Optimal. Leaf size=79 \[ \frac {2 \left (a^2-b^2\right )}{b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac {4 a \sqrt {a+b \sin (c+d x)}}{b^3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2668, 697} \[ \frac {2 \left (a^2-b^2\right )}{b^3 d \sqrt {a+b \sin (c+d x)}}-\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac {4 a \sqrt {a+b \sin (c+d x)}}{b^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {-a^2+b^2}{(a+x)^{3/2}}+\frac {2 a}{\sqrt {a+x}}-\sqrt {a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {2 \left (a^2-b^2\right )}{b^3 d \sqrt {a+b \sin (c+d x)}}+\frac {4 a \sqrt {a+b \sin (c+d x)}}{b^3 d}-\frac {2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 57, normalized size = 0.72 \[ \frac {16 a^2+8 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-7 b^2}{3 b^3 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 67, normalized size = 0.85 \[ \frac {2 \, {\left (b^{2} \cos \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) + 8 \, a^{2} - 4 \, b^{2}\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{3 \, {\left (b^{4} d \sin \left (d x + c\right ) + a b^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.56, size = 72, normalized size = 0.91 \[ \frac {2 \, {\left (\frac {3 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {b \sin \left (d x + c\right ) + a} b^{3}} - \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b \sin \left (d x + c\right ) + a} a b^{6}}{b^{9}}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 54, normalized size = 0.68 \[ \frac {\frac {2 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{3}+\frac {8 a b \sin \left (d x +c \right )}{3}+\frac {16 a^{2}}{3}-\frac {8 b^{2}}{3}}{b^{3} \sqrt {a +b \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 67, normalized size = 0.85 \[ -\frac {2 \, {\left (\frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 6 \, \sqrt {b \sin \left (d x + c\right ) + a} a}{b^{2}} - \frac {3 \, {\left (a^{2} - b^{2}\right )}}{\sqrt {b \sin \left (d x + c\right ) + a} b^{2}}\right )}}{3 \, b d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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