Optimal. Leaf size=145 \[ -\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d} \]
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Rubi [A]
time = 0.17, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2844, 3047,
3102, 2830, 2728, 212} \begin {gather*} -\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {7 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{6 a^2 d}+\frac {\sin ^2(c+d x) \cos (c+d x)}{2 d (a \sin (c+d x)+a)^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a \sin (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2830
Rule 2844
Rule 3047
Rule 3102
Rubi steps
\begin {align*} \int \frac {\sin ^3(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {\sin (c+d x) \left (2 a-\frac {7}{2} a \sin (c+d x)\right )}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 a^2}\\ &=\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {\int \frac {2 a \sin (c+d x)-\frac {7}{2} a \sin ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{2 a^2}\\ &=\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}-\frac {\int \frac {-\frac {7 a^2}{4}+\frac {13}{2} a^2 \sin (c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx}{3 a^3}\\ &=\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}+\frac {11 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{4 a}\\ &=\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}-\frac {11 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{2 a d}\\ &=-\frac {11 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {\cos (c+d x) \sin ^2(c+d x)}{2 d (a+a \sin (c+d x))^{3/2}}+\frac {13 \cos (c+d x)}{3 a d \sqrt {a+a \sin (c+d x)}}-\frac {7 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{6 a^2 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 156, normalized size = 1.08 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (11 \cos \left (\frac {1}{2} (c+d x)\right )+7 \cos \left (\frac {3}{2} (c+d x)\right )+\cos \left (\frac {5}{2} (c+d x)\right )-11 \sin \left (\frac {1}{2} (c+d x)\right )+(33+33 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) (1+\sin (c+d x))+7 \sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {5}{2} (c+d x)\right )\right )}{6 d (a (1+\sin (c+d x)))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.86, size = 183, normalized size = 1.26
method | result | size |
default | \(-\frac {\left (\sin \left (d x +c \right ) \left (33 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-24 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}-8 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}\right )+33 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-30 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {3}{2}}-8 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a}\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{12 a^{\frac {7}{2}} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 295 vs.
\(2 (122) = 244\).
time = 0.35, size = 295, normalized size = 2.03 \begin {gather*} \frac {33 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) - 4 \, {\left (4 \, \cos \left (d x + c\right )^{3} + 16 \, \cos \left (d x + c\right )^{2} - {\left (4 \, \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) + 15 \, \cos \left (d x + c\right ) + 3\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{24 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d \cos \left (d x + c\right ) - 2 \, a^{2} d - {\left (a^{2} d \cos \left (d x + c\right ) + 2 \, a^{2} d\right )} \sin \left (d x + c\right )\right )}} \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 = 0.51, size = 197, normalized size = 1.36 \begin {gather*} \frac {\frac {33 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {33 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {6 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {16 \, \sqrt {2} {\left (2 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a^{\frac {9}{2}} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{24 \, 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 {{\sin \left (c+d\,x\right )}^3}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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