3.3.98 \(\int \frac {(a+b \sin (c+\frac {d}{x}))^2}{(e+f x)^3} \, dx\) [298]

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

[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 = 0.64, 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 = {3512, 3398, 3378, 3384, 3380, 3383, 3395, 31, 3393, 3394, 12} \begin {gather*} \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 {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 {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 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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(a^2*f)/(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 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 \frac {\left (a+b \sin \left (c+\frac {d}{x}\right )\right )^2}{(e+f x)^3} \, dx &=-\text {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 {\text {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e}+\frac {f \text {Subst}\left (\int \frac {(a+b \sin (c+d x))^2}{(f+e x)^3} \, dx,x,\frac {1}{x}\right )}{e}\\ &=-\frac {\text {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 \text {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) \text {Subst}\left (\int \frac {\sin (c+d x)}{(f+e x)^2} \, dx,x,\frac {1}{x}\right )}{e}-\frac {b^2 \text {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) \text {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 ) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 2.22, size = 740, normalized size = 1.57 \begin {gather*} -\frac {2 a^2 e^4+b^2 e^4+4 a b d e^2 f^2 x \cos \left (c+\frac {d}{x}\right )+4 a b d e f^3 x^2 \cos \left (c+\frac {d}{x}\right )+2 b^2 e^3 f x \cos \left (2 \left (c+\frac {d}{x}\right )\right )+b^2 e^2 f^2 x^2 \cos \left (2 \left (c+\frac {d}{x}\right )\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 a b d f (e+f x)^2 \text {Ci}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\right ) \left (2 e \cos \left (c-\frac {d f}{e}\right )+d f \sin \left (c-\frac {d f}{e}\right )\right )-8 a b e^3 f x \sin \left (c+\frac {d}{x}\right )-4 a b e^2 f^2 x^2 \sin \left (c+\frac {d}{x}\right )+2 b^2 d e^2 f^2 x \sin \left (2 \left (c+\frac {d}{x}\right )\right )+2 b^2 d e f^3 x^2 \sin \left (2 \left (c+\frac {d}{x}\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 )+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 )+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 e^3 f \sin \left (c-\frac {d f}{e}\right ) \text {Si}\left (d \left (\frac {f}{e}+\frac {1}{x}\right )\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 )-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 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 )+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 )+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 )+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 )+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^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 )}{4 e^4 f (e+f x)^2} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1138\) vs. \(2(466)=932\).
time = 0.20, size = 1139, normalized size = 2.42

method	result	size
risch	\(\frac {i a b \,d^{2} {\mathrm e}^{-\frac {i \left (c e -d f \right )}{e}} \expIntegral \left (1, \frac {i d}{x}+i c -\frac {i \left (c e -d f \right )}{e}\right ) f}{2 e^{4}}+\frac {a b d \,{\mathrm e}^{-\frac {i \left (c e -d f \right )}{e}} \expIntegral \left (1, \frac {i d}{x}+i c -\frac {i \left (c e -d f \right )}{e}\right )}{e^{3}}-\frac {a^{2}}{2 f \left (f x +e \right )^{2}}-\frac {b^{2}}{4 f \left (f x +e \right )^{2}}-\frac {d^{2} b^{2} {\mathrm e}^{-\frac {2 i \left (c e -d f \right )}{e}} \expIntegral \left (1, \frac {2 i d}{x}+2 i c -\frac {2 i \left (c e -d f \right )}{e}\right ) f}{2 e^{4}}+\frac {i d \,b^{2} {\mathrm e}^{-\frac {2 i \left (c e -d f \right )}{e}} \expIntegral \left (1, \frac {2 i d}{x}+2 i c -\frac {2 i \left (c e -d f \right )}{e}\right )}{2 e^{3}}-\frac {i d \,b^{2} {\mathrm e}^{\frac {2 i \left (c e -d f \right )}{e}} \expIntegral \left (1, -\frac {2 i d}{x}-2 i c -\frac {2 \left (-i c e +i d f \right )}{e}\right )}{2 e^{3}}-\frac {d^{2} b^{2} f \,{\mathrm e}^{\frac {2 i \left (c e -d f \right )}{e}} \expIntegral \left (1, -\frac {2 i d}{x}-2 i c -\frac {2 \left (-i c e +i d f \right )}{e}\right )}{2 e^{4}}+\frac {a b d \,{\mathrm e}^{\frac {i \left (c e -d f \right )}{e}} \expIntegral \left (1, -\frac {i d}{x}-i c -\frac {-i c e +i d f}{e}\right )}{e^{3}}-\frac {i a b \,d^{2} f \,{\mathrm e}^{\frac {i \left (c e -d f \right )}{e}} \expIntegral \left (1, -\frac {i d}{x}-i c -\frac {-i c e +i d f}{e}\right )}{2 e^{4}}+\frac {i a b x \left (2 i d^{3} f^{4} x^{3}+6 i d^{3} e \,f^{3} x^{2}+6 i d^{3} e^{2} f^{2} x +2 i d^{3} e^{3} f \right ) \cos \left (\frac {c x +d}{x}\right )}{2 e^{3} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}\right )}-\frac {a b x \left (-2 d^{2} f^{3} x^{3}-8 d^{2} e \,f^{2} x^{2}-10 d^{2} e^{2} f x -4 d^{2} e^{3}\right ) \sin \left (\frac {c x +d}{x}\right )}{2 e^{2} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}\right )}+\frac {b^{2} x \left (-2 d^{2} f^{3} x^{3}-8 d^{2} e \,f^{2} x^{2}-10 d^{2} e^{2} f x -4 d^{2} e^{3}\right ) \cos \left (\frac {2 c x +2 d}{x}\right )}{8 e^{2} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}\right )}+\frac {i b^{2} x \left (4 i d^{3} f^{4} x^{3}+12 i d^{3} e \,f^{3} x^{2}+12 i d^{3} e^{2} f^{2} x +4 i d^{3} e^{3} f \right ) \sin \left (\frac {2 c x +2 d}{x}\right )}{8 e^{3} \left (f x +e \right )^{2} \left (d^{2} x^{2} f^{2}+2 d^{2} e f x +d^{2} e^{2}\right )}\)	\(866\)
derivativedivides	\(\text {Expression too large to display}\)	\(1139\)
default	\(\text {Expression too large to display}\)	\(1139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/x))^2/(f*x+e)^3,x, algorithm="maxima")

[Out]

-1/2*a^2/(f^3*x^2 + 2*f^2*x*e + f*e^2) - 1/4*(b^2 + 4*(b^2*f^3*x^2 + 2*b^2*f^2*x*e + b^2*f*e^2)*integrate(1/4*
cos(2*(c*x + d)/x)/(f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2 + e^3), x) + 4*(b^2*f^3*x^2 + 2*b^2*f^2*x*e + b^2*f*e^2)
*integrate(1/4*cos(2*(c*x + d)/x)/((f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2 + e^3)*cos(2*(c*x + d)/x)^2 + (f^3*x^3 +
3*f^2*x^2*e + 3*f*x*e^2 + e^3)*sin(2*(c*x + d)/x)^2), x) - 4*(a*b*f^3*x^2 + 2*a*b*f^2*x*e + a*b*f*e^2)*integr
ate(sin((c*x + d)/x)/(f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2 + e^3), x) - 4*(a*b*f^3*x^2 + 2*a*b*f^2*x*e + a*b*f*e^
2)*integrate(sin((c*x + d)/x)/((f^3*x^3 + 3*f^2*x^2*e + 3*f*x*e^2 + e^3)*cos((c*x + d)/x)^2 + (f^3*x^3 + 3*f^2
*x^2*e + 3*f*x*e^2 + e^3)*sin((c*x + d)/x)^2), x))/(f^3*x^2 + 2*f^2*x*e + f*e^2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 910 vs. \(2 (442) = 884\).
time = 0.44, size = 910, normalized size = 1.94 \begin {gather*} \frac {b^{2} f^{2} x^{2} e^{2} + 2 \, b^{2} f x e^{3} - 2 \, {\left (b^{2} f^{2} x^{2} e^{2} + 2 \, b^{2} f x e^{3}\right )} \cos \left (\frac {c x + d}{x}\right )^{2} - 4 \, {\left ({\left (a b d f^{3} x^{2} e + 2 \, a b d f^{2} x e^{2} + a b d f e^{3}\right )} \operatorname {Ci}\left (\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + {\left (a b d f^{3} x^{2} e + 2 \, a b d f^{2} x e^{2} + a b d f e^{3}\right )} \operatorname {Ci}\left (-\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + {\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} f^{3} x e + a b d^{2} f^{2} e^{2}\right )} \operatorname {Si}\left (\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right )\right )} \cos \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) + 2 \, {\left ({\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} f^{3} x e + b^{2} d^{2} f^{2} e^{2}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + {\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} f^{3} x e + b^{2} d^{2} f^{2} e^{2}\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) - 2 \, {\left (b^{2} d f^{3} x^{2} e + 2 \, b^{2} d f^{2} x e^{2} + b^{2} d f e^{3}\right )} \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right )\right )} \cos \left (-2 \, {\left (d f - c e\right )} e^{\left (-1\right )}\right ) - 4 \, {\left (a b d f^{3} x^{2} e + a b d f^{2} x e^{2}\right )} \cos \left (\frac {c x + d}{x}\right ) - {\left (2 \, a^{2} + b^{2}\right )} e^{4} - 2 \, {\left ({\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} f^{3} x e + a b d^{2} f^{2} e^{2}\right )} \operatorname {Ci}\left (\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + {\left (a b d^{2} f^{4} x^{2} + 2 \, a b d^{2} f^{3} x e + a b d^{2} f^{2} e^{2}\right )} \operatorname {Ci}\left (-\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) - 4 \, {\left (a b d f^{3} x^{2} e + 2 \, a b d f^{2} x e^{2} + a b d f e^{3}\right )} \operatorname {Si}\left (\frac {{\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right )\right )} \sin \left (-{\left (d f - c e\right )} e^{\left (-1\right )}\right ) - 2 \, {\left ({\left (b^{2} d f^{3} x^{2} e + 2 \, b^{2} d f^{2} x e^{2} + b^{2} d f e^{3}\right )} \operatorname {Ci}\left (\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + {\left (b^{2} d f^{3} x^{2} e + 2 \, b^{2} d f^{2} x e^{2} + b^{2} d f e^{3}\right )} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right ) + 2 \, {\left (b^{2} d^{2} f^{4} x^{2} + 2 \, b^{2} d^{2} f^{3} x e + b^{2} d^{2} f^{2} e^{2}\right )} \operatorname {Si}\left (\frac {2 \, {\left (d f x + d e\right )} e^{\left (-1\right )}}{x}\right )\right )} \sin \left (-2 \, {\left (d f - c e\right )} e^{\left (-1\right )}\right ) + 4 \, {\left (a b f^{2} x^{2} e^{2} + 2 \, a b f x e^{3} - {\left (b^{2} d f^{3} x^{2} e + b^{2} d f^{2} x e^{2}\right )} \cos \left (\frac {c x + d}{x}\right )\right )} \sin \left (\frac {c x + d}{x}\right )}{4 \, {\left (f^{3} x^{2} e^{4} + 2 \, f^{2} x e^{5} + f e^{6}\right )}} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3062 vs. \(2 (442) = 884\).
time = 7.92, size = 3062, normalized size = 6.51 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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

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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