Optimal. Leaf size=22 \[ -\frac {1}{7 b d (a+b \sin (c+d x))^7} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2668, 32} \[ -\frac {1}{7 b d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^8} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^8} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {1}{7 b d (a+b \sin (c+d x))^7}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 22, normalized size = 1.00 \[ -\frac {1}{7 b d (a+b \sin (c+d x))^7} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 218, normalized size = 9.91 \[ \frac {1}{7 \, {\left (7 \, a b^{7} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{3} + 10 \, a^{3} b^{5} + 3 \, a b^{7}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b + 21 \, a^{5} b^{3} + 35 \, a^{3} b^{5} + 7 \, a b^{7}\right )} d + {\left (b^{8} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{6} + b^{8}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{4} + 42 \, a^{2} b^{6} + 3 \, b^{8}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 21 \, a^{2} b^{6} + b^{8}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.57, size = 20, normalized size = 0.91 \[ -\frac {1}{7 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 21, normalized size = 0.95 \[ -\frac {1}{7 b d \left (a +b \sin \left (d x +c \right )\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 20, normalized size = 0.91 \[ -\frac {1}{7 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.20, size = 119, normalized size = 5.41 \[ -\frac {1}{d\,\left (7\,a^7\,b+49\,a^6\,b^2\,\sin \left (c+d\,x\right )+147\,a^5\,b^3\,{\sin \left (c+d\,x\right )}^2+245\,a^4\,b^4\,{\sin \left (c+d\,x\right )}^3+245\,a^3\,b^5\,{\sin \left (c+d\,x\right )}^4+147\,a^2\,b^6\,{\sin \left (c+d\,x\right )}^5+49\,a\,b^7\,{\sin \left (c+d\,x\right )}^6+7\,b^8\,{\sin \left (c+d\,x\right )}^7\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.65, size = 167, normalized size = 7.59 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a^{8}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{8} d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{8}} & \text {for}\: d = 0 \\- \frac {1}{7 a^{7} b d + 49 a^{6} b^{2} d \sin {\left (c + d x \right )} + 147 a^{5} b^{3} d \sin ^{2}{\left (c + d x \right )} + 245 a^{4} b^{4} d \sin ^{3}{\left (c + d x \right )} + 245 a^{3} b^{5} d \sin ^{4}{\left (c + d x \right )} + 147 a^{2} b^{6} d \sin ^{5}{\left (c + d x \right )} + 49 a b^{7} d \sin ^{6}{\left (c + d x \right )} + 7 b^{8} d \sin ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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