Optimal. Leaf size=68 \[ \frac {\sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) \sec (a+b x) (c \sin (a+b x))^{1+m}}{b c (1+m)} \]
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Rubi [A]
time = 0.03, 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 = {2657}
\begin {gather*} \frac {\sqrt {\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \, _2F_1\left (\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};\sin ^2(a+b x)\right )}{b c (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2657
Rubi steps
\begin {align*} \int \sec ^2(a+b x) (c \sin (a+b x))^m \, dx &=\frac {\sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {3}{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 = 0.03, size = 63, normalized size = 0.93 \begin {gather*} \frac {\sqrt {\cos ^2(a+b x)} \, _2F_1\left (\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (\sec ^{2}\left (b x +a \right )\right ) \left (c \sin \left (b x +a \right )\right )^{m}\, dx\]
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 \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 \left (c \sin {\left (a + b x \right )}\right )^{m} \sec ^{2}{\left (a + b x \right )}\, 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 (c\,\sin \left (a+b\,x\right )\right )}^m}{{\cos \left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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