Optimal. Leaf size=83 \[ -\frac {2 c \sin (a+b x) \cos ^{m+1}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3 m+2);\frac {3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2) \sqrt {\sin ^2(a+b x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3208, 2643} \[ -\frac {2 c \sin (a+b x) \cos ^{m+1}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3 m+2);\frac {3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2) \sqrt {\sin ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2643
Rule 3208
Rubi steps
\begin {align*} \int \left (c \cos ^m(a+b x)\right )^{3/2} \, dx &=\left (c \cos ^{-\frac {m}{2}}(a+b x) \sqrt {c \cos ^m(a+b x)}\right ) \int \cos ^{\frac {3 m}{2}}(a+b x) \, dx\\ &=-\frac {2 c \cos ^{1+m}(a+b x) \sqrt {c \cos ^m(a+b x)} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (2+3 m);\frac {3 (2+m)}{4};\cos ^2(a+b x)\right ) \sin (a+b x)}{b (2+3 m) \sqrt {\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 72, normalized size = 0.87 \[ -\frac {2 \sqrt {\sin ^2(a+b x)} \cot (a+b x) \left (c \cos ^m(a+b x)\right )^{3/2} \, _2F_1\left (\frac {1}{2},\frac {1}{4} (3 m+2);\frac {3 (m+2)}{4};\cos ^2(a+b x)\right )}{b (3 m+2)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \left (c \cos \left (b x + a\right )^{m}\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \left (c \left (\cos ^{m}\left (b x +a \right )\right )\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int {\left (c\,{\cos \left (a+b\,x\right )}^m\right )}^{3/2} \,d x \]
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 \[ \int \left (c \cos ^{m}{\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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