Optimal. Leaf size=16 \[ b x-\frac {a \cot (c+d x)}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3012, 8} \[ b x-\frac {a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3012
Rubi steps
\begin {align*} \int \csc ^2(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {a \cot (c+d x)}{d}+b \int 1 \, dx\\ &=b x-\frac {a \cot (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 1.00 \[ b x-\frac {a \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 32, normalized size = 2.00 \[ \frac {b d x \sin \left (d x + c\right ) - a \cos \left (d x + c\right )}{d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 39, normalized size = 2.44 \[ \frac {2 \, {\left (d x + c\right )} b + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.43, size = 22, normalized size = 1.38 \[ \frac {-\cot \left (d x +c \right ) a +\left (d x +c \right ) b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 23, normalized size = 1.44 \[ \frac {{\left (d x + c\right )} b - \frac {a}{\tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.36, size = 16, normalized size = 1.00 \[ b\,x-\frac {a\,\mathrm {cot}\left (c+d\,x\right )}{d} \]
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 \[ \int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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