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 = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 697
Rule 2668
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*}
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Mathematica [A] time = 0.98, 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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fricas [A] time = 0.48, size = 212, normalized size = 1.67 \[ -\frac {2 \, b^{4} \cos \left (d x + c\right )^{4} + 14 \, a^{4} - 35 \, a^{2} b^{2} - b^{4} + {\left (22 \, a^{2} b^{2} - 3 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, a^{4} + 2 \, a^{2} b^{2} - b^{4} - {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 2 \, {\left (4 \, a b^{3} \cos \left (d x + c\right )^{2} + 2 \, a^{3} b - 13 \, a b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (b^{7} d \cos \left (d x + c\right )^{2} - 2 \, a b^{6} d \sin \left (d x + c\right ) - {\left (a^{2} b^{5} + b^{7}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 142, normalized size = 1.12 \[ \frac {\frac {4 \, {\left (3 \, a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac {b^{3} \sin \left (d x + c\right )^{2} - 6 \, a b^{2} \sin \left (d x + c\right )}{b^{6}} - \frac {18 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 6 \, b^{4} \sin \left (d x + c\right )^{2} + 28 \, a^{3} b \sin \left (d x + c\right ) - 4 \, a b^{3} \sin \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 183, normalized size = 1.44 \[ \frac {\sin ^{2}\left (d x +c \right )}{2 b^{3} d}-\frac {3 a \sin \left (d x +c \right )}{b^{4} d}+\frac {6 \ln \left (a +b \sin \left (d x +c \right )\right ) a^{2}}{d \,b^{5}}-\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} d}+\frac {4 a^{3}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}-\frac {4 a}{b^{3} d \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{4}}{2 d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {1}{2 b d \left (a +b \sin \left (d x +c \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 131, normalized size = 1.03 \[ \frac {\frac {7 \, a^{4} - 6 \, a^{2} b^{2} - b^{4} + 8 \, {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )}{b^{7} \sin \left (d x + c\right )^{2} + 2 \, a b^{6} \sin \left (d x + c\right ) + a^{2} b^{5}} + \frac {b \sin \left (d x + c\right )^{2} - 6 \, a \sin \left (d x + c\right )}{b^{4}} + \frac {4 \, {\left (3 \, a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.14, size = 142, normalized size = 1.12 \[ \frac {{\sin \left (c+d\,x\right )}^2}{2\,b^3\,d}-\frac {\frac {-7\,a^4+6\,a^2\,b^2+b^4}{2\,b}+\sin \left (c+d\,x\right )\,\left (4\,a\,b^2-4\,a^3\right )}{d\,\left (a^2\,b^4+2\,a\,b^5\,\sin \left (c+d\,x\right )+b^6\,{\sin \left (c+d\,x\right )}^2\right )}-\frac {3\,a\,\sin \left (c+d\,x\right )}{b^4\,d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (6\,a^2-2\,b^2\right )}{b^5\,d} \]
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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