Optimal. Leaf size=127 \[ -\frac{\left (a^2-b^2\right )^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left (a^2-b^2\right )}{b^5 d (a+b \sin (c+d x))}+\frac{2 \left (3 a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d} \]
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Rubi [A] time = 0.104938, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2668, 697} \[ -\frac{\left (a^2-b^2\right )^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left (a^2-b^2\right )}{b^5 d (a+b \sin (c+d x))}+\frac{2 \left (3 a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{(a+x)^3} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a+x+\frac{\left (a^2-b^2\right )^2}{(a+x)^3}-\frac{4 \left (a^3-a b^2\right )}{(a+x)^2}+\frac{2 \left (3 a^2-b^2\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{2 \left (3 a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac{3 a \sin (c+d x)}{b^4 d}+\frac{\sin ^2(c+d x)}{2 b^3 d}-\frac{\left (a^2-b^2\right )^2}{2 b^5 d (a+b \sin (c+d x))^2}+\frac{4 a \left (a^2-b^2\right )}{b^5 d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.959813, size = 143, normalized size = 1.13 \[ \frac{2 \left (b^2-a^2\right ) \left (-\frac{3 a^2+4 a b \sin (c+d x)+b^2}{2 (a+b \sin (c+d x))^2}-\log (a+b \sin (c+d x))\right )+\frac{b^4 \cos ^4(c+d x)}{2 (a+b \sin (c+d x))^2}+2 a \left (\frac{(a-b) (a+b)}{a+b \sin (c+d x)}+2 a \log (a+b \sin (c+d x))-b \sin (c+d x)\right )}{b^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.093, size = 183, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{2\,{b}^{3}d}}-3\,{\frac{a\sin \left ( dx+c \right ) }{{b}^{4}d}}+6\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{5}}}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{{b}^{3}d}}-{\frac{{a}^{4}}{2\,d{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}}{{b}^{3}d \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{1}{2\,bd \left ( a+b\sin \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{{a}^{3}}{d{b}^{5} \left ( a+b\sin \left ( dx+c \right ) \right ) }}-4\,{\frac{a}{{b}^{3}d \left ( a+b\sin \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956052, size = 177, normalized size = 1.39 \begin{align*} \frac{\frac{7 \, a^{4} - 6 \, a^{2} b^{2} - b^{4} + 8 \,{\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )}{b^{7} \sin \left (d x + c\right )^{2} + 2 \, a b^{6} \sin \left (d x + c\right ) + a^{2} b^{5}} + \frac{b \sin \left (d x + c\right )^{2} - 6 \, a \sin \left (d x + c\right )}{b^{4}} + \frac{4 \,{\left (3 \, a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.92335, size = 473, normalized size = 3.72 \begin{align*} -\frac{2 \, b^{4} \cos \left (d x + c\right )^{4} + 14 \, a^{4} - 35 \, a^{2} b^{2} - b^{4} +{\left (22 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (3 \, a^{4} + 2 \, a^{2} b^{2} - b^{4} -{\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left (3 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 2 \,{\left (4 \, a b^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} b - 13 \, a b^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (b^{7} d \cos \left (d x + c\right )^{2} - 2 \, a b^{6} d \sin \left (d x + c\right ) -{\left (a^{2} b^{5} + b^{7}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14714, size = 192, normalized size = 1.51 \begin{align*} \frac{\frac{4 \,{\left (3 \, a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac{b^{3} \sin \left (d x + c\right )^{2} - 6 \, a b^{2} \sin \left (d x + c\right )}{b^{6}} - \frac{18 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 6 \, b^{4} \sin \left (d x + c\right )^{2} + 28 \, a^{3} b \sin \left (d x + c\right ) - 4 \, a b^{3} \sin \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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