Optimal. Leaf size=120 \[ -\frac {\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}-\frac {4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac {a \sin ^2(c+d x)}{b^3 d}+\frac {\sin ^3(c+d x)}{3 b^2 d} \]
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Rubi [A] time = 0.10, antiderivative size = 120, 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} \[ \frac {\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac {\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}-\frac {4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}-\frac {a \sin ^2(c+d x)}{b^3 d}+\frac {\sin ^3(c+d x)}{3 b^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (3 a^2 \left (1-\frac {2 b^2}{3 a^2}\right )-2 a x+x^2+\frac {\left (a^2-b^2\right )^2}{(a+x)^2}-\frac {4 \left (a^3-a b^2\right )}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac {4 a \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{b^5 d}+\frac {\left (3 a^2-2 b^2\right ) \sin (c+d x)}{b^4 d}-\frac {a \sin ^2(c+d x)}{b^3 d}+\frac {\sin ^3(c+d x)}{3 b^2 d}-\frac {\left (a^2-b^2\right )^2}{b^5 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 127, normalized size = 1.06 \[ \frac {\left (8 a^2 b-4 b^3\right ) \sin (c+d x)+\frac {b^4 \cos ^4(c+d x)-4 \left (a^2-b^2\right ) \left (3 a^2 \log (a+b \sin (c+d x))+a^2+3 a b \sin (c+d x) \log (a+b \sin (c+d x))-b^2\right )}{a+b \sin (c+d x)}-2 a b^2 \sin ^2(c+d x)}{3 b^5 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 156, normalized size = 1.30 \[ \frac {2 \, b^{4} \cos \left (d x + c\right )^{4} - 6 \, a^{4} + 27 \, a^{2} b^{2} - 16 \, b^{4} - 4 \, {\left (3 \, a^{2} b^{2} - 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 24 \, {\left (a^{4} - a^{2} b^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (4 \, a b^{3} \cos \left (d x + c\right )^{2} + 18 \, a^{3} b - 13 \, a b^{3}\right )} \sin \left (d x + c\right )}{6 \, {\left (b^{6} d \sin \left (d x + c\right ) + a b^{5} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.84, size = 150, normalized size = 1.25 \[ -\frac {\frac {12 \, {\left (a^{3} - a b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{5}} - \frac {b^{4} \sin \left (d x + c\right )^{3} - 3 \, a b^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} \sin \left (d x + c\right ) - 6 \, b^{4} \sin \left (d x + c\right )}{b^{6}} - \frac {3 \, {\left (4 \, a^{3} b \sin \left (d x + c\right ) - 4 \, a b^{3} \sin \left (d x + c\right ) + 3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{5}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 174, normalized size = 1.45 \[ \frac {\sin ^{3}\left (d x +c \right )}{3 b^{2} d}-\frac {a \left (\sin ^{2}\left (d x +c \right )\right )}{b^{3} d}+\frac {3 a^{2} \sin \left (d x +c \right )}{d \,b^{4}}-\frac {2 \sin \left (d x +c \right )}{b^{2} d}-\frac {4 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{5}}+\frac {4 a \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} d}-\frac {a^{4}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {1}{b d \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 116, normalized size = 0.97 \[ -\frac {\frac {3 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}{b^{6} \sin \left (d x + c\right ) + a b^{5}} - \frac {b^{2} \sin \left (d x + c\right )^{3} - 3 \, a b \sin \left (d x + c\right )^{2} + 3 \, {\left (3 \, a^{2} - 2 \, b^{2}\right )} \sin \left (d x + c\right )}{b^{4}} + \frac {12 \, {\left (a^{3} - a b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{5}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 118, normalized size = 0.98 \[ -\frac {\sin \left (c+d\,x\right )\,\left (\frac {2}{b^2}-\frac {3\,a^2}{b^4}\right )-\frac {{\sin \left (c+d\,x\right )}^3}{3\,b^2}+\frac {a\,{\sin \left (c+d\,x\right )}^2}{b^3}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (4\,a\,b^2-4\,a^3\right )}{b^5}+\frac {a^4-2\,a^2\,b^2+b^4}{b\,\left (\sin \left (c+d\,x\right )\,b^5+a\,b^4\right )}}{d} \]
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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