Optimal. Leaf size=69 \[ -\frac {\sqrt {\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac {3}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2576} \[ -\frac {\sqrt {\sin ^2(a+b x)} \csc (a+b x) (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac {3}{2},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin {align*} \int (d \cos (a+b x))^n \csc ^2(a+b x) \, dx &=-\frac {(d \cos (a+b x))^{1+n} \csc (a+b x) \, _2F_1\left (\frac {3}{2},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sqrt {\sin ^2(a+b x)}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 80, normalized size = 1.16 \[ \frac {d \csc (a+b x) \left (-\cot ^2(a+b x)\right )^{\frac {1-n}{2}} (d \cos (a+b x))^{n-1} \, _2F_1\left (\frac {1-n}{2},1-\frac {n}{2};2-\frac {n}{2};\csc ^2(a+b x)\right )}{b (n-2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.41, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x +a \right )\right )^{n} \left (\csc ^{2}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{{\sin \left (a+b\,x\right )}^2} \,d x \]
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 \[ \int \left (d \cos {\left (a + b x \right )}\right )^{n} \csc ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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