Optimal. Leaf size=129 \[ \frac {\sin \left (a+b x^3\right ) \cos ^7\left (a+b x^3\right )}{192 b^2}+\frac {7 \sin \left (a+b x^3\right ) \cos ^5\left (a+b x^3\right )}{1152 b^2}+\frac {35 \sin \left (a+b x^3\right ) \cos ^3\left (a+b x^3\right )}{4608 b^2}+\frac {35 \sin \left (a+b x^3\right ) \cos \left (a+b x^3\right )}{3072 b^2}-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {35 x^3}{3072 b} \]
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Rubi [A] time = 0.14, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3444, 3380, 2635, 8} \[ \frac {\sin \left (a+b x^3\right ) \cos ^7\left (a+b x^3\right )}{192 b^2}+\frac {7 \sin \left (a+b x^3\right ) \cos ^5\left (a+b x^3\right )}{1152 b^2}+\frac {35 \sin \left (a+b x^3\right ) \cos ^3\left (a+b x^3\right )}{4608 b^2}+\frac {35 \sin \left (a+b x^3\right ) \cos \left (a+b x^3\right )}{3072 b^2}-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {35 x^3}{3072 b} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rule 3380
Rule 3444
Rubi steps
\begin {align*} \int x^5 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {\int x^2 \cos ^8\left (a+b x^3\right ) \, dx}{8 b}\\ &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {\operatorname {Subst}\left (\int \cos ^8(a+b x) \, dx,x,x^3\right )}{24 b}\\ &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {\cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{192 b^2}+\frac {7 \operatorname {Subst}\left (\int \cos ^6(a+b x) \, dx,x,x^3\right )}{192 b}\\ &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {7 \cos ^5\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{1152 b^2}+\frac {\cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{192 b^2}+\frac {35 \operatorname {Subst}\left (\int \cos ^4(a+b x) \, dx,x,x^3\right )}{1152 b}\\ &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {35 \cos ^3\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{4608 b^2}+\frac {7 \cos ^5\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{1152 b^2}+\frac {\cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{192 b^2}+\frac {35 \operatorname {Subst}\left (\int \cos ^2(a+b x) \, dx,x,x^3\right )}{1536 b}\\ &=-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {35 \cos \left (a+b x^3\right ) \sin \left (a+b x^3\right )}{3072 b^2}+\frac {35 \cos ^3\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{4608 b^2}+\frac {7 \cos ^5\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{1152 b^2}+\frac {\cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{192 b^2}+\frac {35 \operatorname {Subst}\left (\int 1 \, dx,x,x^3\right )}{3072 b}\\ &=\frac {35 x^3}{3072 b}-\frac {x^3 \cos ^8\left (a+b x^3\right )}{24 b}+\frac {35 \cos \left (a+b x^3\right ) \sin \left (a+b x^3\right )}{3072 b^2}+\frac {35 \cos ^3\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{4608 b^2}+\frac {7 \cos ^5\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{1152 b^2}+\frac {\cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right )}{192 b^2}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 120, normalized size = 0.93 \[ \frac {672 \sin \left (2 \left (a+b x^3\right )\right )+168 \sin \left (4 \left (a+b x^3\right )\right )+32 \sin \left (6 \left (a+b x^3\right )\right )+3 \sin \left (8 \left (a+b x^3\right )\right )-1344 b x^3 \cos \left (2 \left (a+b x^3\right )\right )-672 b x^3 \cos \left (4 \left (a+b x^3\right )\right )-192 b x^3 \cos \left (6 \left (a+b x^3\right )\right )-24 b x^3 \cos \left (8 \left (a+b x^3\right )\right )}{73728 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 85, normalized size = 0.66 \[ -\frac {384 \, b x^{3} \cos \left (b x^{3} + a\right )^{8} - 105 \, b x^{3} - {\left (48 \, \cos \left (b x^{3} + a\right )^{7} + 56 \, \cos \left (b x^{3} + a\right )^{5} + 70 \, \cos \left (b x^{3} + a\right )^{3} + 105 \, \cos \left (b x^{3} + a\right )\right )} \sin \left (b x^{3} + a\right )}{9216 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 126, normalized size = 0.98 \[ -\frac {24 \, b x^{3} \cos \left (8 \, b x^{3} + 8 \, a\right ) + 192 \, b x^{3} \cos \left (6 \, b x^{3} + 6 \, a\right ) + 672 \, b x^{3} \cos \left (4 \, b x^{3} + 4 \, a\right ) + 1344 \, b x^{3} \cos \left (2 \, b x^{3} + 2 \, a\right ) - 3 \, \sin \left (8 \, b x^{3} + 8 \, a\right ) - 32 \, \sin \left (6 \, b x^{3} + 6 \, a\right ) - 168 \, \sin \left (4 \, b x^{3} + 4 \, a\right ) - 672 \, \sin \left (2 \, b x^{3} + 2 \, a\right )}{73728 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.95, size = 403, normalized size = 3.12 \[ \frac {-\frac {4 x^{3}}{3 b}+\frac {4 \tan \left (b \,x^{3}+a \right )}{3 b^{2}}+\frac {4 x^{3} \left (\tan ^{2}\left (b \,x^{3}+a \right )\right )}{3 b}}{128+128 \left (\tan ^{2}\left (b \,x^{3}+a \right )\right )}+\frac {\frac {\tan \left (b \,x^{3}+a \right )}{b^{2}}-\frac {x^{3}}{b}-\frac {\tan ^{3}\left (b \,x^{3}+a \right )}{b^{2}}+\frac {6 x^{3} \left (\tan ^{2}\left (b \,x^{3}+a \right )\right )}{b}-\frac {x^{3} \left (\tan ^{4}\left (b \,x^{3}+a \right )\right )}{b}}{128 \left (1+\tan ^{2}\left (b \,x^{3}+a \right )\right )^{2}}+\frac {-6 x^{3} b \left (\cos ^{2}\left (3 b \,x^{3}+3 a \right )\right )-18 \left (\cos ^{2}\left (b \,x^{3}+a \right )\right ) b \,x^{3}+12 b \,x^{3}+\sin \left (3 b \,x^{3}+3 a \right ) \cos \left (3 b \,x^{3}+3 a \right )+9 \cos \left (b \,x^{3}+a \right ) \sin \left (b \,x^{3}+a \right )}{1152 b^{2}}+\frac {-\frac {x^{3}}{6 b}+\frac {\tan \left (2 b \,x^{3}+2 a \right )}{12 b^{2}}+\frac {x^{3} \left (\tan ^{2}\left (2 b \,x^{3}+2 a \right )\right )}{6 b}}{128+128 \left (\tan ^{2}\left (2 b \,x^{3}+2 a \right )\right )}+\frac {-\frac {x^{3}}{24 b}+\frac {\tan \left (2 b \,x^{3}+2 a \right )}{48 b^{2}}-\frac {\tan ^{3}\left (2 b \,x^{3}+2 a \right )}{48 b^{2}}+\frac {x^{3} \left (\tan ^{2}\left (2 b \,x^{3}+2 a \right )\right )}{4 b}-\frac {x^{3} \left (\tan ^{4}\left (2 b \,x^{3}+2 a \right )\right )}{24 b}}{128 \left (1+\tan ^{2}\left (2 b \,x^{3}+2 a \right )\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 126, normalized size = 0.98 \[ -\frac {24 \, b x^{3} \cos \left (8 \, b x^{3} + 8 \, a\right ) + 192 \, b x^{3} \cos \left (6 \, b x^{3} + 6 \, a\right ) + 672 \, b x^{3} \cos \left (4 \, b x^{3} + 4 \, a\right ) + 1344 \, b x^{3} \cos \left (2 \, b x^{3} + 2 \, a\right ) - 3 \, \sin \left (8 \, b x^{3} + 8 \, a\right ) - 32 \, \sin \left (6 \, b x^{3} + 6 \, a\right ) - 168 \, \sin \left (4 \, b x^{3} + 4 \, a\right ) - 672 \, \sin \left (2 \, b x^{3} + 2 \, a\right )}{73728 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.45, size = 147, normalized size = 1.14 \[ \frac {168\,\sin \left (2\,b\,x^3+2\,a\right )+42\,\sin \left (4\,b\,x^3+4\,a\right )+8\,\sin \left (6\,b\,x^3+6\,a\right )+\frac {3\,\sin \left (8\,b\,x^3+8\,a\right )}{4}+336\,b\,x^3\,\left (2\,{\sin \left (b\,x^3+a\right )}^2-1\right )+168\,b\,x^3\,\left (2\,{\sin \left (2\,b\,x^3+2\,a\right )}^2-1\right )+48\,b\,x^3\,\left (2\,{\sin \left (3\,b\,x^3+3\,a\right )}^2-1\right )+6\,b\,x^3\,\left (2\,{\sin \left (4\,b\,x^3+4\,a\right )}^2-1\right )}{18432\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 74.97, size = 241, normalized size = 1.87 \[ \begin {cases} \frac {35 x^{3} \sin ^{8}{\left (a + b x^{3} \right )}}{3072 b} + \frac {35 x^{3} \sin ^{6}{\left (a + b x^{3} \right )} \cos ^{2}{\left (a + b x^{3} \right )}}{768 b} + \frac {35 x^{3} \sin ^{4}{\left (a + b x^{3} \right )} \cos ^{4}{\left (a + b x^{3} \right )}}{512 b} + \frac {35 x^{3} \sin ^{2}{\left (a + b x^{3} \right )} \cos ^{6}{\left (a + b x^{3} \right )}}{768 b} - \frac {31 x^{3} \cos ^{8}{\left (a + b x^{3} \right )}}{1024 b} + \frac {35 \sin ^{7}{\left (a + b x^{3} \right )} \cos {\left (a + b x^{3} \right )}}{3072 b^{2}} + \frac {385 \sin ^{5}{\left (a + b x^{3} \right )} \cos ^{3}{\left (a + b x^{3} \right )}}{9216 b^{2}} + \frac {511 \sin ^{3}{\left (a + b x^{3} \right )} \cos ^{5}{\left (a + b x^{3} \right )}}{9216 b^{2}} + \frac {31 \sin {\left (a + b x^{3} \right )} \cos ^{7}{\left (a + b x^{3} \right )}}{1024 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{6} \sin {\relax (a )} \cos ^{7}{\relax (a )}}{6} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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