Optimal. Leaf size=126 \[ -\frac {\cot ^4(c+d x) \left (a^2+2 a b \sec (c+d x)+b^2\right )}{4 d}+\frac {a^2 \log (\cos (c+d x))}{d}+\frac {a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac {a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac {a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d} \]
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Rubi [A] time = 0.16, antiderivative size = 126, 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 = {3885, 1805, 823, 801} \[ -\frac {\cot ^4(c+d x) \left (a^2+2 a b \sec (c+d x)+b^2\right )}{4 d}+\frac {a^2 \log (\cos (c+d x))}{d}+\frac {a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac {a (4 a-3 b) \log (\sec (c+d x)+1)}{8 d}+\frac {a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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Rule 801
Rule 823
Rule 1805
Rule 3885
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \sec (c+d x))^2 \, dx &=-\frac {b^6 \operatorname {Subst}\left (\int \frac {(a+x)^2}{x \left (b^2-x^2\right )^3} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac {\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac {b^4 \operatorname {Subst}\left (\int \frac {-4 a^2-6 a x}{x \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{4 d}\\ &=\frac {a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac {\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac {\operatorname {Subst}\left (\int \frac {-8 a^2 b^2-6 a b^2 x}{x \left (b^2-x^2\right )} \, dx,x,b \sec (c+d x)\right )}{8 d}\\ &=\frac {a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac {\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}+\frac {\operatorname {Subst}\left (\int \left (-\frac {a (4 a+3 b)}{b-x}-\frac {8 a^2}{x}+\frac {a (4 a-3 b)}{b+x}\right ) \, dx,x,b \sec (c+d x)\right )}{8 d}\\ &=\frac {a^2 \log (\cos (c+d x))}{d}+\frac {a (4 a+3 b) \log (1-\sec (c+d x))}{8 d}+\frac {a (4 a-3 b) \log (1+\sec (c+d x))}{8 d}+\frac {a \cot ^2(c+d x) (2 a+3 b \sec (c+d x))}{4 d}-\frac {\cot ^4(c+d x) \left (a^2+b^2+2 a b \sec (c+d x)\right )}{4 d}\\ \end {align*}
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Mathematica [A] time = 3.19, size = 148, normalized size = 1.17 \[ \frac {2 \left (7 a^2+10 a b+3 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+2 \left (7 a^2-10 a b+3 b^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )-(a+b)^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )-(a-b)^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )+16 a \left ((4 a+3 b) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+(4 a-3 b) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{64 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 203, normalized size = 1.61 \[ -\frac {10 \, a b \cos \left (d x + c\right )^{3} - 6 \, a b \cos \left (d x + c\right ) + 4 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 6 \, a^{2} - 2 \, b^{2} - {\left ({\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} - 3 \, a b\right )} \cos \left (d x + c\right )^{2} + 4 \, a^{2} - 3 \, a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left ({\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (4 \, a^{2} + 3 \, a b\right )} \cos \left (d x + c\right )^{2} + 4 \, a^{2} + 3 \, a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{8 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 360, normalized size = 2.86 \[ -\frac {64 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) + \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 8 \, {\left (4 \, a^{2} + 3 \, a b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac {{\left (a^{2} + 2 \, a b + b^{2} + \frac {12 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {16 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {48 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {36 \, a b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 169, normalized size = 1.34 \[ -\frac {a^{2} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a^{2} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{4}}+\frac {a b \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{2}}+\frac {a b \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a b \cos \left (d x +c \right )}{4 d}+\frac {3 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{4 d}-\frac {b^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 122, normalized size = 0.97 \[ \frac {{\left (4 \, a^{2} - 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + {\left (4 \, a^{2} + 3 \, a b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (5 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right ) + 2 \, {\left (2 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{2} - 3 \, a^{2} - b^{2}\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.45, size = 164, normalized size = 1.30 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {5\,a^2}{32}-\frac {3\,a\,b}{16}+\frac {b^2}{32}+\frac {{\left (a-b\right )}^2}{32}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\left (a-b\right )}^2}{64\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^2+\frac {3\,b\,a}{4}\right )}{d}-\frac {a^2\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a\,b}{2}+\frac {a^2}{4}+\frac {b^2}{4}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (3\,a^2+4\,a\,b+b^2\right )\right )}{16\,d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right )^{2} \cot ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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