Optimal. Leaf size=108 \[ -\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2679, 2649, 206} \[ \frac {4 \cos (c+d x)}{a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{a^{5/2} d}+\frac {2 \cos ^3(c+d x)}{3 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2679
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{a}\\ &=\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}}-\frac {8 \operatorname {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 )}{a^2 d}\\ &=-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{a^{5/2} d}+\frac {2 \cos ^3(c+d x)}{3 a d (a+a \sin (c+d x))^{3/2}}+\frac {4 \cos (c+d x)}{a^2 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 96, normalized size = 0.89 \[ -\frac {2 \cos (c+d x) \left (\sqrt {1-\sin (c+d x)} (\sin (c+d x)-7)+6 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )\right )}{3 a^2 d \sqrt {1-\sin (c+d x)} \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 215, normalized size = 1.99 \[ \frac {2 \, {\left (\frac {3 \, \sqrt {2} {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} {\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{\sqrt {a}} - {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right ) - 7 \, \cos \left (d x + c\right ) - 8\right )} \sqrt {a \sin \left (d x + c\right ) + a}\right )}}{3 \, {\left (a^{3} d \cos \left (d x + c\right ) + a^{3} d \sin \left (d x + c\right ) + a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.12, size = 255, normalized size = 2.36 \[ -\frac {2 \, {\left (\frac {{\left ({\left (\frac {7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )} - \frac {9}{a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {9}{a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {7}{a \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {3}{2}}} + \frac {4 \, \sqrt {2} {\left (3 \, a \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) + 2 \, \sqrt {-a} \sqrt {a}\right )} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {-a} a^{3}} - \frac {12 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} + \sqrt {a}\right )}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2} \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}\right )}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 112, normalized size = 1.04 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (6 a^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )-\left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-6 a \sqrt {a -a \sin \left (d x +c \right )}\right )}{3 a^{4} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{4}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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