Optimal. Leaf size=73 \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2675, 2667, 63, 206} \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 2667
Rule 2675
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {1}{4} a \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{2 d}\\ &=\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2 \sqrt {2} d}+\frac {\sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 72, normalized size = 0.99 \[ \frac {a \left (\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a (\sin (c+d x)+1)}}{\sqrt {2} \sqrt {a}}\right )-\frac {2 \sqrt {a (\sin (c+d x)+1)}}{\sin (c+d x)-1}\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 99, normalized size = 1.36 \[ \frac {{\left (\sqrt {2} a \sin \left (d x + c\right ) - \sqrt {2} a\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, \sqrt {a \sin \left (d x + c\right ) + a} a}{8 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.19, size = 70, normalized size = 0.96 \[ \frac {2 a^{3} \left (-\frac {\sqrt {a +a \sin \left (d x +c \right )}}{4 a \left (a \sin \left (d x +c \right )-a \right )}+\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 94, normalized size = 1.29 \[ -\frac {\sqrt {2} a^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) + \frac {4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}}{a \sin \left (d x + c\right ) - a}}{8 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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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