Optimal. Leaf size=43 \[ -\frac {(2 a+3 b) \cot (c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
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Rubi [A]
time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3091, 3852, 8}
\begin {gather*} -\frac {(2 a+3 b) \cot (c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3091
Rule 3852
Rubi steps
\begin {align*} \int \csc ^4(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}+\frac {1}{3} (2 a+3 b) \int \csc ^2(c+d x) \, dx\\ &=-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {(2 a+3 b) \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=-\frac {(2 a+3 b) \cot (c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 1.14 \begin {gather*} -\frac {2 a \cot (c+d x)}{3 d}-\frac {b \cot (c+d x)}{d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 35, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-\cot \left (d x +c \right ) b}{d}\) | \(35\) |
default | \(\frac {a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-\cot \left (d x +c \right ) b}{d}\) | \(35\) |
risch | \(-\frac {2 i \left (3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-6 a \,{\mathrm e}^{2 i \left (d x +c \right )}-6 b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a +3 b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}\) | \(63\) |
norman | \(\frac {-\frac {a}{24 d}+\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {\left (5 a +6 b \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {\left (5 a +6 b \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {\left (11 a +12 b \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {\left (11 a +12 b \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\) | \(144\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 28, normalized size = 0.65 \begin {gather*} -\frac {3 \, {\left (a + b\right )} \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 54, normalized size = 1.26 \begin {gather*} -\frac {{\left (2 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \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 ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 37, normalized size = 0.86 \begin {gather*} -\frac {3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.37, size = 29, normalized size = 0.67 \begin {gather*} -\frac {a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {\mathrm {cot}\left (c+d\,x\right )\,\left (a+b\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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