\(\int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx\) [144]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\text {Int}\left (\frac {\sqrt [3]{a+a \sin (c+d x)}}{x},x\right ) \]

[Out]

Unintegrable((a+a*sin(d*x+c))^(1/3)/x,x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx \]

[In]

Int[(a + a*Sin[c + d*x])^(1/3)/x,x]

[Out]

Defer[Int][(a + a*Sin[c + d*x])^(1/3)/x, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx \]

[In]

Integrate[(a + a*Sin[c + d*x])^(1/3)/x,x]

[Out]

Integrate[(a + a*Sin[c + d*x])^(1/3)/x, x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

\[\int \frac {\left (a +a \sin \left (d x +c \right )\right )^{\frac {1}{3}}}{x}d x\]

[In]

int((a+a*sin(d*x+c))^(1/3)/x,x)

[Out]

int((a+a*sin(d*x+c))^(1/3)/x,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+a*sin(d*x+c))^(1/3)/x,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

Sympy [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int \frac {\sqrt [3]{a \left (\sin {\left (c + d x \right )} + 1\right )}}{x}\, dx \]

[In]

integrate((a+a*sin(d*x+c))**(1/3)/x,x)

[Out]

Integral((a*(sin(c + d*x) + 1))**(1/3)/x, x)

Maxima [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{x} \,d x } \]

[In]

integrate((a+a*sin(d*x+c))^(1/3)/x,x, algorithm="maxima")

[Out]

integrate((a*sin(d*x + c) + a)^(1/3)/x, x)

Giac [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int { \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {1}{3}}}{x} \,d x } \]

[In]

integrate((a+a*sin(d*x+c))^(1/3)/x,x, algorithm="giac")

[Out]

integrate((a*sin(d*x + c) + a)^(1/3)/x, x)

Mupad [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{a+a \sin (c+d x)}}{x} \, dx=\int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{1/3}}{x} \,d x \]

[In]

int((a + a*sin(c + d*x))^(1/3)/x,x)

[Out]

int((a + a*sin(c + d*x))^(1/3)/x, x)