Optimal. Leaf size=35 \[ -\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b} \]
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Rubi [A] time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4410, 3767, 8} \[ -\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 4410
Rubi steps
\begin {align*} \int (c+d x) \cot (a+b x) \csc ^2(a+b x) \, dx &=-\frac {(c+d x) \csc ^2(a+b x)}{2 b}+\frac {d \int \csc ^2(a+b x) \, dx}{2 b}\\ &=-\frac {(c+d x) \csc ^2(a+b x)}{2 b}-\frac {d \operatorname {Subst}(\int 1 \, dx,x,\cot (a+b x))}{2 b^2}\\ &=-\frac {d \cot (a+b x)}{2 b^2}-\frac {(c+d x) \csc ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 48, normalized size = 1.37 \[ -\frac {d \cot (a+b x)}{2 b^2}-\frac {c \csc ^2(a+b x)}{2 b}-\frac {d x \csc ^2(a+b x)}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 44, normalized size = 1.26 \[ \frac {b d x + d \cos \left (b x + a\right ) \sin \left (b x + a\right ) + b c}{2 \, {\left (b^{2} \cos \left (b x + a\right )^{2} - b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 526, normalized size = 15.03 \[ -\frac {b d x \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{4} + b c \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{3} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{4} + b d x \tan \left (\frac {1}{2} \, b x\right )^{4} + 4 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + b d x \tan \left (\frac {1}{2} \, a\right )^{4} + b c \tan \left (\frac {1}{2} \, b x\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right ) + 4 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} + 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{2} + 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{3} + b c \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, d \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{4} + 2 \, b d x \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b d x \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b c \tan \left (\frac {1}{2} \, b x\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, b x\right )^{3} - 12 \, d \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right ) + 2 \, b c \tan \left (\frac {1}{2} \, a\right )^{2} - 12 \, d \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, a\right )^{3} + b d x + b c + 2 \, d \tan \left (\frac {1}{2} \, b x\right ) + 2 \, d \tan \left (\frac {1}{2} \, a\right )}{8 \, {\left (b^{2} \tan \left (\frac {1}{2} \, b x\right )^{4} \tan \left (\frac {1}{2} \, a\right )^{2} + 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right )^{3} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{4} - 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{3} \tan \left (\frac {1}{2} \, a\right ) - 4 \, b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} \tan \left (\frac {1}{2} \, a\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right )^{3} + b^{2} \tan \left (\frac {1}{2} \, b x\right )^{2} + 2 \, b^{2} \tan \left (\frac {1}{2} \, b x\right ) \tan \left (\frac {1}{2} \, a\right ) + b^{2} \tan \left (\frac {1}{2} \, a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 61, normalized size = 1.74 \[ \frac {\frac {d \left (-\frac {b x +a}{2 \sin \left (b x +a \right )^{2}}-\frac {\cot \left (b x +a \right )}{2}\right )}{b}+\frac {d a}{2 b \sin \left (b x +a \right )^{2}}-\frac {c}{2 \sin \left (b x +a \right )^{2}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 287, normalized size = 8.20 \[ \frac {\frac {2 \, {\left (4 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - {\left (2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b x + a\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (2 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \sin \left (4 \, b x + 4 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (2 \, {\left (2 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )^{2} - 4 \, \cos \left (2 \, b x + 2 \, a\right )^{2} - \sin \left (4 \, b x + 4 \, a\right )^{2} + 4 \, \sin \left (4 \, b x + 4 \, a\right ) \sin \left (2 \, b x + 2 \, a\right ) - 4 \, \sin \left (2 \, b x + 2 \, a\right )^{2} + 4 \, \cos \left (2 \, b x + 2 \, a\right ) - 1\right )} b} - \frac {c}{\sin \left (b x + a\right )^{2}} + \frac {a d}{b \sin \left (b x + a\right )^{2}}}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.72, size = 53, normalized size = 1.51 \[ \frac {d\,1{}\mathrm {i}-{\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,\left (-b\,\left (2\,c+2\,d\,x\right )+d\,1{}\mathrm {i}\right )}{b^2\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-1\right )}^2} \]
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 \[ \int \left (c + d x\right ) \cos {\left (a + b x \right )} \csc ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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