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 = {3091, 8}
\begin {gather*} b x-\frac {a \cot (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3091
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 = 0.01, size = 16, normalized size = 1.00 \begin {gather*} b x-\frac {a \cot (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 22, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {-\cot \left (d x +c \right ) a +b \left (d x +c \right )}{d}\) | \(22\) |
default | \(\frac {-\cot \left (d x +c \right ) a +b \left (d x +c \right )}{d}\) | \(22\) |
risch | \(b x -\frac {2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}\) | \(25\) |
norman | \(\frac {b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+b x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {a}{2 d}-\frac {a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+2 b x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 23, normalized size = 1.44 \begin {gather*} \frac {{\left (d x + c\right )} b - \frac {a}{\tan \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 32, normalized size = 2.00 \begin {gather*} \frac {b d x \sin \left (d x + c\right ) - a \cos \left (d x + c\right )}{d \sin \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (16) = 32\).
time = 0.48, size = 39, normalized size = 2.44 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} b\,x-\frac {a\,\mathrm {cot}\left (c+d\,x\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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