3.1.26 \(\int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx\) [26]

Optimal. Leaf size=233 \[ \frac {2 \cos (c+d x)}{b^2 d^3}-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}+\frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^6} \]

[Out]

a^4*d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/b^6+2*cos(d*x+c)/b^2/d^3-3*a^2*cos(d*x+c)/b^4/d+2*a*x*cos(d*x+c)/b^3/d-x^2*c
os(d*x+c)/b^2/d-4*a^3*cos(-c+a*d/b)*Si(a*d/b+d*x)/b^5+4*a^3*Ci(a*d/b+d*x)*sin(-c+a*d/b)/b^5+a^4*d*Si(a*d/b+d*x
)*sin(-c+a*d/b)/b^6-2*a*sin(d*x+c)/b^3/d^2+2*x*sin(d*x+c)/b^2/d^2-a^4*sin(d*x+c)/b^5/(b*x+a)

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Rubi [A]
time = 0.36, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {6874, 2718, 3377, 2717, 3378, 3384, 3380, 3383} \begin {gather*} \frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^6}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cos (c+d x)}{b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 a x \cos (c+d x)}{b^3 d}+\frac {2 \cos (c+d x)}{b^2 d^3}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {x^2 \cos (c+d x)}{b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

(2*Cos[c + d*x])/(b^2*d^3) - (3*a^2*Cos[c + d*x])/(b^4*d) + (2*a*x*Cos[c + d*x])/(b^3*d) - (x^2*Cos[c + d*x])/
(b^2*d) + (a^4*d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/b^6 - (4*a^3*CosIntegral[(a*d)/b + d*x]*Sin[c -
(a*d)/b])/b^5 - (2*a*Sin[c + d*x])/(b^3*d^2) + (2*x*Sin[c + d*x])/(b^2*d^2) - (a^4*Sin[c + d*x])/(b^5*(a + b*x
)) - (4*a^3*Cos[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/b^5 - (a^4*d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d
*x])/b^6

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

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 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {x^4 \sin (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac {3 a^2 \sin (c+d x)}{b^4}-\frac {2 a x \sin (c+d x)}{b^3}+\frac {x^2 \sin (c+d x)}{b^2}+\frac {a^4 \sin (c+d x)}{b^4 (a+b x)^2}-\frac {4 a^3 \sin (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac {\left (3 a^2\right ) \int \sin (c+d x) \, dx}{b^4}-\frac {\left (4 a^3\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{b^4}+\frac {a^4 \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac {(2 a) \int x \sin (c+d x) \, dx}{b^3}+\frac {\int x^2 \sin (c+d x) \, dx}{b^2}\\ &=-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {(2 a) \int \cos (c+d x) \, dx}{b^3 d}+\frac {2 \int x \cos (c+d x) \, dx}{b^2 d}+\frac {\left (a^4 d\right ) \int \frac {\cos (c+d x)}{a+b x} \, dx}{b^5}-\frac {\left (4 a^3 \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}-\frac {\left (4 a^3 \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^4}\\ &=-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}-\frac {4 a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {2 \int \sin (c+d x) \, dx}{b^2 d^2}+\frac {\left (a^4 d \cos \left (c-\frac {a d}{b}\right )\right ) \int \frac {\cos \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^5}-\frac {\left (a^4 d \sin \left (c-\frac {a d}{b}\right )\right ) \int \frac {\sin \left (\frac {a d}{b}+d x\right )}{a+b x} \, dx}{b^5}\\ &=\frac {2 \cos (c+d x)}{b^2 d^3}-\frac {3 a^2 \cos (c+d x)}{b^4 d}+\frac {2 a x \cos (c+d x)}{b^3 d}-\frac {x^2 \cos (c+d x)}{b^2 d}+\frac {a^4 d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{b^6}-\frac {4 a^3 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{b^5}-\frac {2 a \sin (c+d x)}{b^3 d^2}+\frac {2 x \sin (c+d x)}{b^2 d^2}-\frac {a^4 \sin (c+d x)}{b^5 (a+b x)}-\frac {4 a^3 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {a^4 d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{b^6}\\ \end {align*}

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Mathematica [A]
time = 0.65, size = 177, normalized size = 0.76 \begin {gather*} \frac {a^3 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (a d \cos \left (c-\frac {a d}{b}\right )-4 b \sin \left (c-\frac {a d}{b}\right )\right )-\frac {b \left (b (a+b x) \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (-2+d^2 x^2\right )\right ) \cos (c+d x)+d \left (2 a^2 b^2+a^4 d^2-2 b^4 x^2\right ) \sin (c+d x)\right )}{d^3 (a+b x)}-a^3 \left (4 b \cos \left (c-\frac {a d}{b}\right )+a d \sin \left (c-\frac {a d}{b}\right )\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b^6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4*Sin[c + d*x])/(a + b*x)^2,x]

[Out]

(a^3*CosIntegral[d*(a/b + x)]*(a*d*Cos[c - (a*d)/b] - 4*b*Sin[c - (a*d)/b]) - (b*(b*(a + b*x)*(3*a^2*d^2 - 2*a
*b*d^2*x + b^2*(-2 + d^2*x^2))*Cos[c + d*x] + d*(2*a^2*b^2 + a^4*d^2 - 2*b^4*x^2)*Sin[c + d*x]))/(d^3*(a + b*x
)) - a^3*(4*b*Cos[c - (a*d)/b] + a*d*Sin[c - (a*d)/b])*SinIntegral[d*(a/b + x)])/b^6

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1215\) vs. \(2(236)=472\).
time = 0.24, size = 1216, normalized size = 5.22

method	result	size
risch	\(-\frac {i \left (-2 i b^{6} d^{5} x^{5}+4 i a \,b^{5} d^{5} x^{4}-6 i b^{6} c \,d^{4} x^{4}-8 i a^{3} b^{3} d^{5} x^{2}+10 i a^{4} b^{2} d^{5} x -24 i a^{3} b^{3} c \,d^{4} x +4 i b^{6} d^{3} x^{3}+12 i a^{5} b \,d^{5}-18 i a^{4} b^{2} c \,d^{4}+12 i b^{6} c \,d^{2} x^{2}-12 i a^{2} b^{4} d^{3} x +24 i a \,b^{5} c \,d^{2} x -8 i a^{3} b^{3} d^{3}+12 i a^{2} b^{4} c \,d^{2}\right ) \cos \left (d x +c \right )}{2 d^{4} b^{5} \left (b x +a \right ) \left (-d x b -d a \right ) \left (-d x b +2 d a -3 c b \right )}+\frac {\left (-2 a^{4} b^{2} d^{6} x^{2}+4 b^{6} d^{4} x^{4}+2 a^{5} b \,d^{6} x -6 a^{4} b^{2} c \,d^{5} x -4 a \,b^{5} d^{4} x^{3}+12 b^{6} c \,d^{3} x^{3}+4 a^{6} d^{6}-6 a^{5} b c \,d^{5}-12 a^{2} b^{4} d^{4} x^{2}+12 a \,b^{5} c \,d^{3} x^{2}+4 a^{3} b^{3} d^{4} x -12 a^{2} b^{4} c \,d^{3} x +8 a^{4} b^{2} d^{4}-12 a^{3} b^{3} c \,d^{3}\right ) \sin \left (d x +c \right )}{2 d^{4} b^{5} \left (b x +a \right ) \left (-d x b -d a \right ) \left (-d x b +2 d a -3 c b \right )}-\frac {2 i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {d \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}+\frac {2 i \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {d \cos \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}-\frac {2 \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}+\frac {i d \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, -\frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}-\frac {2 \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{3}}{b^{5}}-\frac {i d \sin \left (\frac {d a -c b}{b}\right ) \expIntegral \left (1, \frac {i d \left (b x +a \right )}{b}\right ) a^{4}}{2 b^{6}}\)	\(734\)
derivativedivides	\(\text {Expression too large to display}\)	\(1216\)
default	\(\text {Expression too large to display}\)	\(1216\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^5*(d^2*c^4*(-sin(d*x+c)/(d*a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)
/b)*cos((a*d-b*c)/b)/b)/b)+4*d^2*c^3*(a*d-b*c)/b*(-sin(d*x+c)/(d*a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin
((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-4*d^2*c^3/b*(Si(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/
b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)+6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*d^2*c^2/b^2*(-sin(d*x+c)/(d*a-c*b
+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-12/b^2*(a
*d-b*c)*d^2*c^2*(Si(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)-6*d^2*c^2/
b^2*cos(d*x+c)+4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d^2*c/b^3*(-sin(d*x+c)/(d*a-c*b+b*(d*x+c))/b+(S
i(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b)/b)-12/b^3*(a^2*d^2-2*a*b*c*d
+b^2*c^2)*d^2*c*(Si(d*x+c+(a*d-b*c)/b)*cos((a*d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)+4*(2*a*d-2
*b*c-b)*d^2*c/b^3*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c
^4)*d^2/b^4*(-sin(d*x+c)/(d*a-c*b+b*(d*x+c))/b+(Si(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b+Ci(d*x+c+(a*d-b*c)/b)
*cos((a*d-b*c)/b)/b)/b)-4/b^4*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*d^2*(Si(d*x+c+(a*d-b*c)/b)*cos((a*
d-b*c)/b)/b-Ci(d*x+c+(a*d-b*c)/b)*sin((a*d-b*c)/b)/b)+(3*a^2*d^2-6*a*b*c*d+3*b^2*c^2-2*a*b*d+2*b^2*c+b^2)*d^2/
b^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*((2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 +
2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^3*(-
I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(-I*exp_i
ntegral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c - a*d)/b
) + (2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 +
2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2 + (a^
3*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(exp_integ
ral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d)/b))*cos
(d*x + c)^2 + 2*((2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*c
os(c)^2 + 2*a^2*b*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2
+ (a^3*(-I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*
(-I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) + I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*cos(-(b*c
- a*d)/b) + (2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*c
os(c)^2 + 2*a^2*b*(I*exp_integral_e(3, (I*b*d*x + I*a*d)/b) - I*exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c
)^2 + (a^3*(exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*cos(c)^2 + a^3*(
exp_integral_e(3, (I*b*d*x + I*a*d)/b) + exp_integral_e(3, -(I*b*d*x + I*a*d)/b))*sin(c)^2)*d)*sin(-(b*c - a*d
)/b))*sin(d*x + c)^2 - ((b^3*d^2*x^4*cos(c) + 2*b^3*d*x^3*sin(c) + 2*(a*b^2*d*sin(c) - b^3*cos(c))*x^2 - 2*(a^
2*b*d*sin(c) + 2*a*b^2*cos(c))*x)*cos(d*x + c)^2 + (b^3*d^2*x^4*cos(c) + 2*b^3*d*x^3*sin(c) + 2*(a*b^2*d*sin(c
) - b^3*cos(c))*x^2 - 2*(a^2*b*d*sin(c) + 2*a*b^2*cos(c))*x)*sin(d*x + c)^2)*cos(d*x + 2*c) - ((b^3*cos(c)^2 +
b^3*sin(c)^2)*d^2*x^4 - 2*(b^3*cos(c)^2 + b^3*sin(c)^2)*x^2 - 4*(a*b^2*cos(c)^2 + a*b^2*sin(c)^2)*x)*cos(d*x
+ c) + 2*(((a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*d^5*x^2 + 2*(a^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^5*x + (a^
5*b^3*cos(c)^2 + a^5*b^3*sin(c)^2)*d^5)*cos(d*x + c)^2 + ((a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*d^5*x^2 + 2*(a
^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^5*x + (a^5*b^3*cos(c)^2 + a^5*b^3*sin(c)^2)*d^5)*sin(d*x + c)^2)*integra
te(x*cos(d*x + c)/(b^5*d^3*x^3 + 3*a*b^4*d^3*x^2 + 3*a^2*b^3*d^3*x + a^3*b^2*d^3), x) + 2*(((a^3*b^5*cos(c)^2
+ a^3*b^5*sin(c)^2)*d^5*x^2 + 2*(a^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^5*x + (a^5*b^3*cos(c)^2 + a^5*b^3*sin(
c)^2)*d^5)*cos(d*x + c)^2 + ((a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*d^5*x^2 + 2*(a^4*b^4*cos(c)^2 + a^4*b^4*sin
(c)^2)*d^5*x + (a^5*b^3*cos(c)^2 + a^5*b^3*sin(c)^2)*d^5)*sin(d*x + c)^2)*integrate(x*cos(d*x + c)/((b^5*d^3*x
^3 + 3*a*b^4*d^3*x^2 + 3*a^2*b^3*d^3*x + a^3*b^2*d^3)*cos(d*x + c)^2 + (b^5*d^3*x^3 + 3*a*b^4*d^3*x^2 + 3*a^2*
b^3*d^3*x + a^3*b^2*d^3)*sin(d*x + c)^2), x) - 2*(((a^2*b^6*cos(c)^2 + a^2*b^6*sin(c)^2)*d^4*x^2 + 2*(a^3*b^5*
cos(c)^2 + a^3*b^5*sin(c)^2)*d^4*x + (a^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^4)*cos(d*x + c)^2 + ((a^2*b^6*cos
(c)^2 + a^2*b^6*sin(c)^2)*d^4*x^2 + 2*(a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*d^4*x + (a^4*b^4*cos(c)^2 + a^4*b^
4*sin(c)^2)*d^4)*sin(d*x + c)^2)*integrate(x*sin(d*x + c)/(b^5*d^3*x^3 + 3*a*b^4*d^3*x^2 + 3*a^2*b^3*d^3*x + a
^3*b^2*d^3), x) - 2*(((a^2*b^6*cos(c)^2 + a^2*b^6*sin(c)^2)*d^4*x^2 + 2*(a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*
d^4*x + (a^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^4)*cos(d*x + c)^2 + ((a^2*b^6*cos(c)^2 + a^2*b^6*sin(c)^2)*d^4
*x^2 + 2*(a^3*b^5*cos(c)^2 + a^3*b^5*sin(c)^2)*d^4*x + (a^4*b^4*cos(c)^2 + a^4*b^4*sin(c)^2)*d^4)*sin(d*x + c)
^2)*integrate(x*sin(d*x + c)/((b^5*d^3*x^3 + 3*a*b^4*d^3*x^2 + 3*a^2*b^3*d^3*x + a^3*b^2*d^3)*cos(d*x + c)^2 +
(b^5*d^3*x^3 + 3*a*b^4*d^3*x^2 + 3*a^2*b^3*d^3*x + a^3*b^2*d^3)*sin(d*x + c)^2), x) - ((b^3*d^2*x^4*sin(c) -
2*b^3*d*x^3*cos(c) - 2*(a*b^2*d*cos(c) + b^3*sin(c))*x^2 + 2*(a^2*b*d*cos(c) - 2*a*b^2*sin(c))*x)*cos(d*x + c)
^2 + (b^3*d^2*x^4*sin(c) - 2*b^3*d*x^3*cos(c) - 2*(a*b^2*d*cos(c) + b^3*sin(c))*x^2 + 2*(a^2*b*d*cos(c) - 2*a*
b^2*sin(c))*x)*sin(d*x + c)^2)*sin(d*x + 2*c) + 2*((b^3*cos(c)^2 + b^3*sin(c)^2)*d*x^3 + (a*b^2*cos(c)^2 + a*b
^2*sin(c)^2)*d*x^2 - (a^2*b*cos(c)^2 + a^2*b*sin(c)^2)*d*x)*sin(d*x + c))/(((b^5*cos(c)^2 + b^5*sin(c)^2)*d^3*
x^2 + 2*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x + (a^2*b^3*cos(c)^2 + a^2*b^3*sin(c)^2)*d^3)*cos(d*x + c)^2 +
((b^5*cos(c)^2 + b^5*sin(c)^2)*d^3*x^2 + 2*(a*b^4*cos(c)^2 + a*b^4*sin(c)^2)*d^3*x + (a^2*b^3*cos(c)^2 + a^2*b
^3*sin(c)^2)*d^3)*sin(d*x + c)^2)

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Fricas [A]
time = 0.36, size = 357, normalized size = 1.53 \begin {gather*} -\frac {2 \, {\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} - 2 \, a b^{4} + {\left (a^{2} b^{3} d^{2} - 2 \, b^{5}\right )} x\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 8 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (a^{4} b d^{3} - 2 \, b^{5} d x^{2} + 2 \, a^{2} b^{3} d\right )} \sin \left (d x + c\right ) - 2 \, {\left (2 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (a^{3} b^{2} d^{3} x + a^{4} b d^{3}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + {\left (a^{4} b d^{4} x + a^{5} d^{4}\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (b^{7} d^{3} x + a b^{6} d^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*(b^5*d^2*x^3 - a*b^4*d^2*x^2 + 3*a^3*b^2*d^2 - 2*a*b^4 + (a^2*b^3*d^2 - 2*b^5)*x)*cos(d*x + c) - ((a^4
*b*d^4*x + a^5*d^4)*cos_integral((b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*cos_integral(-(b*d*x + a*d)/b) - 8
*(a^3*b^2*d^3*x + a^4*b*d^3)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) + 2*(a^4*b*d^3 - 2*b^5*d*x^2 +
2*a^2*b^3*d)*sin(d*x + c) - 2*(2*(a^3*b^2*d^3*x + a^4*b*d^3)*cos_integral((b*d*x + a*d)/b) + 2*(a^3*b^2*d^3*x
+ a^4*b*d^3)*cos_integral(-(b*d*x + a*d)/b) + (a^4*b*d^4*x + a^5*d^4)*sin_integral((b*d*x + a*d)/b))*sin(-(b*
c - a*d)/b))/(b^7*d^3*x + a*b^6*d^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*sin(d*x+c)/(b*x+a)**2,x)

[Out]

Integral(x**4*sin(c + d*x)/(a + b*x)**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1973 vs. \(2 (236) = 472\).
time = 3.52, size = 1973, normalized size = 8.47 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a^4*b*c*d^4*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a)*a^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^4*sin(-(b*c - a*d)/
b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a^4*b*c*d^4*sin(-(b*c - a*d)/
b)*sin_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^5*d^5*sin(-(b*c - a*d)/b)*s
in_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 4*(b*x + a)*a^3*b*(b*c/(b*x + a)
- a*d/(b*x + a) + d)*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c
- a*d)/b) - 4*a^3*b^2*c*d^3*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-
(b*c - a*d)/b) + 4*a^4*b*d^4*cos_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-
(b*c - a*d)/b) - 4*(b*x + a)*a^3*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(-(b*c - a*d)/b)*sin_integral(((
b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + 4*a^3*b^2*c*d^3*cos(-(b*c - a*d)/b)*sin_integra
l(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - 4*a^4*b*d^4*cos(-(b*c - a*d)/b)*sin_integra
l(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^4*b*d^4*sin(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) - (b*x + a)^3*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^3*cos(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) + 3*(b*x + a)^2*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*c*cos(-(b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d)/b) - 3*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c^2*cos(-(b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d)/b) + b^5*c^3*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + (b*x + a)^2*a*b^2
*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)*a
*b^3*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a*b^4*c^2
*d*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - (b*x + a)*a^2*b^2*(b*c/(b*x + a) - a*d/(b*x + a) +
d)*d^2*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a^2*b^3*c*d^2*cos(-(b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d)/b) - 3*a^3*b^2*d^3*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*(b*x + a)^2*b^3
*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 4*(b*x + a)*b^4
*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) - 2*b^5*c^2*sin(-
(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a^2*b^3*d^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a)
+ d)/b) + 2*(b*x + a)*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d)*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) +
d)/b) - 2*b^5*c*cos(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + 2*a*b^4*d*cos(-(b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)*b^8*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2 - b^9*c*d^2 + a*b^8*d^
3)*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*sin(c + d*x))/(a + b*x)^2,x)

[Out]

int((x^4*sin(c + d*x))/(a + b*x)^2, x)

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