Optimal. Leaf size=111 \[ -\frac {\sqrt {2} F_1\left (\frac {1}{2};\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}}{(a+b) d \sqrt {1+\sin (c+d x)} \sqrt [3]{a+b \sin (c+d x)}} \]
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Rubi [A]
time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2744, 144, 143}
\begin {gather*} -\frac {\sqrt {2} \cos (c+d x) \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}} F_1\left (\frac {1}{2};\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right )}{d (a+b) \sqrt {\sin (c+d x)+1} \sqrt [3]{a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 143
Rule 144
Rule 2744
Rubi steps
\begin {align*} \int \frac {1}{(a+b \sin (c+d x))^{4/3}} \, dx &=\frac {\cos (c+d x) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} (a+b x)^{4/3}} \, dx,x,\sin (c+d x)\right )}{d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)}}\\ &=\frac {\left (\cos (c+d x) \sqrt [3]{-\frac {a+b \sin (c+d x)}{-a-b}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {1+x} \left (-\frac {a}{-a-b}-\frac {b x}{-a-b}\right )^{4/3}} \, dx,x,\sin (c+d x)\right )}{(a+b) d \sqrt {1-\sin (c+d x)} \sqrt {1+\sin (c+d x)} \sqrt [3]{a+b \sin (c+d x)}}\\ &=-\frac {\sqrt {2} F_1\left (\frac {1}{2};\frac {1}{2},\frac {4}{3};\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {b (1-\sin (c+d x))}{a+b}\right ) \cos (c+d x) \sqrt [3]{\frac {a+b \sin (c+d x)}{a+b}}}{(a+b) d \sqrt {1+\sin (c+d x)} \sqrt [3]{a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(262\) vs. \(2(111)=222\).
time = 1.23, size = 262, normalized size = 2.36 \begin {gather*} -\frac {3 \sec (c+d x) \left (5 a F_1\left (\frac {2}{3};\frac {1}{2},\frac {1}{2};\frac {5}{3};\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}} (a+b \sin (c+d x))-2 \left (5 b^2 \cos ^2(c+d x)+2 F_1\left (\frac {5}{3};\frac {1}{2},\frac {1}{2};\frac {8}{3};\frac {a+b \sin (c+d x)}{a-b},\frac {a+b \sin (c+d x)}{a+b}\right ) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {\frac {b (1+\sin (c+d x))}{-a+b}} (a+b \sin (c+d x))^2\right )\right )}{10 b \left (a^2-b^2\right ) d \sqrt [3]{a+b \sin (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \sin \left (d x +c \right )\right )^{\frac {4}{3}}}\, dx\]
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 \frac {1}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {4}{3}}}\, 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 {1}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{4/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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