Optimal. Leaf size=49 \[ -\frac {(d \cos (a+b x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2565, 364} \[ -\frac {(d \cos (a+b x))^{n+1} \, _2F_1\left (2,\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2565
Rubi steps
\begin {align*} \int (d \cos (a+b x))^n \csc ^3(a+b x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {x^n}{\left (1-\frac {x^2}{d^2}\right )^2} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (2,\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right )}{b d (1+n)}\\ \end {align*}
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Mathematica [B] time = 2.70, size = 154, normalized size = 3.14 \[ -\frac {2^{-n-3} \cos (a+b x) (d \cos (a+b x))^n \left (2^{n+1} \, _2F_1(1,n+1;n+2;\cos (a+b x))+2^{n+1} \, _2F_1(2,n+1;n+2;\cos (a+b x))+\sec ^2\left (\frac {1}{2} (a+b x)\right )^{n+1} \left (\, _2F_1\left (n,n+1;n+2;\frac {1}{2} \cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )\right )+\, _2F_1\left (n+1,n+1;n+2;\frac {1}{2} \cos (a+b x) \sec ^2\left (\frac {1}{2} (a+b x)\right )\right )\right )\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{3}, x\right ) \]
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 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \left (d \cos \left (b x +a \right )\right )^{n} \left (\csc ^{3}\left (b x +a \right )\right )\, dx \]
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 \[ \int \left (d \cos \left (b x + a\right )\right )^{n} \csc \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{{\sin \left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \left (d \cos {\left (a + b x \right )}\right )^{n} \csc ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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