Optimal. Leaf size=247 \[ -\frac{b^4 \sin (a) \sqrt [3]{e (c+d x)} \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}-\frac{b^4 \cos (a) \sqrt [3]{e (c+d x)} \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}-\frac{b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}+\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.242208, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3431, 15, 3297, 3303, 3299, 3302} \[ -\frac{b^4 \sin (a) \sqrt [3]{e (c+d x)} \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}-\frac{b^4 \cos (a) \sqrt [3]{e (c+d x)} \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}-\frac{b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}+\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3431
Rule 15
Rule 3297
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \sqrt [3]{c e+d e x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right ) \, dx &=-\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt [3]{\frac{e}{x^3}} \sin (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d}\\ &=-\frac{\left (3 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^5} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{d \sqrt [3]{c+d x}}\\ &=\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{\left (3 b \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^4} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{4 d \sqrt [3]{c+d x}}\\ &=\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}+\frac{\left (b^2 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x^3} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{4 d \sqrt [3]{c+d x}}\\ &=\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}+\frac{\left (b^3 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\cos (a+b x)}{x^2} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}\\ &=-\frac{b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{\left (b^4 \sqrt [3]{e (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{\sin (a+b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}\\ &=-\frac{b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{\left (b^4 \sqrt [3]{e (c+d x)} \cos (a)\right ) \operatorname{Subst}\left (\int \frac{\sin (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}-\frac{\left (b^4 \sqrt [3]{e (c+d x)} \sin (a)\right ) \operatorname{Subst}\left (\int \frac{\cos (b x)}{x} \, dx,x,\frac{1}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}\\ &=-\frac{b^3 \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{b (c+d x)^{2/3} \sqrt [3]{e (c+d x)} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{b^4 \sqrt [3]{e (c+d x)} \text{Ci}\left (\frac{b}{\sqrt [3]{c+d x}}\right ) \sin (a)}{8 d \sqrt [3]{c+d x}}-\frac{b^2 \sqrt [3]{c+d x} \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d}+\frac{3 (c+d x) \sqrt [3]{e (c+d x)} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )}{4 d}-\frac{b^4 \sqrt [3]{e (c+d x)} \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )}{8 d \sqrt [3]{c+d x}}\\ \end{align*}
Mathematica [A] time = 0.340315, size = 208, normalized size = 0.84 \[ -\frac{\sqrt [3]{e (c+d x)} \left (b^4 \sin (a) \text{CosIntegral}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+b^4 \cos (a) \text{Si}\left (\frac{b}{\sqrt [3]{c+d x}}\right )+b^2 (c+d x)^{2/3} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )+b^3 \sqrt [3]{c+d x} \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-6 c \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-6 d x \sqrt [3]{c+d x} \sin \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-2 b c \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )-2 b d x \cos \left (a+\frac{b}{\sqrt [3]{c+d x}}\right )\right )}{8 d \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.042, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{dex+ce}\sin \left ( a+{b{\frac{1}{\sqrt [3]{dx+c}}}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d e x + c e\right )}^{\frac{1}{3}} \sin \left (\frac{a d x + a c +{\left (d x + c\right )}^{\frac{2}{3}} b}{d x + c}\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{e \left (c + d x\right )} \sin{\left (a + \frac{b}{\sqrt [3]{c + d x}} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d e x + c e\right )}^{\frac{1}{3}} \sin \left (a + \frac{b}{{\left (d x + c\right )}^{\frac{1}{3}}}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]