3.3.69 \(\int (a+b \sin (c+d (f+g x)^n)) \, dx\) [269]

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 = 0.04, 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 = {3446, 2239} \begin {gather*} \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 \end {gather*}

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 2239

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 3446

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 = 0.18, size = 126, normalized size = 1.03 \begin {gather*} a x-\frac {i b (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 ) (\cos (c)-i \sin (c))}{2 g n}+\frac {i b (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 ) (\cos (c)+i \sin (c))}{2 g n} \end {gather*}

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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Maple [F]
time = 0.04, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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