Optimal. Leaf size=144 \[ -\frac {25 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {25 a^3 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]
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Rubi [A]
time = 0.19, antiderivative size = 144, 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 = {2841, 3059,
2851, 2852, 212} \begin {gather*} -\frac {25 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{8 d}-\frac {25 a^3 \cot (c+d x)}{8 d \sqrt {a \sin (c+d x)+a}}-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a \sin (c+d x)+a}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2841
Rule 2851
Rule 2852
Rule 3059
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {1}{3} a \int \csc ^3(c+d x) \left (-\frac {13 a}{2}-\frac {9}{2} a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{8} \left (25 a^2\right ) \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {25 a^3 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{16} \left (25 a^2\right ) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {25 a^3 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {\left (25 a^3\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}\\ &=-\frac {25 a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{8 d}-\frac {25 a^3 \cot (c+d x)}{8 d \sqrt {a+a \sin (c+d x)}}-\frac {13 a^3 \cot (c+d x) \csc (c+d x)}{12 d \sqrt {a+a \sin (c+d x)}}-\frac {a^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 288, normalized size = 2.00 \begin {gather*} \frac {a^2 \csc ^{10}\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sin (c+d x))} \left (-228 \cos \left (\frac {1}{2} (c+d x)\right )+14 \cos \left (\frac {3}{2} (c+d x)\right )+150 \cos \left (\frac {5}{2} (c+d x)\right )+228 \sin \left (\frac {1}{2} (c+d x)\right )-225 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+225 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (c+d x)+14 \sin \left (\frac {3}{2} (c+d x)\right )-150 \sin \left (\frac {5}{2} (c+d x)\right )+75 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))-75 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin (3 (c+d x))\right )}{24 d \left (1+\cot \left (\frac {1}{2} (c+d x)\right )\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.12, size = 144, normalized size = 1.00
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (75 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+75 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{4} \left (\sin ^{3}\left (d x +c \right )\right )-184 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}+117 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {7}{2}}\right )}{24 \sin \left (d x +c \right )^{3} a^{\frac {3}{2}} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(144\) |
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. 408 vs.
\(2 (124) = 248\).
time = 0.35, size = 408, normalized size = 2.83 \begin {gather*} \frac {75 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} - {\left (a^{2} \cos \left (d x + c\right )^{3} + a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right ) - a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (75 \, a^{2} \cos \left (d x + c\right )^{3} + 41 \, a^{2} \cos \left (d x + c\right )^{2} - 83 \, a^{2} \cos \left (d x + c\right ) - 49 \, a^{2} - {\left (75 \, a^{2} \cos \left (d x + c\right )^{2} + 34 \, a^{2} \cos \left (d x + c\right ) - 49 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\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 = 0.55, size = 196, normalized size = 1.36 \begin {gather*} -\frac {\sqrt {2} {\left (75 \, \sqrt {2} a^{2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {4 \, {\left (300 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 368 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 117 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}\right )} \sqrt {a}}{96 \, 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}}{{\sin \left (c+d\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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