Optimal. Leaf size=78 \[ -\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (\frac {5}{4},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt {c \sin (a+b x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {2656}
\begin {gather*} -\frac {\sqrt [4]{\sin ^2(a+b x)} (d \cos (a+b x))^{n+1} \, _2F_1\left (\frac {5}{4},\frac {n+1}{2};\frac {n+3}{2};\cos ^2(a+b x)\right )}{b c d (n+1) \sqrt {c \sin (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rubi steps
\begin {align*} \int \frac {(d \cos (a+b x))^n}{(c \sin (a+b x))^{3/2}} \, dx &=-\frac {(d \cos (a+b x))^{1+n} \, _2F_1\left (\frac {5}{4},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sqrt [4]{\sin ^2(a+b x)}}{b c d (1+n) \sqrt {c \sin (a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 79, normalized size = 1.01 \begin {gather*} -\frac {(d \cos (a+b x))^n \cot (a+b x) \, _2F_1\left (\frac {5}{4},\frac {1+n}{2};\frac {3+n}{2};\cos ^2(a+b x)\right ) \sqrt {c \sin (a+b x)} \sqrt [4]{\sin ^2(a+b x)}}{b c^2 (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (d \cos \left (b x +a \right )\right )^{n}}{\left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 \frac {\left (d \cos {\left (a + b x \right )}\right )^{n}}{\left (c \sin {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,\cos \left (a+b\,x\right )\right )}^n}{{\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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