3.3.94 \(\int (e+f x) (a+b \sin (c+\frac {d}{x}))^2 \, dx\) [294]

Optimal. Leaf size=254 \[ a^2 e x+\frac {1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac {d}{x}\right )-2 a b d e \cos (c) \text {Ci}\left (\frac {d}{x}\right )-b^2 d^2 f \cos (2 c) \text {Ci}\left (\frac {2 d}{x}\right )+a b d^2 f \text {Ci}\left (\frac {d}{x}\right ) \sin (c)-b^2 d e \text {Ci}\left (\frac {2 d}{x}\right ) \sin (2 c)+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )+a b d^2 f \cos (c) \text {Si}\left (\frac {d}{x}\right )+2 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )-b^2 d e \cos (2 c) \text {Si}\left (\frac {2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text {Si}\left (\frac {2 d}{x}\right ) \]

[Out]

a^2*e*x+1/2*a^2*f*x^2-2*a*b*d*e*Ci(d/x)*cos(c)-b^2*d^2*f*Ci(2*d/x)*cos(2*c)+a*b*d*f*x*cos(c+d/x)+a*b*d^2*f*cos
(c)*Si(d/x)-b^2*d*e*cos(2*c)*Si(2*d/x)+a*b*d^2*f*Ci(d/x)*sin(c)+2*a*b*d*e*Si(d/x)*sin(c)-b^2*d*e*Ci(2*d/x)*sin
(2*c)+b^2*d^2*f*Si(2*d/x)*sin(2*c)+2*a*b*e*x*sin(c+d/x)+a*b*f*x^2*sin(c+d/x)+b^2*d*f*x*cos(c+d/x)*sin(c+d/x)+b
^2*e*x*sin(c+d/x)^2+1/2*b^2*f*x^2*sin(c+d/x)^2

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Rubi [A]
time = 0.41, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3512, 3398, 3378, 3384, 3380, 3383, 3395, 29, 3393, 3394, 12} \begin {gather*} a^2 e x+\frac {1}{2} a^2 f x^2+a b d^2 f \sin (c) \text {CosIntegral}\left (\frac {d}{x}\right )-2 a b d e \cos (c) \text {CosIntegral}\left (\frac {d}{x}\right )+a b d^2 f \cos (c) \text {Si}\left (\frac {d}{x}\right )+2 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+a b d f x \cos \left (c+\frac {d}{x}\right )-b^2 d^2 f \cos (2 c) \text {CosIntegral}\left (\frac {2 d}{x}\right )-b^2 d e \sin (2 c) \text {CosIntegral}\left (\frac {2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text {Si}\left (\frac {2 d}{x}\right )-b^2 d e \cos (2 c) \text {Si}\left (\frac {2 d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )+b^2 d f x \sin \left (c+\frac {d}{x}\right ) \cos \left (c+\frac {d}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a^2*e*x + (a^2*f*x^2)/2 + a*b*d*f*x*Cos[c + d/x] - 2*a*b*d*e*Cos[c]*CosIntegral[d/x] - b^2*d^2*f*Cos[2*c]*CosI
ntegral[(2*d)/x] + a*b*d^2*f*CosIntegral[d/x]*Sin[c] - b^2*d*e*CosIntegral[(2*d)/x]*Sin[2*c] + 2*a*b*e*x*Sin[c
+ d/x] + a*b*f*x^2*Sin[c + d/x] + b^2*d*f*x*Cos[c + d/x]*Sin[c + d/x] + b^2*e*x*Sin[c + d/x]^2 + (b^2*f*x^2*S
in[c + d/x]^2)/2 + a*b*d^2*f*Cos[c]*SinIntegral[d/x] + 2*a*b*d*e*Sin[c]*SinIntegral[d/x] - b^2*d*e*Cos[2*c]*Si
nIntegral[(2*d)/x] + b^2*d^2*f*Sin[2*c]*SinIntegral[(2*d)/x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 3378

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m
+ 1))), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3383

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3394

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]^
n/(d*(m + 1))), x] - Dist[f*(n/(d*(m + 1))), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

Rule 3395

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(c + d*x)^(m + 1)*((b*Si
n[e + f*x])^n/(d*(m + 1))), x] + (Dist[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[f^2*(n^2/(d^2*(m + 1)*(m + 2))), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1)*(m + 2))), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 3398

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 3512

Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :
> Dist[1/(n*f), Subst[Int[ExpandIntegrand[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m,
x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p, 0] && IntegerQ[1/n]

Rubi steps

\begin {align*} \int (e+f x) \left (a+b \sin \left (c+\frac {d}{x}\right )\right )^2 \, dx &=-\text {Subst}\left (\int \left (\frac {f (a+b \sin (c+d x))^2}{x^3}+\frac {e (a+b \sin (c+d x))^2}{x^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\left (e \text {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{x^2} \, dx,x,\frac {1}{x}\right )\right )-f \text {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=-\left (e \text {Subst}\left (\int \left (\frac {a^2}{x^2}+\frac {2 a b \sin (c+d x)}{x^2}+\frac {b^2 \sin ^2(c+d x)}{x^2}\right ) \, dx,x,\frac {1}{x}\right )\right )-f \text {Subst}\left (\int \left (\frac {a^2}{x^3}+\frac {2 a b \sin (c+d x)}{x^3}+\frac {b^2 \sin ^2(c+d x)}{x^3}\right ) \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2-(2 a b e) \text {Subst}\left (\int \frac {\sin (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-\left (b^2 e\right ) \text {Subst}\left (\int \frac {\sin ^2(c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-(2 a b f) \text {Subst}\left (\int \frac {\sin (c+d x)}{x^3} \, dx,x,\frac {1}{x}\right )-\left (b^2 f\right ) \text {Subst}\left (\int \frac {\sin ^2(c+d x)}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )-(2 a b d e) \text {Subst}\left (\int \frac {\cos (c+d x)}{x} \, dx,x,\frac {1}{x}\right )-\left (2 b^2 d e\right ) \text {Subst}\left (\int \frac {\sin (2 c+2 d x)}{2 x} \, dx,x,\frac {1}{x}\right )-(a b d f) \text {Subst}\left (\int \frac {\cos (c+d x)}{x^2} \, dx,x,\frac {1}{x}\right )-\left (b^2 d^2 f\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )+\left (2 b^2 d^2 f\right ) \text {Subst}\left (\int \frac {\sin ^2(c+d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac {d}{x}\right )+b^2 d^2 f \log (x)+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )-\left (b^2 d e\right ) \text {Subst}\left (\int \frac {\sin (2 c+2 d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (a b d^2 f\right ) \text {Subst}\left (\int \frac {\sin (c+d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (2 b^2 d^2 f\right ) \text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos (2 c+2 d x)}{2 x}\right ) \, dx,x,\frac {1}{x}\right )-(2 a b d e \cos (c)) \text {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )+(2 a b d e \sin (c)) \text {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac {d}{x}\right )-2 a b d e \cos (c) \text {Ci}\left (\frac {d}{x}\right )+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )+2 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )-\left (b^2 d^2 f\right ) \text {Subst}\left (\int \frac {\cos (2 c+2 d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (a b d^2 f \cos (c)\right ) \text {Subst}\left (\int \frac {\sin (d x)}{x} \, dx,x,\frac {1}{x}\right )-\left (b^2 d e \cos (2 c)\right ) \text {Subst}\left (\int \frac {\sin (2 d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (a b d^2 f \sin (c)\right ) \text {Subst}\left (\int \frac {\cos (d x)}{x} \, dx,x,\frac {1}{x}\right )-\left (b^2 d e \sin (2 c)\right ) \text {Subst}\left (\int \frac {\cos (2 d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac {d}{x}\right )-2 a b d e \cos (c) \text {Ci}\left (\frac {d}{x}\right )+a b d^2 f \text {Ci}\left (\frac {d}{x}\right ) \sin (c)-b^2 d e \text {Ci}\left (\frac {2 d}{x}\right ) \sin (2 c)+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )+a b d^2 f \cos (c) \text {Si}\left (\frac {d}{x}\right )+2 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )-b^2 d e \cos (2 c) \text {Si}\left (\frac {2 d}{x}\right )-\left (b^2 d^2 f \cos (2 c)\right ) \text {Subst}\left (\int \frac {\cos (2 d x)}{x} \, dx,x,\frac {1}{x}\right )+\left (b^2 d^2 f \sin (2 c)\right ) \text {Subst}\left (\int \frac {\sin (2 d x)}{x} \, dx,x,\frac {1}{x}\right )\\ &=a^2 e x+\frac {1}{2} a^2 f x^2+a b d f x \cos \left (c+\frac {d}{x}\right )-2 a b d e \cos (c) \text {Ci}\left (\frac {d}{x}\right )-b^2 d^2 f \cos (2 c) \text {Ci}\left (\frac {2 d}{x}\right )+a b d^2 f \text {Ci}\left (\frac {d}{x}\right ) \sin (c)-b^2 d e \text {Ci}\left (\frac {2 d}{x}\right ) \sin (2 c)+2 a b e x \sin \left (c+\frac {d}{x}\right )+a b f x^2 \sin \left (c+\frac {d}{x}\right )+b^2 d f x \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )+b^2 e x \sin ^2\left (c+\frac {d}{x}\right )+\frac {1}{2} b^2 f x^2 \sin ^2\left (c+\frac {d}{x}\right )+a b d^2 f \cos (c) \text {Si}\left (\frac {d}{x}\right )+2 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )-b^2 d e \cos (2 c) \text {Si}\left (\frac {2 d}{x}\right )+b^2 d^2 f \sin (2 c) \text {Si}\left (\frac {2 d}{x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 252, normalized size = 0.99 \begin {gather*} \frac {1}{4} \left (4 a^2 e x+2 b^2 e x+2 a^2 f x^2+b^2 f x^2+4 a b d f x \cos \left (c+\frac {d}{x}\right )-2 b^2 e x \cos \left (2 \left (c+\frac {d}{x}\right )\right )-b^2 f x^2 \cos \left (2 \left (c+\frac {d}{x}\right )\right )+4 a b d \text {Ci}\left (\frac {d}{x}\right ) (-2 e \cos (c)+d f \sin (c))-4 b^2 d \text {Ci}\left (\frac {2 d}{x}\right ) (d f \cos (2 c)+e \sin (2 c))+8 a b e x \sin \left (c+\frac {d}{x}\right )+4 a b f x^2 \sin \left (c+\frac {d}{x}\right )+2 b^2 d f x \sin \left (2 \left (c+\frac {d}{x}\right )\right )+4 a b d^2 f \cos (c) \text {Si}\left (\frac {d}{x}\right )+8 a b d e \sin (c) \text {Si}\left (\frac {d}{x}\right )-4 b^2 d e \cos (2 c) \text {Si}\left (\frac {2 d}{x}\right )+4 b^2 d^2 f \sin (2 c) \text {Si}\left (\frac {2 d}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a^2*e*x + 2*b^2*e*x + 2*a^2*f*x^2 + b^2*f*x^2 + 4*a*b*d*f*x*Cos[c + d/x] - 2*b^2*e*x*Cos[2*(c + d/x)] - b^2
*f*x^2*Cos[2*(c + d/x)] + 4*a*b*d*CosIntegral[d/x]*(-2*e*Cos[c] + d*f*Sin[c]) - 4*b^2*d*CosIntegral[(2*d)/x]*(
d*f*Cos[2*c] + e*Sin[2*c]) + 8*a*b*e*x*Sin[c + d/x] + 4*a*b*f*x^2*Sin[c + d/x] + 2*b^2*d*f*x*Sin[2*(c + d/x)]
+ 4*a*b*d^2*f*Cos[c]*SinIntegral[d/x] + 8*a*b*d*e*Sin[c]*SinIntegral[d/x] - 4*b^2*d*e*Cos[2*c]*SinIntegral[(2*
d)/x] + 4*b^2*d^2*f*Sin[2*c]*SinIntegral[(2*d)/x])/4

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Maple [A]
time = 0.12, size = 265, normalized size = 1.04

method	result	size
derivativedivides	\(-d \left (-\frac {a^{2} f \,x^{2}}{2 d}-\frac {a^{2} e x}{d}+2 a b f d \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x^{2}}{2 d^{2}}-\frac {\cos \left (c +\frac {d}{x}\right ) x}{2 d}-\frac {\sinIntegral \left (\frac {d}{x}\right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (\frac {d}{x}\right ) \sin \left (c \right )}{2}\right )+2 a b e \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x}{d}-\sinIntegral \left (\frac {d}{x}\right ) \sin \left (c \right )+\cosineIntegral \left (\frac {d}{x}\right ) \cos \left (c \right )\right )-\frac {b^{2} f \,x^{2}}{4 d}-\frac {b^{2} f d \left (-\frac {\cos \left (\frac {2 d}{x}+2 c \right ) x^{2}}{d^{2}}+\frac {2 \sin \left (\frac {2 d}{x}+2 c \right ) x}{d}+4 \sinIntegral \left (\frac {2 d}{x}\right ) \sin \left (2 c \right )-4 \cosineIntegral \left (\frac {2 d}{x}\right ) \cos \left (2 c \right )\right )}{4}-\frac {b^{2} e x}{2 d}-\frac {b^{2} e \left (-\frac {2 \cos \left (\frac {2 d}{x}+2 c \right ) x}{d}-4 \sinIntegral \left (\frac {2 d}{x}\right ) \cos \left (2 c \right )-4 \cosineIntegral \left (\frac {2 d}{x}\right ) \sin \left (2 c \right )\right )}{4}\right )\)	\(265\)
default	\(-d \left (-\frac {a^{2} f \,x^{2}}{2 d}-\frac {a^{2} e x}{d}+2 a b f d \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x^{2}}{2 d^{2}}-\frac {\cos \left (c +\frac {d}{x}\right ) x}{2 d}-\frac {\sinIntegral \left (\frac {d}{x}\right ) \cos \left (c \right )}{2}-\frac {\cosineIntegral \left (\frac {d}{x}\right ) \sin \left (c \right )}{2}\right )+2 a b e \left (-\frac {\sin \left (c +\frac {d}{x}\right ) x}{d}-\sinIntegral \left (\frac {d}{x}\right ) \sin \left (c \right )+\cosineIntegral \left (\frac {d}{x}\right ) \cos \left (c \right )\right )-\frac {b^{2} f \,x^{2}}{4 d}-\frac {b^{2} f d \left (-\frac {\cos \left (\frac {2 d}{x}+2 c \right ) x^{2}}{d^{2}}+\frac {2 \sin \left (\frac {2 d}{x}+2 c \right ) x}{d}+4 \sinIntegral \left (\frac {2 d}{x}\right ) \sin \left (2 c \right )-4 \cosineIntegral \left (\frac {2 d}{x}\right ) \cos \left (2 c \right )\right )}{4}-\frac {b^{2} e x}{2 d}-\frac {b^{2} e \left (-\frac {2 \cos \left (\frac {2 d}{x}+2 c \right ) x}{d}-4 \sinIntegral \left (\frac {2 d}{x}\right ) \cos \left (2 c \right )-4 \cosineIntegral \left (\frac {2 d}{x}\right ) \sin \left (2 c \right )\right )}{4}\right )\)	\(265\)
risch	\(a^{2} e x +\frac {a^{2} f \,x^{2}}{2}+a b d e \,{\mathrm e}^{-i c} \expIntegral \left (1, \frac {i d}{x}\right )+\frac {i \expIntegral \left (1, \frac {2 i d}{x}\right ) {\mathrm e}^{-2 i c} b^{2} d e}{2}+\frac {x \,b^{2} e}{2}+\frac {f \,b^{2} x^{2}}{4}+\frac {\expIntegral \left (1, \frac {2 i d}{x}\right ) {\mathrm e}^{-2 i c} b^{2} d^{2} f}{2}-\frac {i {\mathrm e}^{2 i c} \expIntegral \left (1, -\frac {2 i d}{x}\right ) b^{2} d e}{2}+\frac {{\mathrm e}^{2 i c} \expIntegral \left (1, -\frac {2 i d}{x}\right ) b^{2} d^{2} f}{2}+\frac {i a b \,d^{2} f \,{\mathrm e}^{i c} \expIntegral \left (1, -\frac {i d}{x}\right )}{2}+a b d e \,{\mathrm e}^{i c} \expIntegral \left (1, -\frac {i d}{x}\right )-\frac {i a b \,d^{2} f \,{\mathrm e}^{-i c} \expIntegral \left (1, \frac {i d}{x}\right )}{2}+a b d x f \cos \left (\frac {c x +d}{x}\right )+a b x \left (f x +2 e \right ) \sin \left (\frac {c x +d}{x}\right )-\frac {b^{2} x \left (f x +2 e \right ) \cos \left (\frac {2 c x +2 d}{x}\right )}{4}+\frac {d \,b^{2} x f \sin \left (\frac {2 c x +2 d}{x}\right )}{2}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*(a+b*sin(c+d/x))^2,x,method=_RETURNVERBOSE)

[Out]

-d*(-1/2*a^2*f/d*x^2-a^2*e/d*x+2*a*b*f*d*(-1/2*sin(c+d/x)/d^2*x^2-1/2*cos(c+d/x)/d*x-1/2*Si(d/x)*cos(c)-1/2*Ci
(d/x)*sin(c))+2*a*b*e*(-sin(c+d/x)/d*x-Si(d/x)*sin(c)+Ci(d/x)*cos(c))-1/4*b^2*f/d*x^2-1/4*b^2*f*d*(-cos(2*d/x+
2*c)/d^2*x^2+2*sin(2*d/x+2*c)/d*x+4*Si(2*d/x)*sin(2*c)-4*Ci(2*d/x)*cos(2*c))-1/2*b^2*e/d*x-1/4*b^2*e*(-2*cos(2
*d/x+2*c)/d*x-4*Si(2*d/x)*cos(2*c)-4*Ci(2*d/x)*sin(2*c)))

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Maxima [C] Result contains complex when optimal does not.
time = 0.43, size = 324, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, a^{2} f x^{2} + \frac {1}{2} \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \left (c\right ) + {\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \left (c\right )\right )} d^{2} + 2 \, d x \cos \left (\frac {c x + d}{x}\right ) + 2 \, x^{2} \sin \left (\frac {c x + d}{x}\right )\right )} a b f - \frac {1}{4} \, {\left (2 \, {\left ({\left ({\rm Ei}\left (\frac {2 i \, d}{x}\right ) + {\rm Ei}\left (-\frac {2 i \, d}{x}\right )\right )} \cos \left (2 \, c\right ) + {\left (i \, {\rm Ei}\left (\frac {2 i \, d}{x}\right ) - i \, {\rm Ei}\left (-\frac {2 i \, d}{x}\right )\right )} \sin \left (2 \, c\right )\right )} d^{2} + x^{2} \cos \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) - 2 \, d x \sin \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) - x^{2}\right )} b^{2} f - {\left ({\left ({\left ({\rm Ei}\left (\frac {i \, d}{x}\right ) + {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \cos \left (c\right ) - {\left (-i \, {\rm Ei}\left (\frac {i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {i \, d}{x}\right )\right )} \sin \left (c\right )\right )} d - 2 \, x \sin \left (\frac {c x + d}{x}\right )\right )} a b e - \frac {1}{2} \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {2 i \, d}{x}\right ) + i \, {\rm Ei}\left (-\frac {2 i \, d}{x}\right )\right )} \cos \left (2 \, c\right ) + {\left ({\rm Ei}\left (\frac {2 i \, d}{x}\right ) + {\rm Ei}\left (-\frac {2 i \, d}{x}\right )\right )} \sin \left (2 \, c\right )\right )} d + x \cos \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) - x\right )} b^{2} e + a^{2} x e \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*f*x^2 + 1/2*(((-I*Ei(I*d/x) + I*Ei(-I*d/x))*cos(c) + (Ei(I*d/x) + Ei(-I*d/x))*sin(c))*d^2 + 2*d*x*cos(
(c*x + d)/x) + 2*x^2*sin((c*x + d)/x))*a*b*f - 1/4*(2*((Ei(2*I*d/x) + Ei(-2*I*d/x))*cos(2*c) + (I*Ei(2*I*d/x)
- I*Ei(-2*I*d/x))*sin(2*c))*d^2 + x^2*cos(2*(c*x + d)/x) - 2*d*x*sin(2*(c*x + d)/x) - x^2)*b^2*f - (((Ei(I*d/x
) + Ei(-I*d/x))*cos(c) - (-I*Ei(I*d/x) + I*Ei(-I*d/x))*sin(c))*d - 2*x*sin((c*x + d)/x))*a*b*e - 1/2*(((-I*Ei(
2*I*d/x) + I*Ei(-2*I*d/x))*cos(2*c) + (Ei(2*I*d/x) + Ei(-2*I*d/x))*sin(2*c))*d + x*cos(2*(c*x + d)/x) - x)*b^2
*e + a^2*x*e

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Fricas [A]
time = 0.42, size = 309, normalized size = 1.22 \begin {gather*} a b d f x \cos \left (\frac {c x + d}{x}\right ) + \frac {1}{2} \, {\left (a^{2} + b^{2}\right )} f x^{2} - \frac {1}{2} \, {\left (b^{2} f x^{2} + 2 \, b^{2} x e\right )} \cos \left (\frac {c x + d}{x}\right )^{2} + {\left (a^{2} + b^{2}\right )} x e - \frac {1}{2} \, {\left (b^{2} d^{2} f \operatorname {Ci}\left (\frac {2 \, d}{x}\right ) + b^{2} d^{2} f \operatorname {Ci}\left (-\frac {2 \, d}{x}\right ) + 2 \, b^{2} d e \operatorname {Si}\left (\frac {2 \, d}{x}\right )\right )} \cos \left (2 \, c\right ) + {\left (a b d^{2} f \operatorname {Si}\left (\frac {d}{x}\right ) - a b d \operatorname {Ci}\left (\frac {d}{x}\right ) e - a b d \operatorname {Ci}\left (-\frac {d}{x}\right ) e\right )} \cos \left (c\right ) + \frac {1}{2} \, {\left (2 \, b^{2} d^{2} f \operatorname {Si}\left (\frac {2 \, d}{x}\right ) - b^{2} d \operatorname {Ci}\left (\frac {2 \, d}{x}\right ) e - b^{2} d \operatorname {Ci}\left (-\frac {2 \, d}{x}\right ) e\right )} \sin \left (2 \, c\right ) + \frac {1}{2} \, {\left (a b d^{2} f \operatorname {Ci}\left (\frac {d}{x}\right ) + a b d^{2} f \operatorname {Ci}\left (-\frac {d}{x}\right ) + 4 \, a b d e \operatorname {Si}\left (\frac {d}{x}\right )\right )} \sin \left (c\right ) + {\left (b^{2} d f x \cos \left (\frac {c x + d}{x}\right ) + a b f x^{2} + 2 \, a b x e\right )} \sin \left (\frac {c x + d}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b*d*f*x*cos((c*x + d)/x) + 1/2*(a^2 + b^2)*f*x^2 - 1/2*(b^2*f*x^2 + 2*b^2*x*e)*cos((c*x + d)/x)^2 + (a^2 + b
^2)*x*e - 1/2*(b^2*d^2*f*cos_integral(2*d/x) + b^2*d^2*f*cos_integral(-2*d/x) + 2*b^2*d*e*sin_integral(2*d/x))
*cos(2*c) + (a*b*d^2*f*sin_integral(d/x) - a*b*d*cos_integral(d/x)*e - a*b*d*cos_integral(-d/x)*e)*cos(c) + 1/
2*(2*b^2*d^2*f*sin_integral(2*d/x) - b^2*d*cos_integral(2*d/x)*e - b^2*d*cos_integral(-2*d/x)*e)*sin(2*c) + 1/
2*(a*b*d^2*f*cos_integral(d/x) + a*b*d^2*f*cos_integral(-d/x) + 4*a*b*d*e*sin_integral(d/x))*sin(c) + (b^2*d*f
*x*cos((c*x + d)/x) + a*b*f*x^2 + 2*a*b*x*e)*sin((c*x + d)/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 + \frac {d}{x} \right )}\right )^{2} \left (e + f x\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(a+b*sin(c+d/x))**2,x)

[Out]

Integral((a + b*sin(c + d/x))**2*(e + f*x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1145 vs. \(2 (257) = 514\).
time = 5.36, size = 1145, normalized size = 4.51 \begin {gather*} -\frac {4 \, b^{2} c^{2} d^{3} f \cos \left (2 \, c\right ) \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right ) - 4 \, a b c^{2} d^{3} f \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) \sin \left (c\right ) + 4 \, b^{2} c^{2} d^{3} f \sin \left (2 \, c\right ) \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right ) + 4 \, a b c^{2} d^{3} f \cos \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right ) - \frac {8 \, {\left (c x + d\right )} b^{2} c d^{3} f \cos \left (2 \, c\right ) \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x} + 8 \, a b c^{2} d^{2} \cos \left (c\right ) \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) e + 4 \, b^{2} c^{2} d^{2} \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right ) e \sin \left (2 \, c\right ) + \frac {8 \, {\left (c x + d\right )} a b c d^{3} f \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) \sin \left (c\right )}{x} - 4 \, b^{2} c^{2} d^{2} \cos \left (2 \, c\right ) e \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right ) - \frac {8 \, {\left (c x + d\right )} b^{2} c d^{3} f \sin \left (2 \, c\right ) \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x} - \frac {8 \, {\left (c x + d\right )} a b c d^{3} f \cos \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right )}{x} + 8 \, a b c^{2} d^{2} e \sin \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right ) + 4 \, a b c d^{3} f \cos \left (\frac {c x + d}{x}\right ) + \frac {4 \, {\left (c x + d\right )}^{2} b^{2} d^{3} f \cos \left (2 \, c\right ) \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x^{2}} - \frac {16 \, {\left (c x + d\right )} a b c d^{2} \cos \left (c\right ) \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) e}{x} - \frac {8 \, {\left (c x + d\right )} b^{2} c d^{2} \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right ) e \sin \left (2 \, c\right )}{x} - \frac {4 \, {\left (c x + d\right )}^{2} a b d^{3} f \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) \sin \left (c\right )}{x^{2}} + 2 \, b^{2} c d^{3} f \sin \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) + \frac {8 \, {\left (c x + d\right )} b^{2} c d^{2} \cos \left (2 \, c\right ) e \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x} + \frac {4 \, {\left (c x + d\right )}^{2} b^{2} d^{3} f \sin \left (2 \, c\right ) \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x^{2}} + \frac {4 \, {\left (c x + d\right )}^{2} a b d^{3} f \cos \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right )}{x^{2}} - \frac {16 \, {\left (c x + d\right )} a b c d^{2} e \sin \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right )}{x} + b^{2} d^{3} f \cos \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) - \frac {4 \, {\left (c x + d\right )} a b d^{3} f \cos \left (\frac {c x + d}{x}\right )}{x} - 2 \, b^{2} c d^{2} \cos \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) e + \frac {8 \, {\left (c x + d\right )}^{2} a b d^{2} \cos \left (c\right ) \operatorname {Ci}\left (-c + \frac {c x + d}{x}\right ) e}{x^{2}} + \frac {4 \, {\left (c x + d\right )}^{2} b^{2} d^{2} \operatorname {Ci}\left (-2 \, c + \frac {2 \, {\left (c x + d\right )}}{x}\right ) e \sin \left (2 \, c\right )}{x^{2}} - \frac {2 \, {\left (c x + d\right )} b^{2} d^{3} f \sin \left (\frac {2 \, {\left (c x + d\right )}}{x}\right )}{x} - 4 \, a b d^{3} f \sin \left (\frac {c x + d}{x}\right ) + 8 \, a b c d^{2} e \sin \left (\frac {c x + d}{x}\right ) - \frac {4 \, {\left (c x + d\right )}^{2} b^{2} d^{2} \cos \left (2 \, c\right ) e \operatorname {Si}\left (2 \, c - \frac {2 \, {\left (c x + d\right )}}{x}\right )}{x^{2}} + \frac {8 \, {\left (c x + d\right )}^{2} a b d^{2} e \sin \left (c\right ) \operatorname {Si}\left (c - \frac {c x + d}{x}\right )}{x^{2}} - 2 \, a^{2} d^{3} f - b^{2} d^{3} f + 4 \, a^{2} c d^{2} e + 2 \, b^{2} c d^{2} e + \frac {2 \, {\left (c x + d\right )} b^{2} d^{2} \cos \left (\frac {2 \, {\left (c x + d\right )}}{x}\right ) e}{x} - \frac {8 \, {\left (c x + d\right )} a b d^{2} e \sin \left (\frac {c x + d}{x}\right )}{x} - \frac {4 \, {\left (c x + d\right )} a^{2} d^{2} e}{x} - \frac {2 \, {\left (c x + d\right )} b^{2} d^{2} e}{x}}{4 \, {\left (c^{2} - \frac {2 \, {\left (c x + d\right )} c}{x} + \frac {{\left (c x + d\right )}^{2}}{x^{2}}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*b^2*c^2*d^3*f*cos(2*c)*cos_integral(-2*c + 2*(c*x + d)/x) - 4*a*b*c^2*d^3*f*cos_integral(-c + (c*x + d
)/x)*sin(c) + 4*b^2*c^2*d^3*f*sin(2*c)*sin_integral(2*c - 2*(c*x + d)/x) + 4*a*b*c^2*d^3*f*cos(c)*sin_integral
(c - (c*x + d)/x) - 8*(c*x + d)*b^2*c*d^3*f*cos(2*c)*cos_integral(-2*c + 2*(c*x + d)/x)/x + 8*a*b*c^2*d^2*cos(
c)*cos_integral(-c + (c*x + d)/x)*e + 4*b^2*c^2*d^2*cos_integral(-2*c + 2*(c*x + d)/x)*e*sin(2*c) + 8*(c*x + d
)*a*b*c*d^3*f*cos_integral(-c + (c*x + d)/x)*sin(c)/x - 4*b^2*c^2*d^2*cos(2*c)*e*sin_integral(2*c - 2*(c*x + d
)/x) - 8*(c*x + d)*b^2*c*d^3*f*sin(2*c)*sin_integral(2*c - 2*(c*x + d)/x)/x - 8*(c*x + d)*a*b*c*d^3*f*cos(c)*s
in_integral(c - (c*x + d)/x)/x + 8*a*b*c^2*d^2*e*sin(c)*sin_integral(c - (c*x + d)/x) + 4*a*b*c*d^3*f*cos((c*x
+ d)/x) + 4*(c*x + d)^2*b^2*d^3*f*cos(2*c)*cos_integral(-2*c + 2*(c*x + d)/x)/x^2 - 16*(c*x + d)*a*b*c*d^2*co
s(c)*cos_integral(-c + (c*x + d)/x)*e/x - 8*(c*x + d)*b^2*c*d^2*cos_integral(-2*c + 2*(c*x + d)/x)*e*sin(2*c)/
x - 4*(c*x + d)^2*a*b*d^3*f*cos_integral(-c + (c*x + d)/x)*sin(c)/x^2 + 2*b^2*c*d^3*f*sin(2*(c*x + d)/x) + 8*(
c*x + d)*b^2*c*d^2*cos(2*c)*e*sin_integral(2*c - 2*(c*x + d)/x)/x + 4*(c*x + d)^2*b^2*d^3*f*sin(2*c)*sin_integ
ral(2*c - 2*(c*x + d)/x)/x^2 + 4*(c*x + d)^2*a*b*d^3*f*cos(c)*sin_integral(c - (c*x + d)/x)/x^2 - 16*(c*x + d)
*a*b*c*d^2*e*sin(c)*sin_integral(c - (c*x + d)/x)/x + b^2*d^3*f*cos(2*(c*x + d)/x) - 4*(c*x + d)*a*b*d^3*f*cos
((c*x + d)/x)/x - 2*b^2*c*d^2*cos(2*(c*x + d)/x)*e + 8*(c*x + d)^2*a*b*d^2*cos(c)*cos_integral(-c + (c*x + d)/
x)*e/x^2 + 4*(c*x + d)^2*b^2*d^2*cos_integral(-2*c + 2*(c*x + d)/x)*e*sin(2*c)/x^2 - 2*(c*x + d)*b^2*d^3*f*sin
(2*(c*x + d)/x)/x - 4*a*b*d^3*f*sin((c*x + d)/x) + 8*a*b*c*d^2*e*sin((c*x + d)/x) - 4*(c*x + d)^2*b^2*d^2*cos(
2*c)*e*sin_integral(2*c - 2*(c*x + d)/x)/x^2 + 8*(c*x + d)^2*a*b*d^2*e*sin(c)*sin_integral(c - (c*x + d)/x)/x^
2 - 2*a^2*d^3*f - b^2*d^3*f + 4*a^2*c*d^2*e + 2*b^2*c*d^2*e + 2*(c*x + d)*b^2*d^2*cos(2*(c*x + d)/x)*e/x - 8*(
c*x + d)*a*b*d^2*e*sin((c*x + d)/x)/x - 4*(c*x + d)*a^2*d^2*e/x - 2*(c*x + d)*b^2*d^2*e/x)/((c^2 - 2*(c*x + d)
*c/x + (c*x + d)^2/x^2)*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (e+f\,x\right )\,{\left (a+b\,\sin \left (c+\frac {d}{x}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)*(a + b*sin(c + d/x))^2,x)

[Out]

int((e + f*x)*(a + b*sin(c + d/x))^2, x)

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