Optimal. Leaf size=100 \[ -\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^2(c+d x)}{d}+\frac {b \log (\tan (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3872, 2834, 2620, 266, 43, 3768, 3770} \[ -\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {b \cot ^2(c+d x)}{d}+\frac {b \log (\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2620
Rule 2834
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \csc ^5(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^5(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^5(c+d x) \, dx+b \int \csc ^5(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{4} (3 a) \int \csc ^3(c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^5} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {1}{8} (3 a) \int \csc (c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \frac {(1+x)^2}{x^3} \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {2}{x^2}+\frac {1}{x}\right ) \, dx,x,\tan ^2(c+d x)\right )}{2 d}\\ &=-\frac {3 a \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {b \cot ^2(c+d x)}{d}-\frac {b \cot ^4(c+d x)}{4 d}-\frac {3 a \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a \cot (c+d x) \csc ^3(c+d x)}{4 d}+\frac {b \log (\tan (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.65, size = 164, normalized size = 1.64 \[ -\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {3 a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {3 a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {b \left (\csc ^4(c+d x)+2 \csc ^2(c+d x)-4 \log (\sin (c+d x))+4 \log (\cos (c+d x))\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 201, normalized size = 2.01 \[ \frac {6 \, a \cos \left (d x + c\right )^{3} + 8 \, b \cos \left (d x + c\right )^{2} - 10 \, a \cos \left (d x + c\right ) - 16 \, {\left (b \cos \left (d x + c\right )^{4} - 2 \, b \cos \left (d x + c\right )^{2} + b\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left ({\left (3 \, a - 8 \, b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a - 8 \, b\right )} \cos \left (d x + c\right )^{2} + 3 \, a - 8 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (3 \, a + 8 \, b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a + 8 \, b\right )} \cos \left (d x + c\right )^{2} + 3 \, a + 8 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, b}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 266, normalized size = 2.66 \[ \frac {4 \, {\left (3 \, a + 8 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 64 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {{\left (a + b - \frac {8 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {18 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {8 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {12 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 102, normalized size = 1.02 \[ -\frac {a \cot \left (d x +c \right ) \left (\csc ^{3}\left (d x +c \right )\right )}{4 d}-\frac {3 a \cot \left (d x +c \right ) \csc \left (d x +c \right )}{8 d}+\frac {3 a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {b}{4 d \sin \left (d x +c \right )^{4}}-\frac {b}{2 d \sin \left (d x +c \right )^{2}}+\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 110, normalized size = 1.10 \[ -\frac {{\left (3 \, a - 8 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (3 \, a + 8 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 16 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (3 \, a \cos \left (d x + c\right )^{3} + 4 \, b \cos \left (d x + c\right )^{2} - 5 \, a \cos \left (d x + c\right ) - 6 \, b\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 117, normalized size = 1.17 \[ \frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3\,a}{16}+\frac {b}{2}\right )}{d}-\frac {-\frac {3\,a\,{\cos \left (c+d\,x\right )}^3}{8}-\frac {b\,{\cos \left (c+d\,x\right )}^2}{2}+\frac {5\,a\,\cos \left (c+d\,x\right )}{8}+\frac {3\,b}{4}}{d\,\left ({\cos \left (c+d\,x\right )}^4-2\,{\cos \left (c+d\,x\right )}^2+1\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {3\,a}{16}-\frac {b}{2}\right )}{d}-\frac {b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \csc ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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