Optimal. Leaf size=86 \[ \frac {4 \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 a^2 \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 a (a+a \sin (c+d x))^{3/2}}{3 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2746, 52, 65,
212} \begin {gather*} \frac {4 \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 a^2 \sqrt {a \sin (c+d x)+a}}{d}-\frac {2 a (a \sin (c+d x)+a)^{3/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 2746
Rubi steps
\begin {align*} \int \sec (c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {a \text {Subst}\left (\int \frac {(a+x)^{3/2}}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {2 a (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {\sqrt {a+x}}{a-x} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^2 \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 a (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^2 \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 a (a+a \sin (c+d x))^{3/2}}{3 d}+\frac {\left (8 a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{d}\\ &=\frac {4 \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {4 a^2 \sqrt {a+a \sin (c+d x)}}{d}-\frac {2 a (a+a \sin (c+d x))^{3/2}}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.85 \begin {gather*} \frac {12 \sqrt {2} a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a (1+\sin (c+d x))}}{\sqrt {2} \sqrt {a}}\right )-2 a^2 \sqrt {a (1+\sin (c+d x))} (7+\sin (c+d x))}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 66, normalized size = 0.77
method | result | size |
default | \(-\frac {2 a \left (\frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 a \sqrt {a +a \sin \left (d x +c \right )}-2 a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )\right )}{d}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 97, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (3 \, \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 ) + {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} + 6 \, \sqrt {a \sin \left (d x + c\right ) + a} a^{3}\right )}}{3 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 89, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {5}{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 ) - {\left (a^{2} \sin \left (d x + c\right ) + 7 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.81, size = 91, normalized size = 1.06 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (2 \, \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, \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 )} a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{3 \, 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 {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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