Optimal. Leaf size=67 \[ -\frac {\cos \left (a+b x^2\right )}{2 b}+\frac {\cos ^3\left (a+b x^2\right )}{2 b}-\frac {3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac {\cos ^7\left (a+b x^2\right )}{14 b} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3460, 2713}
\begin {gather*} \frac {\cos ^7\left (a+b x^2\right )}{14 b}-\frac {3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac {\cos ^3\left (a+b x^2\right )}{2 b}-\frac {\cos \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3460
Rubi steps
\begin {align*} \int x \sin ^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \sin ^7(a+b x) \, dx,x,x^2\right )\\ &=-\frac {\text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos \left (a+b x^2\right )\right )}{2 b}\\ &=-\frac {\cos \left (a+b x^2\right )}{2 b}+\frac {\cos ^3\left (a+b x^2\right )}{2 b}-\frac {3 \cos ^5\left (a+b x^2\right )}{10 b}+\frac {\cos ^7\left (a+b x^2\right )}{14 b}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 1.00 \begin {gather*} -\frac {35 \cos \left (a+b x^2\right )}{128 b}+\frac {7 \cos \left (3 \left (a+b x^2\right )\right )}{128 b}-\frac {7 \cos \left (5 \left (a+b x^2\right )\right )}{640 b}+\frac {\cos \left (7 \left (a+b x^2\right )\right )}{896 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 50, normalized size = 0.75
method | result | size |
derivativedivides | \(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b \,x^{2}+a \right )+\frac {6 \left (\sin ^{4}\left (b \,x^{2}+a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b \,x^{2}+a \right )\right )}{5}\right ) \cos \left (b \,x^{2}+a \right )}{14 b}\) | \(50\) |
default | \(-\frac {\left (\frac {16}{5}+\sin ^{6}\left (b \,x^{2}+a \right )+\frac {6 \left (\sin ^{4}\left (b \,x^{2}+a \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (b \,x^{2}+a \right )\right )}{5}\right ) \cos \left (b \,x^{2}+a \right )}{14 b}\) | \(50\) |
risch | \(-\frac {35 \cos \left (b \,x^{2}+a \right )}{128 b}+\frac {\cos \left (7 b \,x^{2}+7 a \right )}{896 b}-\frac {7 \cos \left (5 b \,x^{2}+5 a \right )}{640 b}+\frac {7 \cos \left (3 b \,x^{2}+3 a \right )}{128 b}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 55, normalized size = 0.82 \begin {gather*} \frac {5 \, \cos \left (7 \, b x^{2} + 7 \, a\right ) - 49 \, \cos \left (5 \, b x^{2} + 5 \, a\right ) + 245 \, \cos \left (3 \, b x^{2} + 3 \, a\right ) - 1225 \, \cos \left (b x^{2} + a\right )}{4480 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 52, normalized size = 0.78 \begin {gather*} \frac {5 \, \cos \left (b x^{2} + a\right )^{7} - 21 \, \cos \left (b x^{2} + a\right )^{5} + 35 \, \cos \left (b x^{2} + a\right )^{3} - 35 \, \cos \left (b x^{2} + a\right )}{70 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.89, size = 95, normalized size = 1.42 \begin {gather*} \begin {cases} - \frac {\sin ^{6}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{2 b} - \frac {\sin ^{4}{\left (a + b x^{2} \right )} \cos ^{3}{\left (a + b x^{2} \right )}}{b} - \frac {4 \sin ^{2}{\left (a + b x^{2} \right )} \cos ^{5}{\left (a + b x^{2} \right )}}{5 b} - \frac {8 \cos ^{7}{\left (a + b x^{2} \right )}}{35 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \sin ^{7}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.81, size = 52, normalized size = 0.78 \begin {gather*} \frac {5 \, \cos \left (b x^{2} + a\right )^{7} - 21 \, \cos \left (b x^{2} + a\right )^{5} + 35 \, \cos \left (b x^{2} + a\right )^{3} - 35 \, \cos \left (b x^{2} + a\right )}{70 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.99, size = 55, normalized size = 0.82 \begin {gather*} \frac {245\,\cos \left (3\,b\,x^2+3\,a\right )-49\,\cos \left (5\,b\,x^2+5\,a\right )+5\,\cos \left (7\,b\,x^2+7\,a\right )-1225\,\cos \left (b\,x^2+a\right )}{4480\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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