Optimal. Leaf size=64 \[ -\frac {a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {3872, 2834, 2620, 14, 3768, 3770} \[ -\frac {a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2620
Rule 2834
Rule 3768
Rule 3770
Rule 3872
Rubi steps
\begin {align*} \int \csc ^3(c+d x) (a+b \sec (c+d x)) \, dx &=-\int (-b-a \cos (c+d x)) \csc ^3(c+d x) \sec (c+d x) \, dx\\ &=a \int \csc ^3(c+d x) \, dx+b \int \csc ^3(c+d x) \sec (c+d x) \, dx\\ &=-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {1}{2} a \int \csc (c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int \frac {1+x^2}{x^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{x^3}+\frac {1}{x}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b \cot ^2(c+d x)}{2 d}-\frac {a \cot (c+d x) \csc (c+d x)}{2 d}+\frac {b \log (\tan (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 114, normalized size = 1.78 \[ -\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {b \left (\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 123, normalized size = 1.92 \[ \frac {2 \, a \cos \left (d x + c\right ) - 4 \, {\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\cos \left (d x + c\right )\right ) - {\left ({\left (a - 2 \, b\right )} \cos \left (d x + c\right )^{2} - a + 2 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, b}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.25, size = 169, normalized size = 2.64 \[ \frac {2 \, {\left (a + 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 8 \, b \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {{\left (a + b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 68, normalized size = 1.06 \[ -\frac {a \cot \left (d x +c \right ) \csc \left (d x +c \right )}{2 d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-\frac {b}{2 d \sin \left (d x +c \right )^{2}}+\frac {b \ln \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 71, normalized size = 1.11 \[ -\frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) - {\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) + 4 \, b \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (a \cos \left (d x + c\right ) + b\right )}}{\cos \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 76, normalized size = 1.19 \[ \frac {\frac {\frac {b}{2}+\frac {a\,\cos \left (c+d\,x\right )}{2}}{{\cos \left (c+d\,x\right )}^2-1}+\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {a}{4}+\frac {b}{2}\right )-\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (\frac {a}{4}-\frac {b}{2}\right )-b\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \csc ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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