Optimal. Leaf size=77 \[ \frac {a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac {a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac {1}{5 b^3 d (a+b \sin (c+d x))^5} \]
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Rubi [A] time = 0.07, antiderivative size = 77, 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} \[ \frac {a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac {a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac {1}{5 b^3 d (a+b \sin (c+d x))^5} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {-a^2+b^2}{(a+x)^8}+\frac {2 a}{(a+x)^7}-\frac {1}{(a+x)^6}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {a^2-b^2}{7 b^3 d (a+b \sin (c+d x))^7}-\frac {a}{3 b^3 d (a+b \sin (c+d x))^6}+\frac {1}{5 b^3 d (a+b \sin (c+d x))^5}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 54, normalized size = 0.70 \[ \frac {a^2+7 a b \sin (c+d x)+21 b^2 \sin ^2(c+d x)-15 b^2}{105 b^3 d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 254, normalized size = 3.30 \[ \frac {21 \, b^{2} \cos \left (d x + c\right )^{2} - 7 \, a b \sin \left (d x + c\right ) - a^{2} - 6 \, b^{2}}{105 \, {\left (7 \, a b^{9} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{5} + 10 \, a^{3} b^{7} + 3 \, a b^{9}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{3} + 21 \, a^{5} b^{5} + 35 \, a^{3} b^{7} + 7 \, a b^{9}\right )} d + {\left (b^{10} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{6} + 42 \, a^{2} b^{8} + 3 \, b^{10}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{4} + 35 \, a^{4} b^{6} + 21 \, a^{2} b^{8} + b^{10}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 4.46, size = 52, normalized size = 0.68 \[ \frac {21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 67, normalized size = 0.87 \[ \frac {-\frac {-a^{2}+b^{2}}{7 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {1}{5 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {a}{3 b^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.52, size = 151, normalized size = 1.96 \[ \frac {21 \, b^{2} \sin \left (d x + c\right )^{2} + 7 \, a b \sin \left (d x + c\right ) + a^{2} - 15 \, b^{2}}{105 \, {\left (b^{10} \sin \left (d x + c\right )^{7} + 7 \, a b^{9} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{8} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{7} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{6} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{5} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{4} \sin \left (d x + c\right ) + a^{7} b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.22, size = 152, normalized size = 1.97 \[ \frac {\frac {a^2-15\,b^2}{105\,b^3}+\frac {{\sin \left (c+d\,x\right )}^2}{5\,b}+\frac {a\,\sin \left (c+d\,x\right )}{15\,b^2}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 41.33, size = 636, normalized size = 8.26 \[ \begin {cases} \frac {x \cos ^{3}{\relax (c )}}{a^{8}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\frac {2 \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac {\sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d}}{a^{8}} & \text {for}\: b = 0 \\\frac {x \cos ^{3}{\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{8}} & \text {for}\: d = 0 \\\frac {a^{2}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} + \frac {7 a b \sin {\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} + \frac {6 b^{2} \sin ^{2}{\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} - \frac {15 b^{2} \cos ^{2}{\left (c + d x \right )}}{105 a^{7} b^{3} d + 735 a^{6} b^{4} d \sin {\left (c + d x \right )} + 2205 a^{5} b^{5} d \sin ^{2}{\left (c + d x \right )} + 3675 a^{4} b^{6} d \sin ^{3}{\left (c + d x \right )} + 3675 a^{3} b^{7} d \sin ^{4}{\left (c + d x \right )} + 2205 a^{2} b^{8} d \sin ^{5}{\left (c + d x \right )} + 735 a b^{9} d \sin ^{6}{\left (c + d x \right )} + 105 b^{10} d \sin ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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