3.1.56 \(\int \frac {(a+b x^2)^2 \sin (c+d x)}{x^5} \, dx\) [56]

Optimal. Leaf size=177 \[ -\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text {Ci}(d x) \sin (c)-a b d^2 \text {Ci}(d x) \sin (c)+\frac {1}{24} a^2 d^4 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-a b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x) \]

[Out]

-1/12*a^2*d*cos(d*x+c)/x^3-a*b*d*cos(d*x+c)/x+1/24*a^2*d^3*cos(d*x+c)/x+b^2*cos(c)*Si(d*x)-a*b*d^2*cos(c)*Si(d
*x)+1/24*a^2*d^4*cos(c)*Si(d*x)+b^2*Ci(d*x)*sin(c)-a*b*d^2*Ci(d*x)*sin(c)+1/24*a^2*d^4*Ci(d*x)*sin(c)-1/4*a^2*
sin(d*x+c)/x^4-a*b*sin(d*x+c)/x^2+1/24*a^2*d^2*sin(d*x+c)/x^2

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Rubi [A]
time = 0.22, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3420, 3378, 3384, 3380, 3383} \begin {gather*} \frac {1}{24} a^2 d^4 \sin (c) \text {CosIntegral}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)+\frac {a^2 d^3 \cos (c+d x)}{24 x}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a^2 d \cos (c+d x)}{12 x^3}-a b d^2 \sin (c) \text {CosIntegral}(d x)-a b d^2 \cos (c) \text {Si}(d x)-\frac {a b \sin (c+d x)}{x^2}-\frac {a b d \cos (c+d x)}{x}+b^2 \sin (c) \text {CosIntegral}(d x)+b^2 \cos (c) \text {Si}(d x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*Sin[c + d*x])/x^5,x]

[Out]

-1/12*(a^2*d*Cos[c + d*x])/x^3 - (a*b*d*Cos[c + d*x])/x + (a^2*d^3*Cos[c + d*x])/(24*x) + b^2*CosIntegral[d*x]
*Sin[c] - a*b*d^2*CosIntegral[d*x]*Sin[c] + (a^2*d^4*CosIntegral[d*x]*Sin[c])/24 - (a^2*Sin[c + d*x])/(4*x^4)
- (a*b*Sin[c + d*x])/x^2 + (a^2*d^2*Sin[c + d*x])/(24*x^2) + b^2*Cos[c]*SinIntegral[d*x] - a*b*d^2*Cos[c]*SinI
ntegral[d*x] + (a^2*d^4*Cos[c]*SinIntegral[d*x])/24

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 3420

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2 \sin (c+d x)}{x^5} \, dx &=\int \left (\frac {a^2 \sin (c+d x)}{x^5}+\frac {2 a b \sin (c+d x)}{x^3}+\frac {b^2 \sin (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac {\sin (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\sin (c+d x)}{x^3} \, dx+b^2 \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\cos (c+d x)}{x^4} \, dx+(a b d) \int \frac {\cos (c+d x)}{x^2} \, dx+\left (b^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\left (b^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+b^2 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+b^2 \cos (c) \text {Si}(d x)-\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\sin (c+d x)}{x^3} \, dx-\left (a b d^2\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+b^2 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\cos (c+d x)}{x^2} \, dx-\left (a b d^2 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx-\left (a b d^2 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text {Ci}(d x) \sin (c)-a b d^2 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-a b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} \left (a^2 d^4\right ) \int \frac {\sin (c+d x)}{x} \, dx\\ &=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text {Ci}(d x) \sin (c)-a b d^2 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-a b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} \left (a^2 d^4 \cos (c)\right ) \int \frac {\sin (d x)}{x} \, dx+\frac {1}{24} \left (a^2 d^4 \sin (c)\right ) \int \frac {\cos (d x)}{x} \, dx\\ &=-\frac {a^2 d \cos (c+d x)}{12 x^3}-\frac {a b d \cos (c+d x)}{x}+\frac {a^2 d^3 \cos (c+d x)}{24 x}+b^2 \text {Ci}(d x) \sin (c)-a b d^2 \text {Ci}(d x) \sin (c)+\frac {1}{24} a^2 d^4 \text {Ci}(d x) \sin (c)-\frac {a^2 \sin (c+d x)}{4 x^4}-\frac {a b \sin (c+d x)}{x^2}+\frac {a^2 d^2 \sin (c+d x)}{24 x^2}+b^2 \cos (c) \text {Si}(d x)-a b d^2 \cos (c) \text {Si}(d x)+\frac {1}{24} a^2 d^4 \cos (c) \text {Si}(d x)\\ \end {align*}

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Mathematica [A]
time = 0.26, size = 122, normalized size = 0.69 \begin {gather*} \frac {a d x \left (-24 b x^2+a \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+\left (24 b^2-24 a b d^2+a^2 d^4\right ) x^4 \text {Ci}(d x) \sin (c)+a \left (-24 b x^2+a \left (-6+d^2 x^2\right )\right ) \sin (c+d x)+\left (24 b^2-24 a b d^2+a^2 d^4\right ) x^4 \cos (c) \text {Si}(d x)}{24 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*Sin[c + d*x])/x^5,x]

[Out]

(a*d*x*(-24*b*x^2 + a*(-2 + d^2*x^2))*Cos[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*CosIntegral[d*x]*Sin[
c] + a*(-24*b*x^2 + a*(-6 + d^2*x^2))*Sin[c + d*x] + (24*b^2 - 24*a*b*d^2 + a^2*d^4)*x^4*Cos[c]*SinIntegral[d*
x])/(24*x^4)

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Maple [A]
time = 0.23, size = 157, normalized size = 0.89 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*sin(d*x+c)/x^5,x,method=_RETURNVERBOSE)

[Out]

d^4*(a^2*(-1/4*sin(d*x+c)/d^4/x^4-1/12*cos(d*x+c)/d^3/x^3+1/24*sin(d*x+c)/d^2/x^2+1/24*cos(d*x+c)/d/x+1/24*Si(
d*x)*cos(c)+1/24*Ci(d*x)*sin(c))+2/d^2*a*b*(-1/2*sin(d*x+c)/d^2/x^2-1/2*cos(d*x+c)/d/x-1/2*Si(d*x)*cos(c)-1/2*
Ci(d*x)*sin(c))+1/d^4*b^2*(Si(d*x)*cos(c)+Ci(d*x)*sin(c)))

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Maxima [C] Result contains complex when optimal does not.
time = 9.93, size = 222, normalized size = 1.25 \begin {gather*} -\frac {{\left ({\left (a^{2} {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{8} - 24 \, {\left (a b {\left (i \, \Gamma \left (-4, i \, d x\right ) - i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) + a b {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{6} - 24 \, {\left (b^{2} {\left (-i \, \Gamma \left (-4, i \, d x\right ) + i \, \Gamma \left (-4, -i \, d x\right )\right )} \cos \left (c\right ) - b^{2} {\left (\Gamma \left (-4, i \, d x\right ) + \Gamma \left (-4, -i \, d x\right )\right )} \sin \left (c\right )\right )} d^{4}\right )} x^{4} + 2 \, {\left (b^{2} d^{3} x^{3} + 2 \, {\left (a b d^{3} - b^{2} d\right )} x\right )} \cos \left (d x + c\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 6 \, a b d^{2} - 6 \, b^{2}\right )} \sin \left (d x + c\right )}{2 \, d^{4} x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="maxima")

[Out]

-1/2*(((a^2*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a^2*(gamma(-4, I*d*x) + gamma(-4, -I*d*x))*sin
(c))*d^8 - 24*(a*b*(I*gamma(-4, I*d*x) - I*gamma(-4, -I*d*x))*cos(c) + a*b*(gamma(-4, I*d*x) + gamma(-4, -I*d*
x))*sin(c))*d^6 - 24*(b^2*(-I*gamma(-4, I*d*x) + I*gamma(-4, -I*d*x))*cos(c) - b^2*(gamma(-4, I*d*x) + gamma(-
4, -I*d*x))*sin(c))*d^4)*x^4 + 2*(b^2*d^3*x^3 + 2*(a*b*d^3 - b^2*d)*x)*cos(d*x + c) + 2*(b^2*d^2*x^2 + 6*a*b*d
^2 - 6*b^2)*sin(d*x + c))/(d^4*x^4)

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Fricas [A]
time = 0.37, size = 162, normalized size = 0.92 \begin {gather*} \frac {2 \, {\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) - 2 \, {\left (2 \, a^{2} d x - {\left (a^{2} d^{3} - 24 \, a b d\right )} x^{3}\right )} \cos \left (d x + c\right ) + 2 \, {\left ({\left (a^{2} d^{2} - 24 \, a b\right )} x^{2} - 6 \, a^{2}\right )} \sin \left (d x + c\right ) + {\left ({\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \operatorname {Ci}\left (d x\right ) + {\left (a^{2} d^{4} - 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right )}{48 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="fricas")

[Out]

1/48*(2*(a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos(c)*sin_integral(d*x) - 2*(2*a^2*d*x - (a^2*d^3 - 24*a*b*d)*x^3
)*cos(d*x + c) + 2*((a^2*d^2 - 24*a*b)*x^2 - 6*a^2)*sin(d*x + c) + ((a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos_in
tegral(d*x) + (a^2*d^4 - 24*a*b*d^2 + 24*b^2)*x^4*cos_integral(-d*x))*sin(c))/x^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2} \sin {\left (c + d x \right )}}{x^{5}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*sin(d*x+c)/x**5,x)

[Out]

Integral((a + b*x**2)**2*sin(c + d*x)/x**5, x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 4.49, size = 1497, normalized size = 8.46 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*sin(d*x+c)/x^5,x, algorithm="giac")

[Out]

-1/48*(a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integr
al(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^
4*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^4*x^4*real_part(cos_integral(-d*x))*tan
(1/2*d*x)^2*tan(1/2*c) - a^2*d^4*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_i
ntegral(-d*x))*tan(1/2*d*x)^2 - 2*a^2*d^4*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 + a^2*d^4*x^4*imag_part(cos_int
egral(d*x))*tan(1/2*c)^2 - a^2*d^4*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a^2*d^4*x^4*sin_integral
(d*x)*tan(1/2*c)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 24*a*b*d^2*x^4*
imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*ta
n(1/2*c)^2 - 2*a^2*d^4*x^4*real_part(cos_integral(d*x))*tan(1/2*c) - 2*a^2*d^4*x^4*real_part(cos_integral(-d*x
))*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 48*a*b*d^2*x^4*real_pa
rt(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^2*d^3*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 - a^2*d^4*x^4*ima
g_part(cos_integral(d*x)) + a^2*d^4*x^4*imag_part(cos_integral(-d*x)) - 2*a^2*d^4*x^4*sin_integral(d*x) + 24*a
*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*
d*x)^2 + 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2 - 24*a*b*d^2*x^4*imag_part(cos_integral(d*x))*tan(1/2
*c)^2 + 24*a*b*d^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 - 48*a*b*d^2*x^4*sin_integral(d*x)*tan(1/2*c
)^2 + 24*b^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 24*b^2*x^4*imag_part(cos_integral(
-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 48*b^2*x^4*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a^2*d^3*x^3*
tan(1/2*d*x)^2 + 48*a*b*d^2*x^4*real_part(cos_integral(d*x))*tan(1/2*c) + 48*a*b*d^2*x^4*real_part(cos_integra
l(-d*x))*tan(1/2*c) + 8*a^2*d^3*x^3*tan(1/2*d*x)*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(d*x))*tan(1/2*
d*x)^2*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^3*x^3*tan(1/2
*c)^2 + 48*a*b*d*x^3*tan(1/2*d*x)^2*tan(1/2*c)^2 + 24*a*b*d^2*x^4*imag_part(cos_integral(d*x)) - 24*a*b*d^2*x^
4*imag_part(cos_integral(-d*x)) + 48*a*b*d^2*x^4*sin_integral(d*x) - 24*b^2*x^4*imag_part(cos_integral(d*x))*t
an(1/2*d*x)^2 + 24*b^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2 - 48*b^2*x^4*sin_integral(d*x)*tan(1/2
*d*x)^2 + 4*a^2*d^2*x^2*tan(1/2*d*x)^2*tan(1/2*c) + 24*b^2*x^4*imag_part(cos_integral(d*x))*tan(1/2*c)^2 - 24*
b^2*x^4*imag_part(cos_integral(-d*x))*tan(1/2*c)^2 + 48*b^2*x^4*sin_integral(d*x)*tan(1/2*c)^2 + 4*a^2*d^2*x^2
*tan(1/2*d*x)*tan(1/2*c)^2 - 2*a^2*d^3*x^3 - 48*a*b*d*x^3*tan(1/2*d*x)^2 - 48*b^2*x^4*real_part(cos_integral(d
*x))*tan(1/2*c) - 48*b^2*x^4*real_part(cos_integral(-d*x))*tan(1/2*c) - 192*a*b*d*x^3*tan(1/2*d*x)*tan(1/2*c)
- 48*a*b*d*x^3*tan(1/2*c)^2 + 4*a^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 24*b^2*x^4*imag_part(cos_integral(d*x))
+ 24*b^2*x^4*imag_part(cos_integral(-d*x)) - 48*b^2*x^4*sin_integral(d*x) - 4*a^2*d^2*x^2*tan(1/2*d*x) - 4*a^2
*d^2*x^2*tan(1/2*c) - 96*a*b*x^2*tan(1/2*d*x)^2*tan(1/2*c) - 96*a*b*x^2*tan(1/2*d*x)*tan(1/2*c)^2 + 48*a*b*d*x
^3 - 4*a^2*d*x*tan(1/2*d*x)^2 - 16*a^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 4*a^2*d*x*tan(1/2*c)^2 + 96*a*b*x^2*tan(1
/2*d*x) + 96*a*b*x^2*tan(1/2*c) - 24*a^2*tan(1/2*d*x)^2*tan(1/2*c) - 24*a^2*tan(1/2*d*x)*tan(1/2*c)^2 + 4*a^2*
d*x + 24*a^2*tan(1/2*d*x) + 24*a^2*tan(1/2*c))/(x^4*tan(1/2*d*x)^2*tan(1/2*c)^2 + x^4*tan(1/2*d*x)^2 + x^4*tan
(1/2*c)^2 + x^4)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sin \left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(c + d*x)*(a + b*x^2)^2)/x^5,x)

[Out]

int((sin(c + d*x)*(a + b*x^2)^2)/x^5, x)

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