3.274 \(\int (a+b \sin (c+d (f+g x)^n))^2 \, dx\)

Optimal. Leaf size=261 \[ \frac{i a b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )}{g n}-\frac{i a b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )}{g n}+\frac{b^2 e^{2 i c} 2^{-\frac{1}{n}-2} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac{b^2 e^{-2 i c} 2^{-\frac{1}{n}-2} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i d (f+g x)^n\right )}{g n}+\frac{1}{2} x \left (2 a^2+b^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.149213, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {3367, 3366, 2208, 3365} \[ \frac{i a b e^{i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )}{g n}-\frac{i a b e^{-i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )}{g n}+\frac{b^2 e^{2 i c} 2^{-\frac{1}{n}-2} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac{b^2 e^{-2 i c} 2^{-\frac{1}{n}-2} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i d (f+g x)^n\right )}{g n}+\frac{1}{2} x \left (2 a^2+b^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*(f + g*x)^n])^2,x]

[Out]

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

Rule 3367

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Sin[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]

Rule 3366

Int[Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[1/2, Int[E^(-(c*I) - d*I*(e + f*x)^n), x],
x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

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 )^2 \, dx &=\int \left (a^2+\frac{b^2}{2}-\frac{1}{2} b^2 \cos \left (2 c+2 d (f+g x)^n\right )+2 a b \sin \left (c+d (f+g x)^n\right )\right ) \, dx\\ &=\frac{1}{2} \left (2 a^2+b^2\right ) x+(2 a b) \int \sin \left (c+d (f+g x)^n\right ) \, dx-\frac{1}{2} b^2 \int \cos \left (2 c+2 d (f+g x)^n\right ) \, dx\\ &=\frac{1}{2} \left (2 a^2+b^2\right ) x+(i a b) \int e^{-i c-i d (f+g x)^n} \, dx-(i a b) \int e^{i c+i d (f+g x)^n} \, dx-\frac{1}{4} b^2 \int e^{-2 i c-2 i d (f+g x)^n} \, dx-\frac{1}{4} b^2 \int e^{2 i c+2 i d (f+g x)^n} \, dx\\ &=\frac{1}{2} \left (2 a^2+b^2\right ) x+\frac{i a 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 )}{g n}-\frac{i a 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 )}{g n}+\frac{2^{-2-\frac{1}{n}} b^2 e^{2 i c} (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac{1}{n},-2 i d (f+g x)^n\right )}{g n}+\frac{2^{-2-\frac{1}{n}} b^2 e^{-2 i c} (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \Gamma \left (\frac{1}{n},2 i d (f+g x)^n\right )}{g n}\\ \end{align*}

Mathematica [A]  time = 1.96856, size = 277, normalized size = 1.06 \[ \frac{4 i a b (\cos (c)+i \sin (c)) (f+g x) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i d (f+g x)^n\right )-4 i a b (\cos (c)-i \sin (c)) (f+g x) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i d (f+g x)^n\right )+b^2 (\cos (c)+i \sin (c))^2 (f+g x) \left (\cosh \left (\frac{\log (2)}{n}\right )-\sinh \left (\frac{\log (2)}{n}\right )\right ) \left (-i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-2 i d (f+g x)^n\right )+b^2 (\cos (c)-i \sin (c))^2 (f+g x) \left (\cosh \left (\frac{\log (2)}{n}\right )-\sinh \left (\frac{\log (2)}{n}\right )\right ) \left (i d (f+g x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},2 i d (f+g x)^n\right )+4 a^2 g n x+2 b^2 g n x}{4 g n} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Sin[c + d*(f + g*x)^n])^2,x]

[Out]

(4*a^2*g*n*x + 2*b^2*g*n*x - ((4*I)*a*b*(f + g*x)*Gamma[n^(-1), I*d*(f + g*x)^n]*(Cos[c] - I*Sin[c]))/(I*d*(f
+ g*x)^n)^n^(-1) + ((4*I)*a*b*(f + g*x)*Gamma[n^(-1), (-I)*d*(f + g*x)^n]*(Cos[c] + I*Sin[c]))/((-I)*d*(f + g*
x)^n)^n^(-1) + (b^2*(f + g*x)*Gamma[n^(-1), (2*I)*d*(f + g*x)^n]*(Cos[c] - I*Sin[c])^2*(Cosh[Log[2]/n] - Sinh[
Log[2]/n]))/(I*d*(f + g*x)^n)^n^(-1) + (b^2*(f + g*x)*Gamma[n^(-1), (-2*I)*d*(f + g*x)^n]*(Cos[c] + I*Sin[c])^
2*(Cosh[Log[2]/n] - Sinh[Log[2]/n]))/((-I)*d*(f + g*x)^n)^n^(-1))/(4*g*n)

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Maple [F]  time = 0.206, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sin \left ( c+d \left ( gx+f \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(c+d*(g*x+f)^n))^2,x)

[Out]

int((a+b*sin(c+d*(g*x+f)^n))^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} x + \frac{1}{2} \, b^{2} x - \frac{1}{2} \, b^{2} \int \cos \left (2 \,{\left (g x + f\right )}^{n} d + 2 \, c\right )\,{d x} + 2 \, a b \int \sin \left ({\left (g x + f\right )}^{n} d + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="maxima")

[Out]

a^2*x + 1/2*b^2*x - 1/2*b^2*integrate(cos(2*(g*x + f)^n*d + 2*c), x) + 2*a*b*integrate(sin((g*x + f)^n*d + c),
x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-b^{2} \cos \left ({\left (g x + f\right )}^{n} d + c\right )^{2} + 2 \, a b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a^{2} + b^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="fricas")

[Out]

integral(-b^2*cos((g*x + f)^n*d + c)^2 + 2*a*b*sin((g*x + f)^n*d + c) + a^2 + b^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (c + d \left (f + g x\right )^{n} \right )}\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)**n))**2,x)

[Out]

Integral((a + b*sin(c + d*(f + g*x)**n))**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left ({\left (g x + f\right )}^{n} d + c\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d*(g*x+f)^n))^2,x, algorithm="giac")

[Out]

integrate((b*sin((g*x + f)^n*d + c) + a)^2, x)