Optimal. Leaf size=36 \[ \frac {a \sec (c+d x)}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2838, 2622, 321, 207, 3767, 8} \[ \frac {a \sec (c+d x)}{d}-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 207
Rule 321
Rule 2622
Rule 2838
Rule 3767
Rubi steps
\begin {align*} \int \csc (c+d x) \sec ^2(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \csc (c+d x) \sec ^2(c+d x) \, dx+b \int \sec ^2(c+d x) \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}-\frac {b \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d}\\ &=\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}+\frac {a \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a \sec (c+d x)}{d}+\frac {b \tan (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 1.56 \[ \frac {a \sec (c+d x)}{d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 65, normalized size = 1.81 \[ -\frac {a \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - a \cos \left (d x + c\right ) \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, b \sin \left (d x + c\right ) - 2 \, a}{2 \, d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 48, normalized size = 1.33 \[ \frac {a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 47, normalized size = 1.31 \[ \frac {a}{d \cos \left (d x +c \right )}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {b \tan \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 48, normalized size = 1.33 \[ \frac {a {\left (\frac {2}{\cos \left (d x + c\right )} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 2 \, b \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.94, size = 52, normalized size = 1.44 \[ \frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {2\,a+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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 \[ \int \left (a + b \sin {\left (c + d x \right )}\right ) \csc {\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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