Optimal. Leaf size=85 \[ \frac {6 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}+\frac {6 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3361, 3296, 2638} \[ \frac {6 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}+\frac {6 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2638
Rule 3296
Rule 3361
Rubi steps
\begin {align*} \int \sin \left (a+b \sqrt [3]{c+d x}\right ) \, dx &=\frac {3 \operatorname {Subst}\left (\int x^2 \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{d}\\ &=-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {6 \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b d}\\ &=-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {6 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}-\frac {6 \operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,\sqrt [3]{c+d x}\right )}{b^2 d}\\ &=\frac {6 \cos \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d}-\frac {3 (c+d x)^{2/3} \cos \left (a+b \sqrt [3]{c+d x}\right )}{b d}+\frac {6 \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 65, normalized size = 0.76 \[ \frac {\left (6-3 b^2 (c+d x)^{2/3}\right ) \cos \left (a+b \sqrt [3]{c+d x}\right )+6 b \sqrt [3]{c+d x} \sin \left (a+b \sqrt [3]{c+d x}\right )}{b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.73, size = 58, normalized size = 0.68 \[ \frac {3 \, {\left (2 \, {\left (d x + c\right )}^{\frac {1}{3}} b \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - {\left ({\left (d x + c\right )}^{\frac {2}{3}} b^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.36, size = 82, normalized size = 0.96 \[ \frac {3 \, {\left (\frac {2 \, {\left (d x + c\right )}^{\frac {1}{3}} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b} - \frac {{\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} a + a^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}{b^{2}}\right )}}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 134, normalized size = 1.58 \[ \frac {-3 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )+6 \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-6 a \left (\sin \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )-\left (a +b \left (d x +c \right )^{\frac {1}{3}}\right ) \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )\right )-3 a^{2} \cos \left (a +b \left (d x +c \right )^{\frac {1}{3}}\right )}{d \,b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 120, normalized size = 1.41 \[ -\frac {3 \, {\left (a^{2} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )} a + {\left ({\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )}^{2} - 2\right )} \cos \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right ) - 2 \, {\left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )} \sin \left ({\left (d x + c\right )}^{\frac {1}{3}} b + a\right )\right )}}{b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 4.61, size = 69, normalized size = 0.81 \[ \frac {3\,\left (2\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )+2\,b\,\sin \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{1/3}-b^2\,\cos \left (a+b\,{\left (c+d\,x\right )}^{1/3}\right )\,{\left (c+d\,x\right )}^{2/3}\right )}{b^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.42, size = 94, normalized size = 1.11 \[ \begin {cases} x \sin {\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee d = 0\right ) \\x \sin {\left (a + b \sqrt [3]{c} \right )} & \text {for}\: d = 0 \\- \frac {3 \left (c + d x\right )^{\frac {2}{3}} \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b d} + \frac {6 \sqrt [3]{c + d x} \sin {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{2} d} + \frac {6 \cos {\left (a + b \sqrt [3]{c + d x} \right )}}{b^{3} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________