Optimal. Leaf size=68 \[ \frac {\sqrt {\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 68, 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 = {2577} \[ \frac {\sqrt {\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1)} \]
Antiderivative was successfully verified.
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Rule 2577
Rubi steps
\begin {align*} \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx &=\frac {\sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) \sec (a+b x) (c \sin (a+b x))^{1+m}}{b c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.93 \[ \frac {\sqrt {\cos ^2(a+b x)} \tan (a+b x) (c \sin (a+b x))^m \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4}, x\right ) \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.38, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{4}\left (b x +a \right )\right ) \left (c \sin \left (b x +a \right )\right )^{m}\, dx \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4}\,{d x} \]
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 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{{\cos \left (a+b\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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