log (\tan (c+d x))}{d}+\frac {\left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{8 d}-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}+\frac {b^5 \operatorname {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,\cot (c+d x)\right )}{d}-\frac {\left (a \left (3 a^4+10 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{8 d}\\ &=\frac {1}{8} a \left (3 a^4+10 a^2 b^2+15 b^4\right ) x-\frac {b^5 \log (\sin (c+d x))}{d}+\frac {b^5 \log (\tan (c+d x))}{d}+\frac {\left (4 b \left (5 a^4-b^4\right )+5 a \left (a^2-3 b^2\right ) \left (a^2+b^2\right ) \cot (c+d x)\right ) \sin ^2(c+d x)}{8 d}-\frac {\left (b \left (5 a^4-10 a^2 b^2+b^4\right )+a \left (a^4-10 a^2 b^2+5 b^4\right ) \cot (c+d x)\right ) \sin ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [B] time = 6.44, size = 711, normalized size = 4.18 \[ \frac {b^5 \left (\frac {\cos ^4(c+d x) (a+b \tan (c+d x))^6 \left (a b \tan (c+d x)+b^2\right )}{4 b^6 \left (a^2+b^2\right )}-\frac {\frac {\cos ^2(c+d x) (a+b \tan (c+d x))^6 \left (b \left (a \left (2 b^2-3 a^2\right )+3 a b^2\right ) \tan (c+d x)-3 a^2 b^2+b^2 \left (2 b^2-3 a^2\right )\right )}{2 b^4 \left (a^2+b^2\right )}-\frac {\left (3 a^4-29 a^2 b^2+5 a^2 \left (3 a^2-5 b^2\right )+8 b^4\right ) \left (\frac {1}{2} b^2 \left (10 a^2-b^2\right ) \tan ^2(c+d x)+5 a b \left (2 a^2-b^2\right ) \tan (c+d x)+\frac {1}{2} \left (5 a^4-10 a^2 b^2+\frac {a^5-10 a^3 b^2+5 a b^4}{\sqrt {-b^2}}+b^4\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+\frac {1}{2} \left (5 a^4-10 a^2 b^2-\frac {a^5-10 a^3 b^2+5 a b^4}{\sqrt {-b^2}}+b^4\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {5}{3} a b^3 \tan ^3(c+d x)+\frac {1}{4} b^4 \tan ^4(c+d x)\right )-5 a \left (3 a^2-5 b^2\right ) \left (a b^2 \left (10 a^2-3 b^2\right ) \tan ^2(c+d x)+\frac {1}{3} b^3 \left (15 a^2-b^2\right ) \tan ^3(c+d x)+b \left (15 a^4-15 a^2 b^2+b^4\right ) \tan (c+d x)+\frac {1}{2} \left (6 a^5-20 a^3 b^2+\frac {a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt {-b^2}}+6 a b^4\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )+\frac {1}{2} \left (6 a^5-20 a^3 b^2-\frac {a^6-15 a^4 b^2+15 a^2 b^4-b^6}{\sqrt {-b^2}}+6 a b^4\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )+\frac {3}{2} a b^4 \tan ^4(c+d x)+\frac {1}{5} b^5 \tan ^5(c+d x)\right )}{2 b^2 \left (a^2+b^2\right )}}{4 b^2 \left (a^2+b^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 160, normalized size = 0.94 \[ -\frac {8 \, b^{5} \log \left (-\cos \left (d x + c\right )\right ) + 2 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} - {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 8 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{2} - {\left (2 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} - 25 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.85, size = 199, normalized size = 1.17 \[ \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )} - \frac {6 \, b^{5} \tan \left (d x + c\right )^{4} - 3 \, a^{5} \tan \left (d x + c\right )^{3} - 10 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 25 \, a b^{4} \tan \left (d x + c\right )^{3} + 40 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + 4 \, b^{5} \tan \left (d x + c\right )^{2} - 5 \, a^{5} \tan \left (d x + c\right ) + 10 \, a^{3} b^{2} \tan \left (d x + c\right ) + 15 \, a b^{4} \tan \left (d x + c\right ) + 10 \, a^{4} b + 20 \, a^{2} b^{3}}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 272, normalized size = 1.60 \[ \frac {a^{5} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}+\frac {3 a^{5} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {3 a^{5} x}{8}+\frac {3 a^{5} c}{8 d}-\frac {5 a^{4} \left (\cos ^{4}\left (d x +c \right )\right ) b}{4 d}-\frac {5 a^{3} b^{2} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{2 d}+\frac {5 a^{3} b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{4 d}+\frac {5 a^{3} b^{2} x}{4}+\frac {5 a^{3} b^{2} c}{4 d}+\frac {5 a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{2 d}-\frac {5 a \,b^{4} \cos \left (d x +c \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{4 d}-\frac {15 a \,b^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {15 a \,b^{4} x}{8}+\frac {15 a \,b^{4} c}{8 d}-\frac {b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d}-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) b^{5}}{2 d}-\frac {b^{5} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 170, normalized size = 1.00 \[ \frac {80 \, a^{2} b^{3} \sin \left (d x + c\right )^{4} - 40 \, {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2} a^{4} b + {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{5} + 10 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} b^{2} + 5 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a b^{4} - 8 \, {\left (\sin \left (d x + c\right )^{4} + 2 \, \sin \left (d x + c\right )^{2} + 2 \, \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} b^{5}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.65, size = 297, normalized size = 1.75 \[ \frac {4\,b^5\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )-4\,b^5\,\ln \left (\frac {\cos \left (c+d\,x\right )}{\cos \left (c+d\,x\right )+1}\right )+3\,a^5\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )+\frac {3\,b^5\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {b^5\,\cos \left (4\,c+4\,d\,x\right )}{8}+a^5\,\sin \left (2\,c+2\,d\,x\right )+\frac {a^5\,\sin \left (4\,c+4\,d\,x\right )}{8}+15\,a\,b^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-\frac {5\,a^4\,b\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {5\,a^4\,b\,\cos \left (4\,c+4\,d\,x\right )}{8}-5\,a\,b^4\,\sin \left (2\,c+2\,d\,x\right )+\frac {5\,a\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{8}+10\,a^3\,b^2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )-5\,a^2\,b^3\,\cos \left (2\,c+2\,d\,x\right )+\frac {5\,a^2\,b^3\,\cos \left (4\,c+4\,d\,x\right )}{4}-\frac {5\,a^3\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{4}}{4\,d} \]
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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