\(\int \frac {\cos ^3(a+b x^2)}{x^3} \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 91 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 \cos \left (a+b x^2\right )}{8 x^2}-\frac {\cos \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)-\frac {3}{8} b \operatorname {CosIntegral}\left (3 b x^2\right ) \sin (3 a)-\frac {3}{8} b \cos (a) \text {Si}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {Si}\left (3 b x^2\right ) \]

[Out]

-3/8*cos(b*x^2+a)/x^2-1/8*cos(3*b*x^2+3*a)/x^2-3/8*b*cos(a)*Si(b*x^2)-3/8*b*cos(3*a)*Si(3*b*x^2)-3/8*b*Ci(b*x^
2)*sin(a)-3/8*b*Ci(3*b*x^2)*sin(3*a)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {3485, 3461, 3378, 3384, 3380, 3383} \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3}{8} b \sin (a) \operatorname {CosIntegral}\left (b x^2\right )-\frac {3}{8} b \sin (3 a) \operatorname {CosIntegral}\left (3 b x^2\right )-\frac {3}{8} b \cos (a) \text {Si}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {Si}\left (3 b x^2\right )-\frac {3 \cos \left (a+b x^2\right )}{8 x^2}-\frac {\cos \left (3 \left (a+b x^2\right )\right )}{8 x^2} \]

[In]

Int[Cos[a + b*x^2]^3/x^3,x]

[Out]

(-3*Cos[a + b*x^2])/(8*x^2) - Cos[3*(a + b*x^2)]/(8*x^2) - (3*b*CosIntegral[b*x^2]*Sin[a])/8 - (3*b*CosIntegra
l[3*b*x^2]*Sin[3*a])/8 - (3*b*Cos[a]*SinIntegral[b*x^2])/8 - (3*b*Cos[3*a]*SinIntegral[3*b*x^2])/8

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 3461

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rule 3485

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos \left (a+b x^2\right )}{4 x^3}+\frac {\cos \left (3 a+3 b x^2\right )}{4 x^3}\right ) \, dx \\ & = \frac {1}{4} \int \frac {\cos \left (3 a+3 b x^2\right )}{x^3} \, dx+\frac {3}{4} \int \frac {\cos \left (a+b x^2\right )}{x^3} \, dx \\ & = \frac {1}{8} \text {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x^2} \, dx,x,x^2\right )+\frac {3}{8} \text {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^2\right ) \\ & = -\frac {3 \cos \left (a+b x^2\right )}{8 x^2}-\frac {\cos \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b) \text {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x} \, dx,x,x^2\right ) \\ & = -\frac {3 \cos \left (a+b x^2\right )}{8 x^2}-\frac {\cos \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {1}{8} (3 b \cos (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \cos (3 a)) \text {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \sin (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^2\right )-\frac {1}{8} (3 b \sin (3 a)) \text {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^2\right ) \\ & = -\frac {3 \cos \left (a+b x^2\right )}{8 x^2}-\frac {\cos \left (3 \left (a+b x^2\right )\right )}{8 x^2}-\frac {3}{8} b \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)-\frac {3}{8} b \operatorname {CosIntegral}\left (3 b x^2\right ) \sin (3 a)-\frac {3}{8} b \cos (a) \text {Si}\left (b x^2\right )-\frac {3}{8} b \cos (3 a) \text {Si}\left (3 b x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.99 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 \cos \left (a+b x^2\right )+\cos \left (3 \left (a+b x^2\right )\right )+3 b x^2 \operatorname {CosIntegral}\left (b x^2\right ) \sin (a)+3 b x^2 \operatorname {CosIntegral}\left (3 b x^2\right ) \sin (3 a)+3 b x^2 \cos (a) \text {Si}\left (b x^2\right )+3 b x^2 \cos (3 a) \text {Si}\left (3 b x^2\right )}{8 x^2} \]

[In]

Integrate[Cos[a + b*x^2]^3/x^3,x]

[Out]

-1/8*(3*Cos[a + b*x^2] + Cos[3*(a + b*x^2)] + 3*b*x^2*CosIntegral[b*x^2]*Sin[a] + 3*b*x^2*CosIntegral[3*b*x^2]
*Sin[3*a] + 3*b*x^2*Cos[a]*SinIntegral[b*x^2] + 3*b*x^2*Cos[3*a]*SinIntegral[3*b*x^2])/x^2

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.97 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.03

method	result	size
risch	\(-\frac {3 i {\mathrm e}^{3 i a} b \,\operatorname {Ei}_{1}\left (-3 i b \,x^{2}\right ) x^{2}-3 i {\mathrm e}^{-i a} \operatorname {Ei}_{1}\left (-i b \,x^{2}\right ) b \,x^{2}-3 \,{\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{2}\right ) b \,x^{2}+3 i {\mathrm e}^{i a} b \,\operatorname {Ei}_{1}\left (-i b \,x^{2}\right ) x^{2}-3 \pi \,\operatorname {csgn}\left (b \,x^{2}\right ) {\mathrm e}^{-3 i a} b \,x^{2}-3 i {\mathrm e}^{-3 i a} \operatorname {Ei}_{1}\left (-3 i b \,x^{2}\right ) b \,x^{2}+6 \,{\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{2}\right ) b \,x^{2}+6 \,\operatorname {Si}\left (3 b \,x^{2}\right ) {\mathrm e}^{-3 i a} b \,x^{2}+6 \cos \left (b \,x^{2}+a \right )+2 \cos \left (3 b \,x^{2}+3 a \right )}{16 x^{2}}\)	\(185\)

[In]

int(cos(b*x^2+a)^3/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/16*(3*I*exp(3*I*a)*b*Ei(1,-3*I*b*x^2)*x^2-3*I*exp(-I*a)*Ei(1,-I*b*x^2)*b*x^2-3*exp(-I*a)*Pi*csgn(b*x^2)*b*x
^2+3*I*exp(I*a)*b*Ei(1,-I*b*x^2)*x^2-3*Pi*csgn(b*x^2)*exp(-3*I*a)*b*x^2-3*I*exp(-3*I*a)*Ei(1,-3*I*b*x^2)*b*x^2
+6*exp(-I*a)*Si(b*x^2)*b*x^2+6*Si(3*b*x^2)*exp(-3*I*a)*b*x^2+6*cos(b*x^2+a)+2*cos(3*b*x^2+3*a))/x^2

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 \, b x^{2} \operatorname {Ci}\left (3 \, b x^{2}\right ) \sin \left (3 \, a\right ) + 3 \, b x^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) + 3 \, b x^{2} \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{2}\right ) + 3 \, b x^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) + 4 \, \cos \left (b x^{2} + a\right )^{3}}{8 \, x^{2}} \]

[In]

integrate(cos(b*x^2+a)^3/x^3,x, algorithm="fricas")

[Out]

-1/8*(3*b*x^2*cos_integral(3*b*x^2)*sin(3*a) + 3*b*x^2*cos_integral(b*x^2)*sin(a) + 3*b*x^2*cos(3*a)*sin_integ
ral(3*b*x^2) + 3*b*x^2*cos(a)*sin_integral(b*x^2) + 4*cos(b*x^2 + a)^3)/x^2

Sympy [F]

\[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {\cos ^{3}{\left (a + b x^{2} \right )}}{x^{3}}\, dx \]

[In]

integrate(cos(b*x**2+a)**3/x**3,x)

[Out]

Integral(cos(a + b*x**2)**3/x**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.38 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.08 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=\frac {3}{16} \, {\left ({\left (-i \, \Gamma \left (-1, 3 i \, b x^{2}\right ) + i \, \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + {\left (-i \, \Gamma \left (-1, i \, b x^{2}\right ) + i \, \Gamma \left (-1, -i \, b x^{2}\right )\right )} \cos \left (a\right ) - {\left (\Gamma \left (-1, 3 i \, b x^{2}\right ) + \Gamma \left (-1, -3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) - {\left (\Gamma \left (-1, i \, b x^{2}\right ) + \Gamma \left (-1, -i \, b x^{2}\right )\right )} \sin \left (a\right )\right )} b \]

[In]

integrate(cos(b*x^2+a)^3/x^3,x, algorithm="maxima")

[Out]

3/16*((-I*gamma(-1, 3*I*b*x^2) + I*gamma(-1, -3*I*b*x^2))*cos(3*a) + (-I*gamma(-1, I*b*x^2) + I*gamma(-1, -I*b
*x^2))*cos(a) - (gamma(-1, 3*I*b*x^2) + gamma(-1, -3*I*b*x^2))*sin(3*a) - (gamma(-1, I*b*x^2) + gamma(-1, -I*b
*x^2))*sin(a))*b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (80) = 160\).

Time = 0.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.03 \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=-\frac {3 \, {\left (b x^{2} + a\right )} b^{2} \operatorname {Ci}\left (3 \, b x^{2}\right ) \sin \left (3 \, a\right ) - 3 \, a b^{2} \operatorname {Ci}\left (3 \, b x^{2}\right ) \sin \left (3 \, a\right ) + 3 \, {\left (b x^{2} + a\right )} b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) - 3 \, a b^{2} \operatorname {Ci}\left (b x^{2}\right ) \sin \left (a\right ) + 3 \, {\left (b x^{2} + a\right )} b^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) - 3 \, a b^{2} \cos \left (a\right ) \operatorname {Si}\left (b x^{2}\right ) - 3 \, {\left (b x^{2} + a\right )} b^{2} \cos \left (3 \, a\right ) \operatorname {Si}\left (-3 \, b x^{2}\right ) + 3 \, a b^{2} \cos \left (3 \, a\right ) \operatorname {Si}\left (-3 \, b x^{2}\right ) + b^{2} \cos \left (3 \, b x^{2} + 3 \, a\right ) + 3 \, b^{2} \cos \left (b x^{2} + a\right )}{8 \, b^{2} x^{2}} \]

[In]

integrate(cos(b*x^2+a)^3/x^3,x, algorithm="giac")

[Out]

-1/8*(3*(b*x^2 + a)*b^2*cos_integral(3*b*x^2)*sin(3*a) - 3*a*b^2*cos_integral(3*b*x^2)*sin(3*a) + 3*(b*x^2 + a
)*b^2*cos_integral(b*x^2)*sin(a) - 3*a*b^2*cos_integral(b*x^2)*sin(a) + 3*(b*x^2 + a)*b^2*cos(a)*sin_integral(
b*x^2) - 3*a*b^2*cos(a)*sin_integral(b*x^2) - 3*(b*x^2 + a)*b^2*cos(3*a)*sin_integral(-3*b*x^2) + 3*a*b^2*cos(
3*a)*sin_integral(-3*b*x^2) + b^2*cos(3*b*x^2 + 3*a) + 3*b^2*cos(b*x^2 + a))/(b^2*x^2)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3\left (a+b x^2\right )}{x^3} \, dx=\int \frac {{\cos \left (b\,x^2+a\right )}^3}{x^3} \,d x \]

[In]

int(cos(a + b*x^2)^3/x^3,x)

[Out]

int(cos(a + b*x^2)^3/x^3, x)