3.1.28 \(\int \frac {\cos ^4(a+b x)}{x^3} \, dx\) [28]

Optimal. Leaf size=90 \[ -\frac {\cos ^4(a+b x)}{2 x^2}-b^2 \cos (2 a) \text {CosIntegral}(2 b x)-b^2 \cos (4 a) \text {CosIntegral}(4 b x)+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}+b^2 \sin (2 a) \text {Si}(2 b x)+b^2 \sin (4 a) \text {Si}(4 b x) \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3395, 3393, 3384, 3380, 3383} \begin {gather*} -b^2 \cos (2 a) \text {CosIntegral}(2 b x)-b^2 \cos (4 a) \text {CosIntegral}(4 b x)+b^2 \sin (2 a) \text {Si}(2 b x)+b^2 \sin (4 a) \text {Si}(4 b x)-\frac {\cos ^4(a+b x)}{2 x^2}+\frac {2 b \sin (a+b x) \cos ^3(a+b x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4/x^3,x]

[Out]

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

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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^4(a+b x)}{x^3} \, dx &=-\frac {\cos ^4(a+b x)}{2 x^2}+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (6 b^2\right ) \int \frac {\cos ^2(a+b x)}{x} \, dx-\left (8 b^2\right ) \int \frac {\cos ^4(a+b x)}{x} \, dx\\ &=-\frac {\cos ^4(a+b x)}{2 x^2}+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (6 b^2\right ) \int \left (\frac {1}{2 x}+\frac {\cos (2 a+2 b x)}{2 x}\right ) \, dx-\left (8 b^2\right ) \int \left (\frac {3}{8 x}+\frac {\cos (2 a+2 b x)}{2 x}+\frac {\cos (4 a+4 b x)}{8 x}\right ) \, dx\\ &=-\frac {\cos ^4(a+b x)}{2 x^2}+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}-b^2 \int \frac {\cos (4 a+4 b x)}{x} \, dx+\left (3 b^2\right ) \int \frac {\cos (2 a+2 b x)}{x} \, dx-\left (4 b^2\right ) \int \frac {\cos (2 a+2 b x)}{x} \, dx\\ &=-\frac {\cos ^4(a+b x)}{2 x^2}+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}+\left (3 b^2 \cos (2 a)\right ) \int \frac {\cos (2 b x)}{x} \, dx-\left (4 b^2 \cos (2 a)\right ) \int \frac {\cos (2 b x)}{x} \, dx-\left (b^2 \cos (4 a)\right ) \int \frac {\cos (4 b x)}{x} \, dx-\left (3 b^2 \sin (2 a)\right ) \int \frac {\sin (2 b x)}{x} \, dx+\left (4 b^2 \sin (2 a)\right ) \int \frac {\sin (2 b x)}{x} \, dx+\left (b^2 \sin (4 a)\right ) \int \frac {\sin (4 b x)}{x} \, dx\\ &=-\frac {\cos ^4(a+b x)}{2 x^2}-b^2 \cos (2 a) \text {Ci}(2 b x)-b^2 \cos (4 a) \text {Ci}(4 b x)+\frac {2 b \cos ^3(a+b x) \sin (a+b x)}{x}+b^2 \sin (2 a) \text {Si}(2 b x)+b^2 \sin (4 a) \text {Si}(4 b x)\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 119, normalized size = 1.32 \begin {gather*} -\frac {3+4 \cos (2 (a+b x))+\cos (4 (a+b x))+16 b^2 x^2 \cos (2 a) \text {CosIntegral}(2 b x)+16 b^2 x^2 \cos (4 a) \text {CosIntegral}(4 b x)-8 b x \sin (2 (a+b x))-4 b x \sin (4 (a+b x))-16 b^2 x^2 \sin (2 a) \text {Si}(2 b x)-16 b^2 x^2 \sin (4 a) \text {Si}(4 b x)}{16 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4/x^3,x]

[Out]

-1/16*(3 + 4*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 16*b^2*x^2*Cos[2*a]*CosIntegral[2*b*x] + 16*b^2*x^2*Cos[4*a
]*CosIntegral[4*b*x] - 8*b*x*Sin[2*(a + b*x)] - 4*b*x*Sin[4*(a + b*x)] - 16*b^2*x^2*Sin[2*a]*SinIntegral[2*b*x
] - 16*b^2*x^2*Sin[4*a]*SinIntegral[4*b*x])/x^2

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Maple [A]
time = 0.11, size = 124, normalized size = 1.38

method	result	size
derivativedivides	\(b^{2} \left (-\frac {\cos \left (4 b x +4 a \right )}{16 b^{2} x^{2}}+\frac {\sin \left (4 b x +4 a \right )}{4 b x}+\sinIntegral \left (4 b x \right ) \sin \left (4 a \right )-\cosineIntegral \left (4 b x \right ) \cos \left (4 a \right )-\frac {\cos \left (2 b x +2 a \right )}{4 b^{2} x^{2}}+\frac {\sin \left (2 b x +2 a \right )}{2 b x}+\sinIntegral \left (2 b x \right ) \sin \left (2 a \right )-\cosineIntegral \left (2 b x \right ) \cos \left (2 a \right )-\frac {3}{16 x^{2} b^{2}}\right )\)	\(124\)
default	\(b^{2} \left (-\frac {\cos \left (4 b x +4 a \right )}{16 b^{2} x^{2}}+\frac {\sin \left (4 b x +4 a \right )}{4 b x}+\sinIntegral \left (4 b x \right ) \sin \left (4 a \right )-\cosineIntegral \left (4 b x \right ) \cos \left (4 a \right )-\frac {\cos \left (2 b x +2 a \right )}{4 b^{2} x^{2}}+\frac {\sin \left (2 b x +2 a \right )}{2 b x}+\sinIntegral \left (2 b x \right ) \sin \left (2 a \right )-\cosineIntegral \left (2 b x \right ) \cos \left (2 a \right )-\frac {3}{16 x^{2} b^{2}}\right )\)	\(124\)
risch	\(-\frac {3}{16 x^{2}}-\frac {i \pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-4 i a} b^{2}}{2}+i \sinIntegral \left (4 b x \right ) {\mathrm e}^{-4 i a} b^{2}+\frac {\expIntegral \left (1, -4 i b x \right ) {\mathrm e}^{-4 i a} b^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (b x \right ) {\mathrm e}^{-2 i a} b^{2}}{2}+i \sinIntegral \left (2 b x \right ) {\mathrm e}^{-2 i a} b^{2}+\frac {\expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{-2 i a} b^{2}}{2}+\frac {b^{2} \expIntegral \left (1, -2 i b x \right ) {\mathrm e}^{2 i a}}{2}+\frac {b^{2} \expIntegral \left (1, -4 i b x \right ) {\mathrm e}^{4 i a}}{2}-\frac {\cos \left (4 b x +4 a \right )}{16 x^{2}}+\frac {b \sin \left (4 b x +4 a \right )}{4 x}-\frac {\cos \left (2 b x +2 a \right )}{4 x^{2}}+\frac {b \sin \left (2 b x +2 a \right )}{2 x}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4/x^3,x,method=_RETURNVERBOSE)

[Out]

b^2*(-1/16*cos(4*b*x+4*a)/b^2/x^2+1/4*sin(4*b*x+4*a)/b/x+Si(4*b*x)*sin(4*a)-Ci(4*b*x)*cos(4*a)-1/4*cos(2*b*x+2
*a)/b^2/x^2+1/2*sin(2*b*x+2*a)/b/x+Si(2*b*x)*sin(2*a)-Ci(2*b*x)*cos(2*a)-3/16/x^2/b^2)

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Maxima [C] Result contains complex when optimal does not.
time = 0.38, size = 790, normalized size = 8.78 \begin {gather*} -\frac {{\left ({\left ({\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{3} + {\left ({\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )^{3} + 2 \, {\left (2 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} + 2 \, {\left (-i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + {\left (2 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3\right )} \sin \left (2 \, a\right )^{2} + 2 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3 \, \cos \left (2 \, a\right )^{2} + 2 \, {\left ({\left (-i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} \cos \left (4 \, a\right )^{2} + {\left (4 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{3} + 4 \, {\left (-i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )^{3} + 2 \, {\left (2 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 3\right )} \sin \left (2 \, a\right )^{2} + {\left ({\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right ) + 4 \, {\left (E_{3}\left (2 i \, b x\right ) + E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right ) + 6 \, \cos \left (2 \, a\right )^{2} + 4 \, {\left ({\left (-i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} - i \, E_{3}\left (2 i \, b x\right ) + i \, E_{3}\left (-2 i \, b x\right )\right )} \sin \left (2 \, a\right )\right )} \sin \left (4 \, a\right )^{2} + {\left ({\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (E_{3}\left (4 i \, b x\right ) + E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right ) + {\left ({\left ({\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{2} + {\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \cos \left (2 \, a\right )^{2} + {\left (-i \, E_{3}\left (4 i \, b x\right ) + i \, E_{3}\left (-4 i \, b x\right )\right )} \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )\right )} b^{2}}{32 \, {\left ({\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 )}^{2} + {\left (a^{2} \cos \left (2 \, a\right )^{2} + a^{2} \sin \left (2 \, a\right )^{2}\right )} \cos \left (4 \, a\right )^{2} + {\left (a^{2} \cos \left (2 \, a\right )^{2} + a^{2} \sin \left (2 \, a\right )^{2}\right )} \sin \left (4 \, a\right )^{2} - 2 \, {\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}\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^3,x, algorithm="maxima")

[Out]

-1/32*(((exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(3, 4*I*b*x) +
exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^3 + ((-I*exp_integral_e(3, 4*I*b*x) + I*exp_integral_e(3, -4
*I*b*x))*cos(2*a)^2 + (-I*exp_integral_e(3, 4*I*b*x) + I*exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*sin(4*a)^3 +
2*(2*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a)^3 + 2*(-I*exp_integral_e(3, 2*I*b*x)
+ I*exp_integral_e(3, -2*I*b*x))*sin(2*a)^3 + (2*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*c
os(2*a) + 3)*sin(2*a)^2 + 2*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 3*cos(2*a)^2
+ 2*((-I*exp_integral_e(3, 2*I*b*x) + I*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 - I*exp_integral_e(3, 2*I*b*x
) + I*exp_integral_e(3, -2*I*b*x))*sin(2*a))*cos(4*a)^2 + (4*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -
2*I*b*x))*cos(2*a)^3 + 4*(-I*exp_integral_e(3, 2*I*b*x) + I*exp_integral_e(3, -2*I*b*x))*sin(2*a)^3 + 2*(2*(ex
p_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 3)*sin(2*a)^2 + ((exp_integral_e(3, 4*I*b*x
) + exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*sin(2
*a)^2)*cos(4*a) + 4*(exp_integral_e(3, 2*I*b*x) + exp_integral_e(3, -2*I*b*x))*cos(2*a) + 6*cos(2*a)^2 + 4*((-
I*exp_integral_e(3, 2*I*b*x) + I*exp_integral_e(3, -2*I*b*x))*cos(2*a)^2 - I*exp_integral_e(3, 2*I*b*x) + I*ex
p_integral_e(3, -2*I*b*x))*sin(2*a))*sin(4*a)^2 + ((exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*
cos(2*a)^2 + (exp_integral_e(3, 4*I*b*x) + exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*cos(4*a) + (((-I*exp_integ
ral_e(3, 4*I*b*x) + I*exp_integral_e(3, -4*I*b*x))*cos(2*a)^2 + (-I*exp_integral_e(3, 4*I*b*x) + I*exp_integra
l_e(3, -4*I*b*x))*sin(2*a)^2)*cos(4*a)^2 + (-I*exp_integral_e(3, 4*I*b*x) + I*exp_integral_e(3, -4*I*b*x))*cos
(2*a)^2 + (-I*exp_integral_e(3, 4*I*b*x) + I*exp_integral_e(3, -4*I*b*x))*sin(2*a)^2)*sin(4*a))*b^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)^2 + (a^2*cos(2*a)^2 + a^2*sin(2
*a)^2)*cos(4*a)^2 + (a^2*cos(2*a)^2 + a^2*sin(2*a)^2)*sin(4*a)^2 - 2*((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)*(b*x + a))

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Fricas [A]
time = 0.37, size = 130, normalized size = 1.44 \begin {gather*} \frac {4 \, b x \cos \left (b x + a\right )^{3} \sin \left (b x + a\right ) + 2 \, b^{2} x^{2} \sin \left (4 \, a\right ) \operatorname {Si}\left (4 \, b x\right ) + 2 \, b^{2} x^{2} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x\right ) - \cos \left (b x + a\right )^{4} - {\left (b^{2} x^{2} \operatorname {Ci}\left (4 \, b x\right ) + b^{2} x^{2} \operatorname {Ci}\left (-4 \, b x\right )\right )} \cos \left (4 \, a\right ) - {\left (b^{2} x^{2} \operatorname {Ci}\left (2 \, b x\right ) + b^{2} x^{2} \operatorname {Ci}\left (-2 \, b x\right )\right )} \cos \left (2 \, a\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4/x^3,x, algorithm="fricas")

[Out]

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

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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^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**4/x**3,x)

[Out]

Integral(cos(a + b*x)**4/x**3, x)

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.50, size = 3920, normalized size = 43.56 \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^3,x, algorithm="giac")

[Out]

1/8*(4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_par
t(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))
*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)
^2*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) -
8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) + 16*b^2*x^2*sin_integral(
2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b
*x)^2*tan(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2
+ 16*b^2*x^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(
4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*
tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_par
t(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*
x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x
^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x)
)*tan(2*b*x)^2*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2
+ 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integra
l(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)^
2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos
_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(2*
a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a)^2 + 8*b^2*x^2*imag_part
(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^
2*tan(b*x)^2*tan(2*a) + 16*b^2*x^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(2*a) + 8*b^2*x^2*imag_part(
cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*t
an(b*x)^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(2*b*x)^2*tan(b*x)^2*tan(a) + 8*b^2*x^2*imag_part(cos_int
egral(2*b*x))*tan(2*b*x)^2*tan(2*a)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)
^2*tan(a) + 16*b^2*x^2*sin_integral(2*b*x)*tan(2*b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(2
*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) - 8*b^2*x^2*imag_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(2*a)^2*tan(a) +
16*b^2*x^2*sin_integral(2*b*x)*tan(b*x)^2*tan(2*a)^2*tan(a) + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(2*
b*x)^2*tan(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(2*a)*tan(a)^2 + 16*b^2*x
^2*sin_integral(4*b*x)*tan(2*b*x)^2*tan(2*a)*tan(a)^2 + 8*b^2*x^2*imag_part(cos_integral(4*b*x))*tan(b*x)^2*ta
n(2*a)*tan(a)^2 - 8*b^2*x^2*imag_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)*tan(a)^2 + 16*b^2*x^2*sin_inte
gral(4*b*x)*tan(b*x)^2*tan(2*a)*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(b*x)^2 -
4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*t
an(2*b*x)^2*tan(b*x)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(b*x)^2 + 4*b^2*x^2*real_pa
rt(cos_integral(4*b*x))*tan(2*b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(2*b*x)^2*tan(2*
a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_part(cos_integral(-4
*b*x))*tan(2*b*x)^2*tan(2*a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*re
al_part(cos_integral(2*b*x))*tan(b*x)^2*tan(2*a)^2 - 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(b*x)^2*tan(
2*a)^2 + 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(2*a)^2 - 8*b*x*tan(2*b*x)^2*tan(b*x)^2*tan(2
*a)^2*tan(a) - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integr
al(2*b*x))*tan(2*b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*b*x)^2*tan(a)^2 - 4*b^2*x^2
*real_part(cos_integral(-4*b*x))*tan(2*b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(b*x)^2*t
an(a)^2 + 4*b^2*x^2*real_part(cos_integral(2*b*x))*tan(b*x)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b
*x))*tan(b*x)^2*tan(a)^2 - 4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(b*x)^2*tan(a)^2 - 4*b*x*tan(2*b*x)^2*
tan(b*x)^2*tan(2*a)*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(4*b*x))*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_p
art(cos_integral(2*b*x))*tan(2*a)^2*tan(a)^2 + 4*b^2*x^2*real_part(cos_integral(-2*b*x))*tan(2*a)^2*tan(a)^2 +
4*b^2*x^2*real_part(cos_integral(-4*b*x))*tan(...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^4}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^4/x^3,x)

[Out]

int(cos(a + b*x)^4/x^3, x)

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