Optimal. Leaf size=26 \[ \frac {b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3517, 3770} \[ \frac {b \tanh ^{-1}(\sin (c+d x))}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3517
Rule 3770
Rubi steps
\begin {align*} \int \csc (c+d x) (a+b \tan (c+d x)) \, dx &=\int (a \csc (c+d x)+b \sec (c+d x)) \, dx\\ &=a \int \csc (c+d x) \, dx+b \int \sec (c+d x) \, dx\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {b \tanh ^{-1}(\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 52, normalized size = 2.00 \[ \frac {a \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}-\frac {a \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d}+\frac {b \tanh ^{-1}(\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 58, normalized size = 2.23 \[ -\frac {a \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.49, size = 49, normalized size = 1.88 \[ \frac {b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 42, normalized size = 1.62 \[ \frac {b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 46, normalized size = 1.77 \[ \frac {b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 2 \, a \log \left (\cot \left (d x + c\right ) + \csc \left (d x + c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.75, size = 86, normalized size = 3.31 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,b\,\mathrm {atanh}\left (\frac {b\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right ) \csc {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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