Optimal. Leaf size=95 \[ \frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} d}-\frac {3 a}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d} \]
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Rubi [A]
time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 2746, 53,
65, 212} \begin {gather*} -\frac {3 a}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} d}+\frac {\sec ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 65
Rule 212
Rule 2746
Rule 2754
Rubi steps
\begin {align*} \int \sec ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {1}{4} (3 a) \int \frac {\sec (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=-\frac {3 a}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=-\frac {3 a}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{4 d}\\ &=\frac {3 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} d}-\frac {3 a}{4 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{2 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.26, size = 271, normalized size = 2.85 \begin {gather*} \frac {\left (-2-(3-3 i) \sqrt [4]{-1} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {d x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 c+d x)\right )+\sin \left (\frac {1}{4} (2 c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {2 \sin \left (\frac {d x}{2}\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {\left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right ) \sqrt {a (1+\sin (c+d x))}}{4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.57, size = 90, normalized size = 0.95
method | result | size |
default | \(\frac {2 a^{3} \left (-\frac {\frac {\sqrt {a +a \sin \left (d x +c \right )}}{2 a \sin \left (d x +c \right )-2 a}-\frac {3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{4 \sqrt {a}}}{4 a^{2}}-\frac {1}{4 a^{2} \sqrt {a +a \sin \left (d x +c \right )}}\right )}{d}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 117, normalized size = 1.23 \begin {gather*} -\frac {3 \, \sqrt {2} a^{\frac {3}{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 \, {\left (3 \, {\left (a \sin \left (d x + c\right ) + a\right )} a^{2} - 4 \, a^{3}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} a}}{16 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 99, normalized size = 1.04 \begin {gather*} \frac {3 \, \sqrt {2} \sqrt {a} \cos \left (d x + c\right )^{2} \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} {\left (3 \, \sin \left (d x + c\right ) - 1\right )}}{16 \, d \cos \left (d x + c\right )^{2}} \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 \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sec ^{3}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.33, size = 112, normalized size = 1.18 \begin {gather*} -\frac {\sqrt {2} \sqrt {a} {\left (\frac {2 \, {\left (3 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - 3 \, \log \left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 3 \, \log \left (-\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{16 \, d} \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 {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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