Optimal. Leaf size=69 \[ \frac {a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d} \]
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Rubi [A] time = 0.11, antiderivative size = 69, 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 = {2676, 2667, 63, 206} \[ \frac {a \sec ^2(c+d x) (a \sin (c+d x)+a)^{3/2}}{d}-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rule 2667
Rule 2676
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {1}{2} a^2 \int \sec (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{d}\\ &=-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d}+\frac {a \sec ^2(c+d x) (a+a \sin (c+d x))^{3/2}}{d}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 75, normalized size = 1.09 \[ \frac {a^2 \left (-\frac {2 \sqrt {a (\sin (c+d x)+1)}}{\sin (c+d x)-1}-\sqrt {2} \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a (\sin (c+d x)+1)}}{\sqrt {2} \sqrt {a}}\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 102, normalized size = 1.48 \[ \frac {\sqrt {2} {\left (a^{2} \sin \left (d x + c\right ) - a^{2}\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{2}}{4 \, {\left (d \sin \left (d x + c\right ) - d\right )}} \]
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 = 0.20, size = 66, normalized size = 0.96 \[ -\frac {a^{3} \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}}{a \sin \left (d x +c \right )-a}+\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 \sqrt {a}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 94, normalized size = 1.36 \[ \frac {\sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right ) - \frac {4 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{4}}{a \sin \left (d x + c\right ) - a}}{4 \, a d} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
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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