Optimal. Leaf size=184 \[ \frac {\left (a^2-b^2\right )^3}{b^7 d (a+b \sin (c+d x))}+\frac {6 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x)}{b^4 d}-\frac {\left (5 a^4-9 a^2 b^2+3 b^4\right ) \sin (c+d x)}{b^6 d}+\frac {a \sin ^4(c+d x)}{2 b^3 d}-\frac {\sin ^5(c+d x)}{5 b^2 d} \]
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Rubi [A] time = 0.17, antiderivative size = 184, 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} \[ -\frac {\left (a^2-b^2\right ) \sin ^3(c+d x)}{b^4 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {\left (-9 a^2 b^2+5 a^4+3 b^4\right ) \sin (c+d x)}{b^6 d}+\frac {\left (a^2-b^2\right )^3}{b^7 d (a+b \sin (c+d x))}+\frac {6 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}+\frac {a \sin ^4(c+d x)}{2 b^3 d}-\frac {\sin ^5(c+d x)}{5 b^2 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^3}{(a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-5 a^4 \left (1+\frac {3 b^2 \left (-3 a^2+b^2\right )}{5 a^4}\right )+2 a \left (2 a^2-3 b^2\right ) x-3 \left (a^2-b^2\right ) x^2+2 a x^3-x^4-\frac {\left (a^2-b^2\right )^3}{(a+x)^2}+\frac {6 a \left (a^2-b^2\right )^2}{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d}\\ &=\frac {6 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac {\left (5 a^4-9 a^2 b^2+3 b^4\right ) \sin (c+d x)}{b^6 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x)}{b^4 d}+\frac {a \sin ^4(c+d x)}{2 b^3 d}-\frac {\sin ^5(c+d x)}{5 b^2 d}+\frac {\left (a^2-b^2\right )^3}{b^7 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 235, normalized size = 1.28 \[ \frac {-4 a^2 b^4 \sin ^4(c+d x)+4 \left (a^2-b^2\right )^2 \left (15 a^2 \log (a+b \sin (c+d x))+4 a^2-4 b^2\right )+b^4 \cos ^4(c+d x) \left (-a^2+3 a b \sin (c+d x)+4 b^2\right )+2 a b^3 \left (5 a^2-7 b^2\right ) \sin ^3(c+d x)-2 b^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)+4 a b \sin (c+d x) \left (-11 a^4+15 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+18 a^2 b^2-4 b^4\right )+2 b^6 \cos ^6(c+d x)}{10 b^7 d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 243, normalized size = 1.32 \[ \frac {16 \, b^{6} \cos \left (d x + c\right )^{6} + 80 \, a^{6} - 560 \, a^{4} b^{2} + 785 \, a^{2} b^{4} - 256 \, b^{6} - 8 \, {\left (5 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, a^{4} b^{2} - 25 \, a^{2} b^{4} + 8 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (24 \, a b^{5} \cos \left (d x + c\right )^{4} - 400 \, a^{5} b + 720 \, a^{3} b^{3} - 271 \, a b^{5} - 16 \, {\left (5 \, a^{3} b^{3} - 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{80 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 251, normalized size = 1.36 \[ \frac {\frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {10 \, {\left (6 \, a^{5} b \sin \left (d x + c\right ) - 12 \, a^{3} b^{3} \sin \left (d x + c\right ) + 6 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - 9 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{7}} - \frac {2 \, b^{8} \sin \left (d x + c\right )^{5} - 5 \, a b^{7} \sin \left (d x + c\right )^{4} + 10 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} - 10 \, b^{8} \sin \left (d x + c\right )^{3} - 20 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 30 \, a b^{7} \sin \left (d x + c\right )^{2} + 50 \, a^{4} b^{4} \sin \left (d x + c\right ) - 90 \, a^{2} b^{6} \sin \left (d x + c\right ) + 30 \, b^{8} \sin \left (d x + c\right )}{b^{10}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 305, normalized size = 1.66 \[ -\frac {\sin ^{5}\left (d x +c \right )}{5 b^{2} d}+\frac {a \left (\sin ^{4}\left (d x +c \right )\right )}{2 b^{3} d}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) a^{2}}{d \,b^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{b^{2} d}+\frac {2 \left (\sin ^{2}\left (d x +c \right )\right ) a^{3}}{d \,b^{5}}-\frac {3 a \left (\sin ^{2}\left (d x +c \right )\right )}{b^{3} d}-\frac {5 a^{4} \sin \left (d x +c \right )}{d \,b^{6}}+\frac {9 a^{2} \sin \left (d x +c \right )}{d \,b^{4}}-\frac {3 \sin \left (d x +c \right )}{b^{2} d}+\frac {6 a^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{7}}-\frac {12 a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{5}}+\frac {6 a \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{3} d}+\frac {a^{6}}{d \,b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 a^{4}}{d \,b^{5} \left (a +b \sin \left (d x +c \right )\right )}+\frac {3 a^{2}}{d \,b^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {1}{b d \left (a +b \sin \left (d x +c \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 190, normalized size = 1.03 \[ \frac {\frac {10 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}}{b^{8} \sin \left (d x + c\right ) + a b^{7}} - \frac {2 \, b^{4} \sin \left (d x + c\right )^{5} - 5 \, a b^{3} \sin \left (d x + c\right )^{4} + 10 \, {\left (a^{2} b^{2} - b^{4}\right )} \sin \left (d x + c\right )^{3} - 10 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \sin \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{4} - 9 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{7}}}{10 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 259, normalized size = 1.41 \[ \frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {1}{b^2}-\frac {a^2}{b^4}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,b^2\,d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3}{b^5}+\frac {a\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {3}{b^2}+\frac {a^2\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{d}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2\,b^3\,d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (6\,a^5-12\,a^3\,b^2+6\,a\,b^4\right )}{b^7\,d}+\frac {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^7+a\,b^6\right )} \]
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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