Optimal. Leaf size=67 \[ \frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sin ^5\left (a+b x^2\right )}{10 b}-\frac {\sin ^7\left (a+b x^2\right )}{14 b} \]
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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 = {3461, 2713}
\begin {gather*} -\frac {\sin ^7\left (a+b x^2\right )}{14 b}+\frac {3 \sin ^5\left (a+b x^2\right )}{10 b}-\frac {\sin ^3\left (a+b x^2\right )}{2 b}+\frac {\sin \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 3461
Rubi steps
\begin {align*} \int x \cos ^7\left (a+b x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \cos ^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,-\sin \left (a+b x^2\right )\right )}{2 b}\\ &=\frac {\sin \left (a+b x^2\right )}{2 b}-\frac {\sin ^3\left (a+b x^2\right )}{2 b}+\frac {3 \sin ^5\left (a+b x^2\right )}{10 b}-\frac {\sin ^7\left (a+b x^2\right )}{14 b}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 67, normalized size = 1.00 \begin {gather*} \frac {35 \sin \left (a+b x^2\right )}{128 b}+\frac {7 \sin \left (3 \left (a+b x^2\right )\right )}{128 b}+\frac {7 \sin \left (5 \left (a+b x^2\right )\right )}{640 b}+\frac {\sin \left (7 \left (a+b x^2\right )\right )}{896 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 50, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {\left (\frac {16}{5}+\cos ^{6}\left (b \,x^{2}+a \right )+\frac {6 \left (\cos ^{4}\left (b \,x^{2}+a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b \,x^{2}+a \right )\right )}{5}\right ) \sin \left (b \,x^{2}+a \right )}{14 b}\) | \(50\) |
default | \(\frac {\left (\frac {16}{5}+\cos ^{6}\left (b \,x^{2}+a \right )+\frac {6 \left (\cos ^{4}\left (b \,x^{2}+a \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (b \,x^{2}+a \right )\right )}{5}\right ) \sin \left (b \,x^{2}+a \right )}{14 b}\) | \(50\) |
risch | \(\frac {35 \sin \left (b \,x^{2}+a \right )}{128 b}+\frac {\sin \left (7 b \,x^{2}+7 a \right )}{896 b}+\frac {7 \sin \left (5 b \,x^{2}+5 a \right )}{640 b}+\frac {7 \sin \left (3 b \,x^{2}+3 a \right )}{128 b}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 55, normalized size = 0.82 \begin {gather*} \frac {5 \, \sin \left (7 \, b x^{2} + 7 \, a\right ) + 49 \, \sin \left (5 \, b x^{2} + 5 \, a\right ) + 245 \, \sin \left (3 \, b x^{2} + 3 \, a\right ) + 1225 \, \sin \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 = 0.38, size = 51, normalized size = 0.76 \begin {gather*} \frac {{\left (5 \, \cos \left (b x^{2} + a\right )^{6} + 6 \, \cos \left (b x^{2} + a\right )^{4} + 8 \, \cos \left (b x^{2} + a\right )^{2} + 16\right )} \sin \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 = 0.90, size = 94, normalized size = 1.40 \begin {gather*} \begin {cases} \frac {8 \sin ^{7}{\left (a + b x^{2} \right )}}{35 b} + \frac {4 \sin ^{5}{\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{5 b} + \frac {\sin ^{3}{\left (a + b x^{2} \right )} \cos ^{4}{\left (a + b x^{2} \right )}}{b} + \frac {\sin {\left (a + b x^{2} \right )} \cos ^{6}{\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cos ^{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 = 0.43, size = 52, normalized size = 0.78 \begin {gather*} -\frac {5 \, \sin \left (b x^{2} + a\right )^{7} - 21 \, \sin \left (b x^{2} + a\right )^{5} + 35 \, \sin \left (b x^{2} + a\right )^{3} - 35 \, \sin \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 = 0.75, size = 55, normalized size = 0.82 \begin {gather*} \frac {245\,\sin \left (3\,b\,x^2+3\,a\right )+49\,\sin \left (5\,b\,x^2+5\,a\right )+5\,\sin \left (7\,b\,x^2+7\,a\right )+1225\,\sin \left (b\,x^2+a\right )}{4480\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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