Optimal. Leaf size=76 \[ -\frac {c \sin ^2(a+b x)^{3/4} (b \sec (a+b x))^{n-1} \, _2F_1\left (\frac {3}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right )}{(1-n) (c \sin (a+b x))^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2587, 2576} \[ -\frac {c \sin ^2(a+b x)^{3/4} (b \sec (a+b x))^{n-1} \, _2F_1\left (\frac {3}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right )}{(1-n) (c \sin (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2576
Rule 2587
Rubi steps
\begin {align*} \int \frac {(b \sec (a+b x))^n}{\sqrt {c \sin (a+b x)}} \, dx &=\left (b^2 (b \cos (a+b x))^{-1+n} (b \sec (a+b x))^{-1+n}\right ) \int \frac {(b \cos (a+b x))^{-n}}{\sqrt {c \sin (a+b x)}} \, dx\\ &=-\frac {c \, _2F_1\left (\frac {3}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right ) (b \sec (a+b x))^{-1+n} \sin ^2(a+b x)^{3/4}}{(1-n) (c \sin (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 72, normalized size = 0.95 \[ \frac {\sin (2 (a+b x)) \cos ^2(a+b x)^{\frac {n-1}{2}} (b \sec (a+b x))^n \, _2F_1\left (\frac {1}{4},\frac {n+1}{2};\frac {5}{4};\sin ^2(a+b x)\right )}{b \sqrt {c \sin (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sin \left (b x + a\right )} \left (b \sec \left (b x + a\right )\right )^{n}}{c \sin \left (b x + a\right )}, x\right ) \]
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 \[ \int \frac {\left (b \sec \left (b x + a\right )\right )^{n}}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \sec \left (b x +a \right )\right )^{n}}{\sqrt {c \sin \left (b x +a \right )}}\, dx \]
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 \[ \int \frac {\left (b \sec \left (b x + a\right )\right )^{n}}{\sqrt {c \sin \left (b x + a\right )}}\,{d x} \]
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 \[ \int \frac {{\left (\frac {b}{\cos \left (a+b\,x\right )}\right )}^n}{\sqrt {c\,\sin \left (a+b\,x\right )}} \,d x \]
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 \[ \int \frac {\left (b \sec {\left (a + b x \right )}\right )^{n}}{\sqrt {c \sin {\left (a + b x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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