3.269 \(\int (a+b \sin (c+d (f+g x)^n)) \, dx\)

Optimal. Leaf size=122 \[ a x+\frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g n} \]

[Out]

a*x+1/2*I*b*exp(I*c)*(g*x+f)*GAMMA(1/n,-I*d*(g*x+f)^n)/g/n/((-I*d*(g*x+f)^n)^(1/n))-1/2*I*b*(g*x+f)*GAMMA(1/n,
I*d*(g*x+f)^n)/exp(I*c)/g/n/((I*d*(g*x+f)^n)^(1/n))

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Rubi [A]  time = 0.06, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3365, 2208} \[ \frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i d (f+g x)^n\right )}{2 g n}+a x \]

Antiderivative was successfully verified.

[In]

Int[a + b*Sin[c + d*(f + g*x)^n],x]

[Out]

a*x + ((I/2)*b*E^(I*c)*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^n])/(g*n*((-I)*d*(f + g*x)^n)^n^(-1)) - ((I/2)
*b*(f + g*x)*Gamma[n^(-1), I*d*(f + g*x)^n])/(E^(I*c)*g*n*(I*d*(f + g*x)^n)^n^(-1))

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 3365

Int[Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[I/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] - Dist[I/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rubi steps

\begin {align*} \int \left (a+b \sin \left (c+d (f+g x)^n\right )\right ) \, dx &=a x+b \int \sin \left (c+d (f+g x)^n\right ) \, dx\\ &=a x+\frac {1}{2} (i b) \int e^{-i c-i d (f+g x)^n} \, dx-\frac {1}{2} (i b) \int e^{i c+i d (f+g x)^n} \, dx\\ &=a x+\frac {i b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac {i b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g n}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 126, normalized size = 1.03 \[ a x+\frac {i b (\cos (c)+i \sin (c)) (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i d (f+g x)^n\right )}{2 g n}-\frac {i b (\cos (c)-i \sin (c)) (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i d (f+g x)^n\right )}{2 g n} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*Sin[c + d*(f + g*x)^n],x]

[Out]

a*x - ((I/2)*b*(f + g*x)*Gamma[n^(-1), I*d*(f + g*x)^n]*(Cos[c] - I*Sin[c]))/(g*n*(I*d*(f + g*x)^n)^n^(-1)) +
((I/2)*b*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^n]*(Cos[c] + I*Sin[c]))/(g*n*((-I)*d*(f + g*x)^n)^n^(-1))

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fricas [F]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sin(c+d*(g*x+f)^n),x, algorithm="fricas")

[Out]

integral(b*sin((g*x + f)^n*d + c) + a, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sin(c+d*(g*x+f)^n),x, algorithm="giac")

[Out]

integrate(b*sin((g*x + f)^n*d + c) + a, x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int a +b \sin \left (c +d \left (g x +f \right )^{n}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a+b*sin(c+d*(g*x+f)^n),x)

[Out]

int(a+b*sin(c+d*(g*x+f)^n),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a x + b \int \sin \left ({\left (g x + f\right )}^{n} d + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sin(c+d*(g*x+f)^n),x, algorithm="maxima")

[Out]

a*x + b*integrate(sin((g*x + f)^n*d + c), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int a+b\,\sin \left (c+d\,{\left (f+g\,x\right )}^n\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*sin(c + d*(f + g*x)^n),x)

[Out]

int(a + b*sin(c + d*(f + g*x)^n), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d \left (f + g x\right )^{n} \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(a+b*sin(c+d*(g*x+f)**n),x)

[Out]

Integral(a + b*sin(c + d*(f + g*x)**n), x)

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