Optimal. Leaf size=118 \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d} \]
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Rubi [A] time = 0.105313, antiderivative size = 118, 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} \[ \frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d} \]
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)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{a+x} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-a^3 \left (1-\frac{2 b^2}{a^2}\right )+\left (a^2-2 b^2\right ) x-a x^2+x^3+\frac{\left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^5 d}-\frac{a \left (a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}+\frac{\left (a^2-2 b^2\right ) \sin ^2(c+d x)}{2 b^3 d}-\frac{a \sin ^3(c+d x)}{3 b^2 d}+\frac{\sin ^4(c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.195237, size = 103, normalized size = 0.87 \[ \frac{6 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)-12 a b \left (a^2-2 b^2\right ) \sin (c+d x)+12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-4 a b^3 \sin ^3(c+d x)+3 b^4 \cos ^4(c+d x)}{12 b^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 163, normalized size = 1.4 \begin{align*}{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,bd}}-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}{a}^{2}}{2\,d{b}^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{bd}}-{\frac{{a}^{3}\sin \left ( dx+c \right ) }{d{b}^{4}}}+2\,{\frac{a\sin \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{4}}{d{b}^{5}}}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{bd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.963665, size = 146, normalized size = 1.24 \begin{align*} \frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \,{\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \,{\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.71323, size = 250, normalized size = 2.12 \begin{align*} \frac{3 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \,{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )}{12 \, b^{5} d} \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 = 1.125, size = 162, normalized size = 1.37 \begin{align*} \frac{\frac{3 \, b^{3} \sin \left (d x + c\right )^{4} - 4 \, a b^{2} \sin \left (d x + c\right )^{3} + 6 \, a^{2} b \sin \left (d x + c\right )^{2} - 12 \, b^{3} \sin \left (d x + c\right )^{2} - 12 \, a^{3} \sin \left (d x + c\right ) + 24 \, a b^{2} \sin \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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