\(\int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx\) [1191]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\text {Int}\left (\frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}},x\right ) \]

[Out]

Unintegrable(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 33, 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 = {} \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx \]

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

Defer[Int][(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 91.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx \]

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]

Maple [N/A] (verified)

Not integrable

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88

\[\int \frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{\frac {1}{3}}\left (d x +c \right )\right )}{\sqrt {a +b \sin \left (d x +c \right )}}d x\]

[In]

int(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x)

[Out]

int(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x)

Fricas [N/A]

Not integrable

Time = 13.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), x)

Sympy [N/A]

Not integrable

Time = 23.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {\sqrt [3]{\sin {\left (c + d x \right )}} \cos ^{4}{\left (c + d x \right )}}{\sqrt {a + b \sin {\left (c + d x \right )}}}\, dx \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**(1/3)/(a+b*sin(d*x+c))**(1/2),x)

[Out]

Integral(sin(c + d*x)**(1/3)*cos(c + d*x)**4/sqrt(a + b*sin(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), x)

Giac [N/A]

Not integrable

Time = 216.55 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int { \frac {\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac {1}{3}}}{\sqrt {b \sin \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), x)

Mupad [N/A]

Not integrable

Time = 15.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt {a+b \sin (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^{1/3}}{\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \]

[In]

int((cos(c + d*x)^4*sin(c + d*x)^(1/3))/(a + b*sin(c + d*x))^(1/2),x)

[Out]

int((cos(c + d*x)^4*sin(c + d*x)^(1/3))/(a + b*sin(c + d*x))^(1/2), x)