Optimal. Leaf size=48 \[ \frac {(c \sin (a+b x))^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 48, 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 = {2564, 364} \[ \frac {(c \sin (a+b x))^{m+1} \, _2F_1\left (2,\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 2564
Rubi steps
\begin {align*} \int \sec ^3(a+b x) (c \sin (a+b x))^m \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^m}{\left (1-\frac {x^2}{c^2}\right )^2} \, dx,x,c \sin (a+b x)\right )}{b c}\\ &=\frac {\, _2F_1\left (2,\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^{1+m}}{b c (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.06 \[ \frac {\sin (a+b x) (c \sin (a+b x))^m \, _2F_1\left (2,\frac {m+1}{2};\frac {m+1}{2}+1;\sin ^2(a+b x)\right )}{b (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3}, x\right ) \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{3}\left (b x +a \right )\right ) \left (c \sin \left (b x +a \right )\right )^{m}\, dx \]
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 \[ \int \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{3}\,{d x} \]
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 \[ \int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{{\cos \left (a+b\,x\right )}^3} \,d x \]
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 \[ \int \left (c \sin {\left (a + b x \right )}\right )^{m} \sec ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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