3.1.27 \(\int \frac {\cos ^4(a+b x)}{x^2} \, dx\) [27]

Optimal. Leaf size=66 \[ -\frac {\cos ^4(a+b x)}{x}-b \text {CosIntegral}(2 b x) \sin (2 a)-\frac {1}{2} b \text {CosIntegral}(4 b x) \sin (4 a)-b \cos (2 a) \text {Si}(2 b x)-\frac {1}{2} b \cos (4 a) \text {Si}(4 b x) \]

[Out]

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

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Rubi [A]
time = 0.10, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3394, 3384, 3380, 3383} \begin {gather*} -b \sin (2 a) \text {CosIntegral}(2 b x)-\frac {1}{2} b \sin (4 a) \text {CosIntegral}(4 b x)-b \cos (2 a) \text {Si}(2 b x)-\frac {1}{2} b \cos (4 a) \text {Si}(4 b x)-\frac {\cos ^4(a+b x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4/x^2,x]

[Out]

-(Cos[a + b*x]^4/x) - b*CosIntegral[2*b*x]*Sin[2*a] - (b*CosIntegral[4*b*x]*Sin[4*a])/2 - b*Cos[2*a]*SinIntegr
al[2*b*x] - (b*Cos[4*a]*SinIntegral[4*b*x])/2

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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^4(a+b x)}{x^2} \, dx &=-\frac {\cos ^4(a+b x)}{x}+(4 b) \int \left (-\frac {\sin (2 a+2 b x)}{4 x}-\frac {\sin (4 a+4 b x)}{8 x}\right ) \, dx\\ &=-\frac {\cos ^4(a+b x)}{x}-\frac {1}{2} b \int \frac {\sin (4 a+4 b x)}{x} \, dx-b \int \frac {\sin (2 a+2 b x)}{x} \, dx\\ &=-\frac {\cos ^4(a+b x)}{x}-(b \cos (2 a)) \int \frac {\sin (2 b x)}{x} \, dx-\frac {1}{2} (b \cos (4 a)) \int \frac {\sin (4 b x)}{x} \, dx-(b \sin (2 a)) \int \frac {\cos (2 b x)}{x} \, dx-\frac {1}{2} (b \sin (4 a)) \int \frac {\cos (4 b x)}{x} \, dx\\ &=-\frac {\cos ^4(a+b x)}{x}-b \text {Ci}(2 b x) \sin (2 a)-\frac {1}{2} b \text {Ci}(4 b x) \sin (4 a)-b \cos (2 a) \text {Si}(2 b x)-\frac {1}{2} b \cos (4 a) \text {Si}(4 b x)\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 79, normalized size = 1.20 \begin {gather*} -\frac {3+4 \cos (2 (a+b x))+\cos (4 (a+b x))+8 b x \text {CosIntegral}(2 b x) \sin (2 a)+4 b x \text {CosIntegral}(4 b x) \sin (4 a)+8 b x \cos (2 a) \text {Si}(2 b x)+4 b x \cos (4 a) \text {Si}(4 b x)}{8 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4/x^2,x]

[Out]

-1/8*(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 8*b*x*CosIntegral[2*b*x]*Sin[2*a] + 4*b*x*CosIntegral[4*b*x]
*Sin[4*a] + 8*b*x*Cos[2*a]*SinIntegral[2*b*x] + 4*b*x*Cos[4*a]*SinIntegral[4*b*x])/x

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Maple [A]
time = 0.12, size = 90, normalized size = 1.36

method	result	size
derivativedivides	\(b \left (-\frac {\cos \left (4 b x +4 a \right )}{8 b x}-\frac {\sinIntegral \left (4 b x \right ) \cos \left (4 a \right )}{2}-\frac {\cosineIntegral \left (4 b x \right ) \sin \left (4 a \right )}{2}-\frac {\cos \left (2 b x +2 a \right )}{2 b x}-\sinIntegral \left (2 b x \right ) \cos \left (2 a \right )-\cosineIntegral \left (2 b x \right ) \sin \left (2 a \right )-\frac {3}{8 b x}\right )\)	\(90\)
default	\(b \left (-\frac {\cos \left (4 b x +4 a \right )}{8 b x}-\frac {\sinIntegral \left (4 b x \right ) \cos \left (4 a \right )}{2}-\frac {\cosineIntegral \left (4 b x \right ) \sin \left (4 a \right )}{2}-\frac {\cos \left (2 b x +2 a \right )}{2 b x}-\sinIntegral \left (2 b x \right ) \cos \left (2 a \right )-\cosineIntegral \left (2 b x \right ) \sin \left (2 a \right )-\frac {3}{8 b x}\right )\)	\(90\)
risch	\(\frac {\pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-2 i a} b}{2}-\sinIntegral \left (2 b x \right ) {\mathrm e}^{-2 i a} b +\frac {i \expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{-2 i a} b}{2}-\frac {i b \expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{2 i a}}{2}+\frac {\pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-4 i a} b}{4}-\frac {\sinIntegral \left (4 b x \right ) {\mathrm e}^{-4 i a} b}{2}+\frac {i \expIntegral \left (1, -4 i b x \right ) {\mathrm e}^{-4 i a} b}{4}-\frac {i b \expIntegral \left (1, -4 i b x \right ) {\mathrm e}^{4 i a}}{4}-\frac {3}{8 x}-\frac {\cos \left (4 b x +4 a \right )}{8 x}-\frac {\cos \left (2 b x +2 a \right )}{2 x}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4/x^2,x,method=_RETURNVERBOSE)

[Out]

b*(-1/8*cos(4*b*x+4*a)/b/x-1/2*Si(4*b*x)*cos(4*a)-1/2*Ci(4*b*x)*sin(4*a)-1/2*cos(2*b*x+2*a)/b/x-Si(2*b*x)*cos(
2*a)-Ci(2*b*x)*sin(2*a)-3/8/b/x)

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Maxima [C] Result contains complex when optimal does not.
time = 0.38, size = 717, normalized size = 10.86 \begin {gather*} \frac {{\left ({\left ({\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{3} + {\left ({\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )^{3} + 4 \, {\left ({\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} + {\left (-i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + {\left ({\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3\right )} \sin \left (2 \, a\right )^{2} + {\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3 \, \cos \left (2 \, a\right )^{2} + {\left ({\left (-i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} \cos \left (4 \, a\right )^{2} + {\left (4 \, {\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} + 4 \, {\left (-i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + 4 \, {\left ({\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3\right )} \sin \left (2 \, a\right )^{2} + {\left ({\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right ) + 4 \, {\left (E_{2}\left (2 i \, b x\right ) + E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 12 \, \cos \left (2 \, a\right )^{2} + 4 \, {\left ({\left (-i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - i \, E_{2}\left (2 i \, b x\right ) + i \, E_{2}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} \sin \left (4 \, a\right )^{2} + {\left ({\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{2}\left (4 i \, b x\right ) + E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right ) + {\left ({\left ({\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{2} + {\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{2}\left (4 i \, b x\right ) + i \, E_{2}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )\right )} b}{32 \, {\left ({\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{2} + {\left (a \cos \left (2 \, a\right )^{2} + a \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )^{2} - {\left ({\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{2} + {\left (\cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )^{2}\right )} {\left (b x + a\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/x^2,x, algorithm="maxima")

[Out]

1/32*(((exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4*I*b*x) + e
xp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^3 + ((-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*
I*b*x))*cos(2*a)^2 + (-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*sin(4*a)^3 +
4*((exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a)^3 + (-I*exp_integral_e(2, 2*I*b*x) + I*
exp_integral_e(2, -2*I*b*x))*sin(2*a)^3 + ((exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a)
+ 3)*sin(2*a)^2 + (exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 3*cos(2*a)^2 + ((-I*e
xp_integral_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 - I*exp_integral_e(2, 2*I*b*x) + I*exp_i
ntegral_e(2, -2*I*b*x))*sin(2*a))*cos(4*a)^2 + (4*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*c
os(2*a)^3 + 4*(-I*exp_integral_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*sin(2*a)^3 + 4*((exp_integral_e(
2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + ((exp_integral_e(2, 4*I*b*x) + exp_integ
ral_e(2, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*cos(4*
a) + 4*(exp_integral_e(2, 2*I*b*x) + exp_integral_e(2, -2*I*b*x))*cos(2*a) + 12*cos(2*a)^2 + 4*((-I*exp_integr
al_e(2, 2*I*b*x) + I*exp_integral_e(2, -2*I*b*x))*cos(2*a)^2 - I*exp_integral_e(2, 2*I*b*x) + I*exp_integral_e
(2, -2*I*b*x))*sin(2*a))*sin(4*a)^2 + ((exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 +
(exp_integral_e(2, 4*I*b*x) + exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*cos(4*a) + (((-I*exp_integral_e(2, 4*I
*b*x) + I*exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*I*
b*x))*sin(2*a)^2)*cos(4*a)^2 + (-I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*I*b*x))*cos(2*a)^2 + (-
I*exp_integral_e(2, 4*I*b*x) + I*exp_integral_e(2, -4*I*b*x))*sin(2*a)^2)*sin(4*a))*b/((a*cos(2*a)^2 + a*sin(2
*a)^2)*cos(4*a)^2 + (a*cos(2*a)^2 + a*sin(2*a)^2)*sin(4*a)^2 - ((cos(2*a)^2 + sin(2*a)^2)*cos(4*a)^2 + (cos(2*
a)^2 + sin(2*a)^2)*sin(4*a)^2)*(b*x + a))

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Fricas [A]
time = 0.36, size = 87, normalized size = 1.32 \begin {gather*} -\frac {4 \, \cos \left (b x + a\right )^{4} + 2 \, b x \cos \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x\right ) + 4 \, b x \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right ) + {\left (b x \operatorname {Ci}\left (4 \, b x\right ) + b x \operatorname {Ci}\left (-4 \, b x\right )\right )} \sin \left (4 \, a\right ) + 2 \, {\left (b x \operatorname {Ci}\left (2 \, b x\right ) + b x \operatorname {Ci}\left (-2 \, b x\right )\right )} \sin \left (2 \, a\right )}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/x^2,x, algorithm="fricas")

[Out]

-1/4*(4*cos(b*x + a)^4 + 2*b*x*cos(4*a)*sin_integral(4*b*x) + 4*b*x*cos(2*a)*sin_integral(2*b*x) + (b*x*cos_in
tegral(4*b*x) + b*x*cos_integral(-4*b*x))*sin(4*a) + 2*(b*x*cos_integral(2*b*x) + b*x*cos_integral(-2*b*x))*si
n(2*a))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{4}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**4/x**2,x)

[Out]

Integral(cos(a + b*x)**4/x**2, x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.47, size = 3220, normalized size = 48.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/x^2,x, algorithm="giac")

[Out]

1/4*(b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_inte
gral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*
tan(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)
^2 + 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b
*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2
*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) - 2*b*x*real_part(co
s_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(-4*b*x))*tan(2*b*x
)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 2*b
*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*t
an(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 2
*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^
2*tan(2*a)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integ
ral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*t
an(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 2*b*x*sin_integral(4*b*x)*tan
(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + b*x*imag_part(cos
_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a
)^2*tan(a)^2 - 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_inte
gral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 +
4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(b*x)^2*tan
(2*a)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - 2*b*x*imag_part(cos_i
ntegral(-2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2*ta
n(a)^2 + 2*b*x*sin_integral(4*b*x)*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(b*x)^2*tan(2
*a)^2*tan(a)^2 - 2*b*x*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 2*b*x*real_part(cos_i
ntegral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 4*b*x*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)
^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a) - 4*b*x*real_part(cos_integra
l(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a
) - 4*b*x*real_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) - 4*b*x*real_part(cos_integral(-2*b*x))*
tan(b*x)^2*tan(2*a)^2*tan(a) - 2*b*x*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*rea
l_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(4*b*x))*tan(b*x)^2*
tan(2*a)*tan(a)^2 - 2*b*x*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)*tan(a)^2 - 4*tan(2*b*x)^2*tan(b*
x)^2*tan(2*a)^2*tan(a)^2 - b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 2*b*x*imag_part(cos_in
tegral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2 + b*x*i
mag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 2*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2 - 4
*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2 + b*x*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2
- 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integral(-2*b*x))*tan(2*b
*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 2*b*x*sin_integral(4*b*x)*tan
(2*b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2 + b*x*imag_part(cos_integral(4*b*x))*
tan(b*x)^2*tan(2*a)^2 - 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2 + 2*b*x*imag_part(cos_integ
ral(-2*b*x))*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2 + 2*b*x*sin_int
egral(4*b*x)*tan(b*x)^2*tan(2*a)^2 - 4*b*x*sin_integral(2*b*x)*tan(b*x)^2*tan(2*a)^2 - b*x*imag_part(cos_integ
ral(4*b*x))*tan(2*b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(a)^2 - 2*b*x*imag_pa
rt(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(a)^2 - 2
*b*x*sin_integral(4*b*x)*tan(2*b*x)^2*tan(a)^2 + 4*b*x*sin_integral(2*b*x)*tan(2*b*x)^2*tan(a)^2 - b*x*imag_pa
rt(cos_integral(4*b*x))*tan(b*x)^2*tan(a)^2 + 2*b*x*imag_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a)^2 - 2*b*x
*imag_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(a)^2 + b*x*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(a)^2
- 2*b*x*sin_integral(4*b*x)*tan(b*x)^2*tan(a)^...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^4}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^4/x^2,x)

[Out]

int(cos(a + b*x)^4/x^2, x)

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