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

Optimal. Leaf size=145 \[ -\frac {\cos ^3(a+b x)}{d (c+d x)}-\frac {3 b \text {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}-\frac {3 b \text {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d^2}-\frac {3 b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d^2}-\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3394, 3384, 3380, 3383} \begin {gather*} -\frac {3 b \sin \left (3 a-\frac {3 b c}{d}\right ) \text {CosIntegral}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {3 b \sin \left (a-\frac {b c}{d}\right ) \text {CosIntegral}\left (\frac {b c}{d}+b x\right )}{4 d^2}-\frac {3 b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d^2}-\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}-\frac {\cos ^3(a+b x)}{d (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^3/(c + d*x)^2,x]

[Out]

-(Cos[a + b*x]^3/(d*(c + d*x))) - (3*b*CosIntegral[(3*b*c)/d + 3*b*x]*Sin[3*a - (3*b*c)/d])/(4*d^2) - (3*b*Cos
Integral[(b*c)/d + b*x]*Sin[a - (b*c)/d])/(4*d^2) - (3*b*Cos[a - (b*c)/d]*SinIntegral[(b*c)/d + b*x])/(4*d^2)
- (3*b*Cos[3*a - (3*b*c)/d]*SinIntegral[(3*b*c)/d + 3*b*x])/(4*d^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 ^3(a+b x)}{(c+d x)^2} \, dx &=-\frac {\cos ^3(a+b x)}{d (c+d x)}+\frac {(3 b) \int \left (-\frac {\sin (a+b x)}{4 (c+d x)}-\frac {\sin (3 a+3 b x)}{4 (c+d x)}\right ) \, dx}{d}\\ &=-\frac {\cos ^3(a+b x)}{d (c+d x)}-\frac {(3 b) \int \frac {\sin (a+b x)}{c+d x} \, dx}{4 d}-\frac {(3 b) \int \frac {\sin (3 a+3 b x)}{c+d x} \, dx}{4 d}\\ &=-\frac {\cos ^3(a+b x)}{d (c+d x)}-\frac {\left (3 b \cos \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \cos \left (a-\frac {b c}{d}\right )\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \sin \left (3 a-\frac {3 b c}{d}\right )\right ) \int \frac {\cos \left (\frac {3 b c}{d}+3 b x\right )}{c+d x} \, dx}{4 d}-\frac {\left (3 b \sin \left (a-\frac {b c}{d}\right )\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{c+d x} \, dx}{4 d}\\ &=-\frac {\cos ^3(a+b x)}{d (c+d x)}-\frac {3 b \text {Ci}\left (\frac {3 b c}{d}+3 b x\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{4 d^2}-\frac {3 b \text {Ci}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{4 d^2}-\frac {3 b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{4 d^2}-\frac {3 b \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b c}{d}+3 b x\right )}{4 d^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.46, size = 200, normalized size = 1.38 \begin {gather*} -\frac {3 d \cos (a+b x)+d \cos (3 (a+b x))+3 b (c+d x) \text {CosIntegral}\left (\frac {3 b (c+d x)}{d}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )+3 b (c+d x) \text {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+3 b c \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+3 b d x \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )+3 b c \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )+3 b d x \cos \left (3 a-\frac {3 b c}{d}\right ) \text {Si}\left (\frac {3 b (c+d x)}{d}\right )}{4 d^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^3/(c + d*x)^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 247, normalized size = 1.70

method	result	size
derivativedivides	\(\frac {\frac {b^{2} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {3 \left (-\frac {3 \sinIntegral \left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right ) \cos \left (\frac {-3 d a +3 b c}{d}\right )}{d}-\frac {3 \cosineIntegral \left (3 b x +3 a +\frac {-3 d a +3 b c}{d}\right ) \sin \left (\frac {-3 d a +3 b c}{d}\right )}{d}\right )}{d}\right )}{12}+\frac {3 b^{2} \left (-\frac {\cos \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}}{d}\right )}{4}}{b}\)	\(247\)
default	\(\frac {\frac {b^{2} \left (-\frac {3 \cos \left (3 b x +3 a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {3 \left (-\frac {3 \sinIntegral \left (-3 b x -3 a -\frac {3 \left (-d a +b c \right )}{d}\right ) \cos \left (\frac {-3 d a +3 b c}{d}\right )}{d}-\frac {3 \cosineIntegral \left (3 b x +3 a +\frac {-3 d a +3 b c}{d}\right ) \sin \left (\frac {-3 d a +3 b c}{d}\right )}{d}\right )}{d}\right )}{12}+\frac {3 b^{2} \left (-\frac {\cos \left (b x +a \right )}{\left (-d a +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\sinIntegral \left (-b x -a -\frac {-d a +b c}{d}\right ) \cos \left (\frac {-d a +b c}{d}\right )}{d}-\frac {\cosineIntegral \left (b x +a +\frac {-d a +b c}{d}\right ) \sin \left (\frac {-d a +b c}{d}\right )}{d}}{d}\right )}{4}}{b}\)	\(247\)
risch	\(\frac {3 i b \,{\mathrm e}^{-\frac {3 i \left (d a -b c \right )}{d}} \expIntegral \left (1, 3 i b x +3 i a -\frac {3 i \left (d a -b c \right )}{d}\right )}{8 d^{2}}+\frac {3 i b \,{\mathrm e}^{-\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, i b x +i a -\frac {i \left (d a -b c \right )}{d}\right )}{8 d^{2}}-\frac {3 i b \,{\mathrm e}^{\frac {i \left (d a -b c \right )}{d}} \expIntegral \left (1, -i b x -i a -\frac {-i a d +i b c}{d}\right )}{8 d^{2}}-\frac {3 i b \,{\mathrm e}^{\frac {3 i \left (d a -b c \right )}{d}} \expIntegral \left (1, -3 i b x -3 i a -\frac {3 \left (-i a d +i b c \right )}{d}\right )}{8 d^{2}}-\frac {3 \left (-2 d x b -2 b c \right ) \cos \left (b x +a \right )}{8 d \left (d x +c \right ) \left (-d x b -b c \right )}-\frac {\left (-2 d x b -2 b c \right ) \cos \left (3 b x +3 a \right )}{8 d \left (d x +c \right ) \left (-d x b -b c \right )}\)	\(281\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [C] Result contains complex when optimal does not.
time = 0.44, size = 304, normalized size = 2.10 \begin {gather*} -\frac {3 \, b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (E_{2}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{2}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{2} {\left (-i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (i \, E_{2}\left (\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) - i \, E_{2}\left (-\frac {3 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*b^2*(exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e(2, -(I*b*c + I*(b*x + a)*d
- I*a*d)/d))*cos(-(b*c - a*d)/d) + b^2*(exp_integral_e(2, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) + exp_integral
_e(2, -3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d))*cos(-3*(b*c - a*d)/d) + 3*b^2*(-I*exp_integral_e(2, (I*b*c + I*(
b*x + a)*d - I*a*d)/d) + I*exp_integral_e(2, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d) + b^2*(I
*exp_integral_e(2, 3*(-I*b*c - I*(b*x + a)*d + I*a*d)/d) - I*exp_integral_e(2, -3*(-I*b*c - I*(b*x + a)*d + I*
a*d)/d))*sin(-3*(b*c - a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)

________________________________________________________________________________________

Fricas [A]
time = 0.46, size = 227, normalized size = 1.57 \begin {gather*} -\frac {8 \, d \cos \left (b x + a\right )^{3} + 6 \, {\left (b d x + b c\right )} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 6 \, {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + 3 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {b d x + b c}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, {\left ({\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b d x + b c\right )} \operatorname {Ci}\left (-\frac {3 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right )}{8 \, {\left (d^{3} x + c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*d*cos(b*x + a)^3 + 6*(b*d*x + b*c)*cos(-3*(b*c - a*d)/d)*sin_integral(3*(b*d*x + b*c)/d) + 6*(b*d*x +
b*c)*cos(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) + 3*((b*d*x + b*c)*cos_integral((b*d*x + b*c)/d) + (b*d
*x + b*c)*cos_integral(-(b*d*x + b*c)/d))*sin(-(b*c - a*d)/d) + 3*((b*d*x + b*c)*cos_integral(3*(b*d*x + b*c)/
d) + (b*d*x + b*c)*cos_integral(-3*(b*d*x + b*c)/d))*sin(-3*(b*c - a*d)/d))/(d^3*x + c*d^2)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cos ^{3}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos(a + b*x)**3/(c + d*x)**2, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1000 vs. \(2 (137) = 274\).
time = 0.57, size = 1000, normalized size = 6.90 \begin {gather*} -\frac {{\left (3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, b^{3} c \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - 3 \, a b^{2} d \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \operatorname {Ci}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 3 \, b^{3} c \operatorname {Ci}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, a b^{2} d \operatorname {Ci}\left (\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) - 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) - 3 \, b^{3} c \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + 3 \, a b^{2} d \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) - 3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) - 3 \, b^{3} c \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) + 3 \, a b^{2} d \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {3 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d\right )}}{d}\right ) + 3 \, b^{2} d \cos \left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right ) + b^{2} d \cos \left (-\frac {3 \, {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (a+b\,x\right )}^3}{{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)^3/(c + d*x)^2,x)

[Out]

int(cos(a + b*x)^3/(c + d*x)^2, x)

________________________________________________________________________________________