\(\int (a+b \sin ^4(c+d x))^p \, dx\) [571]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\text {Int}\left (\left (a+b \sin ^4(c+d x)\right )^p,x\right ) \]

[Out]

Unintegrable((a+b*sin(d*x+c)^4)^p,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, 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 = {} \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int \left (a+b \sin ^4(c+d x)\right )^p \, dx \]

[In]

Int[(a + b*Sin[c + d*x]^4)^p,x]

[Out]

Defer[Int][(a + b*Sin[c + d*x]^4)^p, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (a+b \sin ^4(c+d x)\right )^p \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int \left (a+b \sin ^4(c+d x)\right )^p \, dx \]

[In]

Integrate[(a + b*Sin[c + d*x]^4)^p,x]

[Out]

Integrate[(a + b*Sin[c + d*x]^4)^p, x]

Maple [N/A] (verified)

Not integrable

Time = 0.94 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int {\left (a +b \left (\sin ^{4}\left (d x +c \right )\right )\right )}^{p}d x\]

[In]

int((a+b*sin(d*x+c)^4)^p,x)

[Out]

int((a+b*sin(d*x+c)^4)^p,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^4)^p,x, algorithm="fricas")

[Out]

integral((b*cos(d*x + c)^4 - 2*b*cos(d*x + c)^2 + a + b)^p, x)

Sympy [F(-1)]

Timed out. \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(d*x+c)**4)**p,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 1.09 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^4)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p, x)

Giac [N/A]

Not integrable

Time = 0.85 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int { {\left (b \sin \left (d x + c\right )^{4} + a\right )}^{p} \,d x } \]

[In]

integrate((a+b*sin(d*x+c)^4)^p,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c)^4 + a)^p, x)

Mupad [N/A]

Not integrable

Time = 14.78 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sin ^4(c+d x)\right )^p \, dx=\int {\left (b\,{\sin \left (c+d\,x\right )}^4+a\right )}^p \,d x \]

[In]

int((a + b*sin(c + d*x)^4)^p,x)

[Out]

int((a + b*sin(c + d*x)^4)^p, x)