Optimal. Leaf size=167 \[ -\frac {12 \cos \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}-\frac {12 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
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Rubi [A] time = 0.14, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3432, 3296, 2638} \[ -\frac {12 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \cos \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3432
Rubi steps
\begin {align*} \int x \cos \left (a+b \sqrt {c+d x}\right ) \, dx &=\frac {2 \operatorname {Subst}\left (\int \left (-\frac {c x \cos (a+b x)}{d}+\frac {x^3 \cos (a+b x)}{d}\right ) \, dx,x,\sqrt {c+d x}\right )}{d}\\ &=\frac {2 \operatorname {Subst}\left (\int x^3 \cos (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}-\frac {(2 c) \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt {c+d x}\right )}{d^2}\\ &=-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {6 \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}+\frac {(2 c) \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b d^2}\\ &=-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}-\frac {12 \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^2 d^2}\\ &=-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {12 \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt {c+d x}\right )}{b^3 d^2}\\ &=-\frac {12 \cos \left (a+b \sqrt {c+d x}\right )}{b^4 d^2}-\frac {2 c \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}+\frac {6 (c+d x) \cos \left (a+b \sqrt {c+d x}\right )}{b^2 d^2}-\frac {12 \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b^3 d^2}-\frac {2 c \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}+\frac {2 (c+d x)^{3/2} \sin \left (a+b \sqrt {c+d x}\right )}{b d^2}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 71, normalized size = 0.43 \[ \frac {2 \left (b \left (b^2 d x-6\right ) \sqrt {c+d x} \sin \left (a+b \sqrt {c+d x}\right )+\left (b^2 (2 c+3 d x)-6\right ) \cos \left (a+b \sqrt {c+d x}\right )\right )}{b^4 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 67, normalized size = 0.40 \[ \frac {2 \, {\left ({\left (b^{3} d x - 6 \, b\right )} \sqrt {d x + c} \sin \left (\sqrt {d x + c} b + a\right ) + {\left (3 \, b^{2} d x + 2 \, b^{2} c - 6\right )} \cos \left (\sqrt {d x + c} b + a\right )\right )}}{b^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 166, normalized size = 0.99 \[ -\frac {2 \, {\left (\frac {{\left (b^{2} c - 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} + 6 \, {\left (\sqrt {d x + c} b + a\right )} a - 3 \, a^{2} + 6\right )} \cos \left (\sqrt {d x + c} b + a\right )}{b^{2}} + \frac {{\left ({\left (\sqrt {d x + c} b + a\right )} b^{2} c - a b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{3} + 3 \, {\left (\sqrt {d x + c} b + a\right )}^{2} a - 3 \, {\left (\sqrt {d x + c} b + a\right )} a^{2} + a^{3} + 6 \, \sqrt {d x + c} b\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 299, normalized size = 1.79 \[ \frac {-2 c \left (\cos \left (a +b \sqrt {d x +c}\right )+\left (a +b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {d x +c}\right )\right )+2 a c \sin \left (a +b \sqrt {d x +c}\right )+\frac {2 \left (\left (a +b \sqrt {d x +c}\right )^{3} \sin \left (a +b \sqrt {d x +c}\right )+3 \left (a +b \sqrt {d x +c}\right )^{2} \cos \left (a +b \sqrt {d x +c}\right )-6 \cos \left (a +b \sqrt {d x +c}\right )-6 \left (a +b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {6 a \left (\left (a +b \sqrt {d x +c}\right )^{2} \sin \left (a +b \sqrt {d x +c}\right )-2 \sin \left (a +b \sqrt {d x +c}\right )+2 \cos \left (a +b \sqrt {d x +c}\right ) \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}+\frac {6 a^{2} \left (\cos \left (a +b \sqrt {d x +c}\right )+\left (a +b \sqrt {d x +c}\right ) \sin \left (a +b \sqrt {d x +c}\right )\right )}{b^{2}}-\frac {2 a^{3} \sin \left (a +b \sqrt {d x +c}\right )}{b^{2}}}{d^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 263, normalized size = 1.57 \[ \frac {2 \, {\left (a c \sin \left (\sqrt {d x + c} b + a\right ) - {\left ({\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )} c - \frac {a^{3} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}} + \frac {3 \, {\left ({\left (\sqrt {d x + c} b + a\right )} \sin \left (\sqrt {d x + c} b + a\right ) + \cos \left (\sqrt {d x + c} b + a\right )\right )} a^{2}}{b^{2}} - \frac {3 \, {\left (2 \, {\left (\sqrt {d x + c} b + a\right )} \cos \left (\sqrt {d x + c} b + a\right ) + {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \sin \left (\sqrt {d x + c} b + a\right )\right )} a}{b^{2}} + \frac {3 \, {\left ({\left (\sqrt {d x + c} b + a\right )}^{2} - 2\right )} \cos \left (\sqrt {d x + c} b + a\right ) + {\left ({\left (\sqrt {d x + c} b + a\right )}^{3} - 6 \, \sqrt {d x + c} b - 6 \, a\right )} \sin \left (\sqrt {d x + c} b + a\right )}{b^{2}}\right )}}{b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\cos \left (a+b\,\sqrt {c+d\,x}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 151, normalized size = 0.90 \[ \begin {cases} \frac {x^{2} \cos {\relax (a )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\\frac {x^{2} \cos {\left (a + b \sqrt {c} \right )}}{2} & \text {for}\: d = 0 \\\frac {2 x \sqrt {c + d x} \sin {\left (a + b \sqrt {c + d x} \right )}}{b d} + \frac {4 c \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d^{2}} + \frac {6 x \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{2} d} - \frac {12 \sqrt {c + d x} \sin {\left (a + b \sqrt {c + d x} \right )}}{b^{3} d^{2}} - \frac {12 \cos {\left (a + b \sqrt {c + d x} \right )}}{b^{4} d^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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