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 = 0.04, 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 = {3012, 3767, 8} \[ -\frac {(2 a+3 b) \cot (c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3012
Rule 3767
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) \operatorname {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 = 0.03, size = 49, normalized size = 1.14 \[ -\frac {2 a \cot (c+d x)}{3 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {b \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 54, normalized size = 1.26 \[ -\frac {{\left (2 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (a + b\right )} \cos \left (d x + c\right )}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.86 \[ -\frac {3 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 35, normalized size = 0.81 \[ \frac {a \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )-b \cot \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 28, normalized size = 0.65 \[ -\frac {3 \, {\left (a + b\right )} \tan \left (d x + c\right )^{2} + a}{3 \, d \tan \left (d x + c\right )^{3}} \]
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 \[ -\frac {a\,{\mathrm {cot}\left (c+d\,x\right )}^3}{3\,d}-\frac {\mathrm {cot}\left (c+d\,x\right )\,\left (a+b\right )}{d} \]
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 \[ \int \left (a + b \sin ^{2}{\left (c + d x \right )}\right ) \csc ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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