3.298 \(\int \frac {(a+b \sin (c+\frac {d}{x}))^2}{(e+f x)^3} \, dx\)

Optimal. Leaf size=470 \[ \frac {a^2}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a^2 f}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {a b d^2 f \sin \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {2 a b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {a b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}+\frac {2 a b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (\frac {e}{x}+f\right )}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )^2}+\frac {b^2 d^2 f \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Ci}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {b^2 d \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Ci}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b^2 d^2 f \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {b^2 d \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b^2 d f \sin \left (c+\frac {d}{x}\right ) \cos \left (c+\frac {d}{x}\right )}{e^3 \left (\frac {e}{x}+f\right )}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (\frac {e}{x}+f\right )^2} \]

[Out]

-1/2*a^2*f/e^2/(f+e/x)^2+a^2/e^2/(f+e/x)+b^2*d^2*f*Ci(2*d*(f/e+1/x))*cos(2*c-2*d*f/e)/e^4-2*a*b*d*Ci(d*(f/e+1/
x))*cos(c-d*f/e)/e^3-a*b*d*f*cos(c+d/x)/e^3/(f+e/x)-a*b*d^2*f*cos(c-d*f/e)*Si(d*(f/e+1/x))/e^4-b^2*d*cos(2*c-2
*d*f/e)*Si(2*d*(f/e+1/x))/e^3-b^2*d*Ci(2*d*(f/e+1/x))*sin(2*c-2*d*f/e)/e^3-b^2*d^2*f*Si(2*d*(f/e+1/x))*sin(2*c
-2*d*f/e)/e^4-a*b*d^2*f*Ci(d*(f/e+1/x))*sin(c-d*f/e)/e^4+2*a*b*d*Si(d*(f/e+1/x))*sin(c-d*f/e)/e^3-a*b*f*sin(c+
d/x)/e^2/(f+e/x)^2+2*a*b*sin(c+d/x)/e^2/(f+e/x)-b^2*d*f*cos(c+d/x)*sin(c+d/x)/e^3/(f+e/x)-1/2*b^2*f*sin(c+d/x)
^2/e^2/(f+e/x)^2+b^2*sin(c+d/x)^2/e^2/(f+e/x)

________________________________________________________________________________________

Rubi [A]  time = 0.96, antiderivative size = 470, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3431, 3317, 3297, 3303, 3299, 3302, 3314, 31, 3312, 3313, 12} \[ \frac {a^2}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a^2 f}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {a b d^2 f \sin \left (c-\frac {d f}{e}\right ) \text {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {2 a b d \cos \left (c-\frac {d f}{e}\right ) \text {CosIntegral}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {a b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}+\frac {2 a b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )^2}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (\frac {e}{x}+f\right )}+\frac {b^2 d^2 f \cos \left (2 c-\frac {2 d f}{e}\right ) \text {CosIntegral}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {b^2 d \sin \left (2 c-\frac {2 d f}{e}\right ) \text {CosIntegral}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}-\frac {b^2 d^2 f \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^4}-\frac {b^2 d \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )}{e^3}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (\frac {e}{x}+f\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (\frac {e}{x}+f\right )^2}-\frac {b^2 d f \sin \left (c+\frac {d}{x}\right ) \cos \left (c+\frac {d}{x}\right )}{e^3 \left (\frac {e}{x}+f\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d/x])^2/(e + f*x)^3,x]

[Out]

-(a^2*f)/(2*e^2*(f + e/x)^2) + a^2/(e^2*(f + e/x)) - (a*b*d*f*Cos[c + d/x])/(e^3*(f + e/x)) - (2*a*b*d*Cos[c -
(d*f)/e]*CosIntegral[d*(f/e + x^(-1))])/e^3 + (b^2*d^2*f*Cos[2*c - (2*d*f)/e]*CosIntegral[2*d*(f/e + x^(-1))]
)/e^4 - (b^2*d*CosIntegral[2*d*(f/e + x^(-1))]*Sin[2*c - (2*d*f)/e])/e^3 - (a*b*d^2*f*CosIntegral[d*(f/e + x^(
-1))]*Sin[c - (d*f)/e])/e^4 - (a*b*f*Sin[c + d/x])/(e^2*(f + e/x)^2) + (2*a*b*Sin[c + d/x])/(e^2*(f + e/x)) -
(b^2*d*f*Cos[c + d/x]*Sin[c + d/x])/(e^3*(f + e/x)) - (b^2*f*Sin[c + d/x]^2)/(2*e^2*(f + e/x)^2) + (b^2*Sin[c
+ d/x]^2)/(e^2*(f + e/x)) - (a*b*d^2*f*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))])/e^4 + (2*a*b*d*Sin[c -
(d*f)/e]*SinIntegral[d*(f/e + x^(-1))])/e^3 - (b^2*d*Cos[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))])/e^3
- (b^2*d^2*f*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))])/e^4

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3297

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 3299

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 3302

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 3303

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 3312

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 3313

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 3314

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 3317

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 3431

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 \frac {\left (a+b \sin \left (c+\frac {d}{x}\right )\right )^2}{(e+f x)^3} \, dx &=-\operatorname {Subst}\left (\int \left (-\frac {f (a+b \sin (c+d x))^2}{e (f+e x)^3}+\frac {(a+b \sin (c+d x))^2}{e (f+e x)^2}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {f \operatorname {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{(f+e x)^3} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^2}{(f+e x)^2}+\frac {2 a b \sin (c+d x)}{(f+e x)^2}+\frac {b^2 \sin ^2(c+d x)}{(f+e x)^2}\right ) \, dx,x,\frac {1}{x}\right )}{e}+\frac {f \operatorname {Subst}\left (\int \left (\frac {a^2}{(f+e x)^3}+\frac {2 a b \sin (c+d x)}{(f+e x)^3}+\frac {b^2 \sin ^2(c+d x)}{(f+e x)^3}\right ) \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\sin ^2(c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {(2 a b f) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{(f+e x)^3} \, dx,x,\frac {1}{x}\right )}{e}+\frac {\left (b^2 f\right ) \operatorname {Subst}\left (\int \frac {\sin ^2(c+d x)}{(f+e x)^3} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )^2}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b^2 d f \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {(2 a b d) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \frac {\sin (2 c+2 d x)}{2 (f+e x)} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {\left (b^2 d^2 f\right ) \operatorname {Subst}\left (\int \frac {1}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (2 b^2 d^2 f\right ) \operatorname {Subst}\left (\int \frac {\sin ^2(c+d x)}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}+\frac {(a b d f) \operatorname {Subst}\left (\int \frac {\cos (c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e^2}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}+\frac {b^2 d^2 f \log \left (f+\frac {e}{x}\right )}{e^4}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )^2}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b^2 d f \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {\left (b^2 d\right ) \operatorname {Subst}\left (\int \frac {\sin (2 c+2 d x)}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {\left (a b d^2 f\right ) \operatorname {Subst}\left (\int \frac {\sin (c+d x)}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (2 b^2 d^2 f\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2 (f+e x)}-\frac {\cos (2 c+2 d x)}{2 (f+e x)}\right ) \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (2 a b d \cos \left (c-\frac {d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {d f}{e}+d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}+\frac {\left (2 a b d \sin \left (c-\frac {d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {d f}{e}+d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {2 a b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )^2}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b^2 d f \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}+\frac {2 a b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}+\frac {\left (b^2 d^2 f\right ) \operatorname {Subst}\left (\int \frac {\cos (2 c+2 d x)}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (b^2 d \cos \left (2 c-\frac {2 d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 d f}{e}+2 d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {\left (a b d^2 f \cos \left (c-\frac {d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {d f}{e}+d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (b^2 d \sin \left (2 c-\frac {2 d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 d f}{e}+2 d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^2}-\frac {\left (a b d^2 f \sin \left (c-\frac {d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {d f}{e}+d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {2 a b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}-\frac {b^2 d \text {Ci}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right ) \sin \left (2 c-\frac {2 d f}{e}\right )}{e^3}-\frac {a b d^2 f \text {Ci}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right ) \sin \left (c-\frac {d f}{e}\right )}{e^4}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )^2}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b^2 d f \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^4}+\frac {2 a b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}-\frac {b^2 d \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}+\frac {\left (b^2 d^2 f \cos \left (2 c-\frac {2 d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 d f}{e}+2 d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}-\frac {\left (b^2 d^2 f \sin \left (2 c-\frac {2 d f}{e}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 d f}{e}+2 d x\right )}{f+e x} \, dx,x,\frac {1}{x}\right )}{e^3}\\ &=-\frac {a^2 f}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {a^2}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d f \cos \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {2 a b d \cos \left (c-\frac {d f}{e}\right ) \text {Ci}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}+\frac {b^2 d^2 f \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Ci}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right )}{e^4}-\frac {b^2 d \text {Ci}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right ) \sin \left (2 c-\frac {2 d f}{e}\right )}{e^3}-\frac {a b d^2 f \text {Ci}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right ) \sin \left (c-\frac {d f}{e}\right )}{e^4}-\frac {a b f \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )^2}+\frac {2 a b \sin \left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {b^2 d f \cos \left (c+\frac {d}{x}\right ) \sin \left (c+\frac {d}{x}\right )}{e^3 \left (f+\frac {e}{x}\right )}-\frac {b^2 f \sin ^2\left (c+\frac {d}{x}\right )}{2 e^2 \left (f+\frac {e}{x}\right )^2}+\frac {b^2 \sin ^2\left (c+\frac {d}{x}\right )}{e^2 \left (f+\frac {e}{x}\right )}-\frac {a b d^2 f \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^4}+\frac {2 a b d \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (\frac {d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}-\frac {b^2 d \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right )}{e^3}-\frac {b^2 d^2 f \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (\frac {2 d \left (f+\frac {e}{x}\right )}{e}\right )}{e^4}\\ \end {align*}

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Mathematica [A]  time = 3.49, size = 740, normalized size = 1.57 \[ -\frac {2 a^2 e^4+4 a b d f (e+f x)^2 \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (d f \sin \left (c-\frac {d f}{e}\right )+2 e \cos \left (c-\frac {d f}{e}\right )\right )+4 a b d^2 e^2 f^2 \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+4 a b d^2 f^4 x^2 \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+8 a b d^2 e f^3 x \cos \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )-8 a b d e^3 f \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )-8 a b e^3 f x \sin \left (c+\frac {d}{x}\right )-16 a b d e^2 f^2 x \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )-4 a b e^2 f^2 x^2 \sin \left (c+\frac {d}{x}\right )+4 a b d e^2 f^2 x \cos \left (c+\frac {d}{x}\right )-8 a b d e f^3 x^2 \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+4 a b d e f^3 x^2 \cos \left (c+\frac {d}{x}\right )-4 b^2 d f (e+f x)^2 \text {Ci}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (d f \cos \left (2 c-\frac {2 d f}{e}\right )-e \sin \left (2 c-\frac {2 d f}{e}\right )\right )+4 b^2 d^2 e^2 f^2 \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+4 b^2 d^2 f^4 x^2 \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+8 b^2 d^2 e f^3 x \sin \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+4 b^2 d e^3 f \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+2 b^2 e^3 f x \cos \left (2 \left (c+\frac {d}{x}\right )\right )+8 b^2 d e^2 f^2 x \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+b^2 e^2 f^2 x^2 \cos \left (2 \left (c+\frac {d}{x}\right )\right )+2 b^2 d e^2 f^2 x \sin \left (2 \left (c+\frac {d}{x}\right )\right )+4 b^2 d e f^3 x^2 \cos \left (2 c-\frac {2 d f}{e}\right ) \text {Si}\left (2 d \left (\frac {f}{e}+\frac {1}{x}\right )\right )+2 b^2 d e f^3 x^2 \sin \left (2 \left (c+\frac {d}{x}\right )\right )+b^2 e^4}{4 e^4 f (e+f x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d/x])^2/(e + f*x)^3,x]

[Out]

-1/4*(2*a^2*e^4 + b^2*e^4 + 4*a*b*d*e^2*f^2*x*Cos[c + d/x] + 4*a*b*d*e*f^3*x^2*Cos[c + d/x] + 2*b^2*e^3*f*x*Co
s[2*(c + d/x)] + b^2*e^2*f^2*x^2*Cos[2*(c + d/x)] - 4*b^2*d*f*(e + f*x)^2*CosIntegral[2*d*(f/e + x^(-1))]*(d*f
*Cos[2*c - (2*d*f)/e] - e*Sin[2*c - (2*d*f)/e]) + 4*a*b*d*f*(e + f*x)^2*CosIntegral[d*(f/e + x^(-1))]*(2*e*Cos
[c - (d*f)/e] + d*f*Sin[c - (d*f)/e]) - 8*a*b*e^3*f*x*Sin[c + d/x] - 4*a*b*e^2*f^2*x^2*Sin[c + d/x] + 2*b^2*d*
e^2*f^2*x*Sin[2*(c + d/x)] + 2*b^2*d*e*f^3*x^2*Sin[2*(c + d/x)] + 4*a*b*d^2*e^2*f^2*Cos[c - (d*f)/e]*SinIntegr
al[d*(f/e + x^(-1))] + 8*a*b*d^2*e*f^3*x*Cos[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))] + 4*a*b*d^2*f^4*x^2*Co
s[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))] - 8*a*b*d*e^3*f*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))] -
16*a*b*d*e^2*f^2*x*Sin[c - (d*f)/e]*SinIntegral[d*(f/e + x^(-1))] - 8*a*b*d*e*f^3*x^2*Sin[c - (d*f)/e]*SinInte
gral[d*(f/e + x^(-1))] + 4*b^2*d*e^3*f*Cos[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))] + 8*b^2*d*e^2*f^2*
x*Cos[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))] + 4*b^2*d*e*f^3*x^2*Cos[2*c - (2*d*f)/e]*SinIntegral[2*
d*(f/e + x^(-1))] + 4*b^2*d^2*e^2*f^2*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))] + 8*b^2*d^2*e*f^3*x
*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d*(f/e + x^(-1))] + 4*b^2*d^2*f^4*x^2*Sin[2*c - (2*d*f)/e]*SinIntegral[2*d
*(f/e + x^(-1))])/(e^4*f*(e + f*x)^2)

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fricas [A]  time = 0.88, size = 926, normalized size = 1.97 \[ \frac {b^{2} e^{2} f^{2} x^{2} + 2 \, b^{2} e^{3} f x - {\left (2 \, a^{2} + b^{2}\right )} e^{4} - 2 \, {\left (b^{2} e^{2} f^{2} x^{2} + 2 \, b^{2} e^{3} f x\right )} \cos \left (\frac {c x + d}{x}\right )^{2} - 4 \, {\left ({\left (a b d e f^{3} x^{2} + 2 \, a b d e^{2} f^{2} x + a b d e^{3} f\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (a b d e f^{3} x^{2} + 2 \, a b d e^{2} f^{2} x + a b d e^{3} f\right )} \operatorname {Ci}\left (-\frac {d f x + d e}{e x}\right ) + {\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} e f^{3} x + a b d^{2} e^{2} f^{2}\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \cos \left (-\frac {c e - d f}{e}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} e f^{3} x + b^{2} d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{e x}\right ) + {\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} e f^{3} x + b^{2} d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + d e\right )}}{e x}\right ) - 2 \, {\left (b^{2} d e f^{3} x^{2} + 2 \, b^{2} d e^{2} f^{2} x + b^{2} d e^{3} f\right )} \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{e x}\right )\right )} \cos \left (-\frac {2 \, {\left (c e - d f\right )}}{e}\right ) - 4 \, {\left (a b d e f^{3} x^{2} + a b d e^{2} f^{2} x\right )} \cos \left (\frac {c x + d}{x}\right ) + 2 \, {\left ({\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} e f^{3} x + a b d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (\frac {d f x + d e}{e x}\right ) + {\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} e f^{3} x + a b d^{2} e^{2} f^{2}\right )} \operatorname {Ci}\left (-\frac {d f x + d e}{e x}\right ) - 4 \, {\left (a b d e f^{3} x^{2} + 2 \, a b d e^{2} f^{2} x + a b d e^{3} f\right )} \operatorname {Si}\left (\frac {d f x + d e}{e x}\right )\right )} \sin \left (-\frac {c e - d f}{e}\right ) + 2 \, {\left ({\left (b^{2} d e f^{3} x^{2} + 2 \, b^{2} d e^{2} f^{2} x + b^{2} d e^{3} f\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )}}{e x}\right ) + {\left (b^{2} d e f^{3} x^{2} + 2 \, b^{2} d e^{2} f^{2} x + b^{2} d e^{3} f\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + d e\right )}}{e x}\right ) + 2 \, {\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} e f^{3} x + b^{2} d^{2} e^{2} f^{2}\right )} \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )}}{e x}\right )\right )} \sin \left (-\frac {2 \, {\left (c e - d f\right )}}{e}\right ) + 4 \, {\left (a b e^{2} f^{2} x^{2} + 2 \, a b e^{3} f x - {\left (b^{2} d e f^{3} x^{2} + b^{2} d e^{2} f^{2} x\right )} \cos \left (\frac {c x + d}{x}\right )\right )} \sin \left (\frac {c x + d}{x}\right )}{4 \, {\left (e^{4} f^{3} x^{2} + 2 \, e^{5} f^{2} x + e^{6} f\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d/x))^2/(f*x+e)^3,x, algorithm="fricas")

[Out]

1/4*(b^2*e^2*f^2*x^2 + 2*b^2*e^3*f*x - (2*a^2 + b^2)*e^4 - 2*(b^2*e^2*f^2*x^2 + 2*b^2*e^3*f*x)*cos((c*x + d)/x
)^2 - 4*((a*b*d*e*f^3*x^2 + 2*a*b*d*e^2*f^2*x + a*b*d*e^3*f)*cos_integral((d*f*x + d*e)/(e*x)) + (a*b*d*e*f^3*
x^2 + 2*a*b*d*e^2*f^2*x + a*b*d*e^3*f)*cos_integral(-(d*f*x + d*e)/(e*x)) + (a*b*d^2*f^4*x^2 + 2*a*b*d^2*e*f^3
*x + a*b*d^2*e^2*f^2)*sin_integral((d*f*x + d*e)/(e*x)))*cos(-(c*e - d*f)/e) + 2*((b^2*d^2*f^4*x^2 + 2*b^2*d^2
*e*f^3*x + b^2*d^2*e^2*f^2)*cos_integral(2*(d*f*x + d*e)/(e*x)) + (b^2*d^2*f^4*x^2 + 2*b^2*d^2*e*f^3*x + b^2*d
^2*e^2*f^2)*cos_integral(-2*(d*f*x + d*e)/(e*x)) - 2*(b^2*d*e*f^3*x^2 + 2*b^2*d*e^2*f^2*x + b^2*d*e^3*f)*sin_i
ntegral(2*(d*f*x + d*e)/(e*x)))*cos(-2*(c*e - d*f)/e) - 4*(a*b*d*e*f^3*x^2 + a*b*d*e^2*f^2*x)*cos((c*x + d)/x)
+ 2*((a*b*d^2*f^4*x^2 + 2*a*b*d^2*e*f^3*x + a*b*d^2*e^2*f^2)*cos_integral((d*f*x + d*e)/(e*x)) + (a*b*d^2*f^4
*x^2 + 2*a*b*d^2*e*f^3*x + a*b*d^2*e^2*f^2)*cos_integral(-(d*f*x + d*e)/(e*x)) - 4*(a*b*d*e*f^3*x^2 + 2*a*b*d*
e^2*f^2*x + a*b*d*e^3*f)*sin_integral((d*f*x + d*e)/(e*x)))*sin(-(c*e - d*f)/e) + 2*((b^2*d*e*f^3*x^2 + 2*b^2*
d*e^2*f^2*x + b^2*d*e^3*f)*cos_integral(2*(d*f*x + d*e)/(e*x)) + (b^2*d*e*f^3*x^2 + 2*b^2*d*e^2*f^2*x + b^2*d*
e^3*f)*cos_integral(-2*(d*f*x + d*e)/(e*x)) + 2*(b^2*d^2*f^4*x^2 + 2*b^2*d^2*e*f^3*x + b^2*d^2*e^2*f^2)*sin_in
tegral(2*(d*f*x + d*e)/(e*x)))*sin(-2*(c*e - d*f)/e) + 4*(a*b*e^2*f^2*x^2 + 2*a*b*e^3*f*x - (b^2*d*e*f^3*x^2 +
b^2*d*e^2*f^2*x)*cos((c*x + d)/x))*sin((c*x + d)/x))/(e^4*f^3*x^2 + 2*e^5*f^2*x + e^6*f)

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giac [B]  time = 0.62, size = 3062, normalized size = 6.51 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d/x))^2/(f*x+e)^3,x, algorithm="giac")

[Out]

1/4*(4*b^2*d^5*f^3*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1)) - 8*b^2*c*d^4
*f^2*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e - 4*a*b*d^5*f^3*cos_integ
ral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*sin(-(d*f - c*e)*e^(-1)) + 8*a*b*c*d^4*f^2*cos_integral((d*f - c*e + (
c*x + d)*e/x)*e^(-1))*e*sin(-(d*f - c*e)*e^(-1)) + 4*a*b*d^5*f^3*cos(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f -
c*e + (c*x + d)*e/x)*e^(-1)) - 8*a*b*c*d^4*f^2*cos(-(d*f - c*e)*e^(-1))*e*sin_integral(-(d*f - c*e + (c*x + d
)*e/x)*e^(-1)) + 4*b^2*d^5*f^3*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))
- 8*b^2*c*d^4*f^2*e*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1)) + 4*b^2*c^2
*d^3*f*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2 + 8*(c*x + d)*b^2*d^4
*f^2*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e/x - 8*a*b*d^4*f^2*cos(-(d
*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e - 4*a*b*c^2*d^3*f*cos_integral((d*f - c*e
+ (c*x + d)*e/x)*e^(-1))*e^2*sin(-(d*f - c*e)*e^(-1)) - 8*(c*x + d)*a*b*d^4*f^2*cos_integral((d*f - c*e + (c*
x + d)*e/x)*e^(-1))*e*sin(-(d*f - c*e)*e^(-1))/x - 4*b^2*d^4*f^2*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^
(-1))*e*sin(-2*(d*f - c*e)*e^(-1)) + 4*a*b*c^2*d^3*f*cos(-(d*f - c*e)*e^(-1))*e^2*sin_integral(-(d*f - c*e + (
c*x + d)*e/x)*e^(-1)) + 8*(c*x + d)*a*b*d^4*f^2*cos(-(d*f - c*e)*e^(-1))*e*sin_integral(-(d*f - c*e + (c*x + d
)*e/x)*e^(-1))/x - 8*a*b*d^4*f^2*e*sin(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f - c*e + (c*x + d)*e/x)*e^(-1))
+ 4*b^2*d^4*f^2*cos(-2*(d*f - c*e)*e^(-1))*e*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1)) + 4*b^2*c^2*d
^3*f*e^2*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1)) + 8*(c*x + d)*b^2*d^4*
f^2*e*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x - 8*(c*x + d)*b^2*c*d^3
*f*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2/x + 16*a*b*c*d^3*f*cos(-(
d*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2 - 4*a*b*d^4*f^2*cos((c*x + d)/x)*e + 8
*(c*x + d)*a*b*c*d^3*f*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2*sin(-(d*f - c*e)*e^(-1))/x + 8*b^2
*c*d^3*f*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2*sin(-2*(d*f - c*e)*e^(-1)) - 2*b^2*d^4*f^2*e*s
in(2*(c*x + d)/x) - 8*(c*x + d)*a*b*c*d^3*f*cos(-(d*f - c*e)*e^(-1))*e^2*sin_integral(-(d*f - c*e + (c*x + d)*
e/x)*e^(-1))/x + 16*a*b*c*d^3*f*e^2*sin(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f - c*e + (c*x + d)*e/x)*e^(-1))
- 8*b^2*c*d^3*f*cos(-2*(d*f - c*e)*e^(-1))*e^2*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1)) - 8*(c*x +
d)*b^2*c*d^3*f*e^2*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x - 8*a*b*c
^2*d^2*cos(-(d*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3 + 4*a*b*c*d^3*f*cos((c*x
+ d)/x)*e^2 + 4*(c*x + d)^2*b^2*d^3*f*cos(-2*(d*f - c*e)*e^(-1))*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^
(-1))*e^2/x^2 - 16*(c*x + d)*a*b*d^3*f*cos(-(d*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1
))*e^2/x - 4*(c*x + d)^2*a*b*d^3*f*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2*sin(-(d*f - c*e)*e^(-1
))/x^2 - 4*b^2*c^2*d^2*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3*sin(-2*(d*f - c*e)*e^(-1)) - 8*(
c*x + d)*b^2*d^3*f*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^2*sin(-2*(d*f - c*e)*e^(-1))/x + 2*b^2
*c*d^3*f*e^2*sin(2*(c*x + d)/x) + 4*(c*x + d)^2*a*b*d^3*f*cos(-(d*f - c*e)*e^(-1))*e^2*sin_integral(-(d*f - c*
e + (c*x + d)*e/x)*e^(-1))/x^2 - 8*a*b*c^2*d^2*e^3*sin(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f - c*e + (c*x +
d)*e/x)*e^(-1)) - 16*(c*x + d)*a*b*d^3*f*e^2*sin(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f - c*e + (c*x + d)*e/x
)*e^(-1))/x + 4*b^2*c^2*d^2*cos(-2*(d*f - c*e)*e^(-1))*e^3*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))
+ 8*(c*x + d)*b^2*d^3*f*cos(-2*(d*f - c*e)*e^(-1))*e^2*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x
+ 4*(c*x + d)^2*b^2*d^3*f*e^2*sin(-2*(d*f - c*e)*e^(-1))*sin_integral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x
^2 + 16*(c*x + d)*a*b*c*d^2*cos(-(d*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3/x -
b^2*d^3*f*cos(2*(c*x + d)/x)*e^2 - 4*(c*x + d)*a*b*d^3*f*cos((c*x + d)/x)*e^2/x + 8*(c*x + d)*b^2*c*d^2*cos_in
tegral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3*sin(-2*(d*f - c*e)*e^(-1))/x - 2*(c*x + d)*b^2*d^3*f*e^2*sin(
2*(c*x + d)/x)/x + 4*a*b*d^3*f*e^2*sin((c*x + d)/x) + 16*(c*x + d)*a*b*c*d^2*e^3*sin(-(d*f - c*e)*e^(-1))*sin_
integral(-(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x - 8*(c*x + d)*b^2*c*d^2*cos(-2*(d*f - c*e)*e^(-1))*e^3*sin_int
egral(-2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))/x + 2*b^2*c*d^2*cos(2*(c*x + d)/x)*e^3 - 8*(c*x + d)^2*a*b*d^2*co
s(-(d*f - c*e)*e^(-1))*cos_integral((d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3/x^2 + 2*a^2*d^3*f*e^2 + b^2*d^3*f*
e^2 - 4*(c*x + d)^2*b^2*d^2*cos_integral(2*(d*f - c*e + (c*x + d)*e/x)*e^(-1))*e^3*sin(-2*(d*f - c*e)*e^(-1))/
x^2 - 8*a*b*c*d^2*e^3*sin((c*x + d)/x) - 8*(c*x + d)^2*a*b*d^2*e^3*sin(-(d*f - c*e)*e^(-1))*sin_integral(-(d*f
- c*e + (c*x + d)*e/x)*e^(-1))/x^2 + 4*(c*x + d)^2*b^2*d^2*cos(-2*(d*f - c*e)*e^(-1))*e^3*sin_integral(-2*(d*
f - c*e + (c*x + d)*e/x)*e^(-1))/x^2 - 4*a^2*c*d^2*e^3 - 2*b^2*c*d^2*e^3 - 2*(c*x + d)*b^2*d^2*cos(2*(c*x + d)
/x)*e^3/x + 8*(c*x + d)*a*b*d^2*e^3*sin((c*x + d)/x)/x + 4*(c*x + d)*a^2*d^2*e^3/x + 2*(c*x + d)*b^2*d^2*e^3/x
)/((d^2*f^2*e^4 - 2*c*d*f*e^5 + c^2*e^6 + 2*(c*x + d)*d*f*e^5/x - 2*(c*x + d)*c*e^6/x + (c*x + d)^2*e^6/x^2)*d
)

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maple [B]  time = 0.10, size = 1124, normalized size = 2.39 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(c+d/x))^2/(f*x+e)^3,x)

[Out]

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

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d/x))^2/(f*x+e)^3,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (c+\frac {d}{x}\right )\right )}^2}{{\left (e+f\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d/x))^2/(e + f*x)^3,x)

[Out]

int((a + b*sin(c + d/x))^2/(e + f*x)^3, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(c+d/x))**2/(f*x+e)**3,x)

[Out]

Timed out

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