Optimal. Leaf size=76 \[ -\frac {c \, _2F_1\left (-\frac {1}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right ) (b \sec (a+b x))^{-1+n} \sqrt {c \sin (a+b x)}}{(1-n) \sqrt [4]{\sin ^2(a+b x)}} \]
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Rubi [A]
time = 0.08, 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 = {2667, 2656}
\begin {gather*} -\frac {c \sqrt {c \sin (a+b x)} (b \sec (a+b x))^{n-1} \, _2F_1\left (-\frac {1}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right )}{(1-n) \sqrt [4]{\sin ^2(a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2656
Rule 2667
Rubi steps
\begin {align*} \int (b \sec (a+b x))^n (c \sin (a+b x))^{3/2} \, dx &=\left (b^2 (b \cos (a+b x))^{-1+n} (b \sec (a+b x))^{-1+n}\right ) \int (b \cos (a+b x))^{-n} (c \sin (a+b x))^{3/2} \, dx\\ &=-\frac {c \, _2F_1\left (-\frac {1}{4},\frac {1-n}{2};\frac {3-n}{2};\cos ^2(a+b x)\right ) (b \sec (a+b x))^{-1+n} \sqrt {c \sin (a+b x)}}{(1-n) \sqrt [4]{\sin ^2(a+b x)}}\\ \end {align*}
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Mathematica [A]
time = 42.08, size = 104, normalized size = 1.37 \begin {gather*} \frac {2 \cos ^2(a+b x)^{\frac {1}{2} (-1+n)} (b \sec (a+b x))^{-1+n} (c \sin (a+b x))^{5/2} \left (9 \, _2F_1\left (\frac {5}{4},\frac {1}{2} (-1+n);\frac {9}{4};\sin ^2(a+b x)\right )+5 \, _2F_1\left (\frac {9}{4},\frac {1+n}{2};\frac {13}{4};\sin ^2(a+b x)\right ) \sin ^2(a+b x)\right )}{45 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \left (b \sec \left (b x +a \right )\right )^{n} \left (c \sin \left (b x +a \right )\right )^{\frac {3}{2}}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (c\,\sin \left (a+b\,x\right )\right )}^{3/2}\,{\left (\frac {b}{\cos \left (a+b\,x\right )}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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