3.1.37 \(\int \frac {\sin (c+d x)}{x (a+b x)^3} \, dx\) [37]

Optimal. Leaf size=261 \[ \frac {d \cos (c+d x)}{2 a b (a+b x)}-\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^2 b}+\frac {\text {Ci}(d x) \sin (c)}{a^3}-\frac {\text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}+\frac {d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 a b^2}+\frac {\sin (c+d x)}{2 a (a+b x)^2}+\frac {\sin (c+d x)}{a^2 (a+b x)}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 a b^2}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2 b} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6874, 3384, 3380, 3383, 3378} \begin {gather*} -\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {\sin (c) \text {CosIntegral}(d x)}{a^3}+\frac {\cos (c) \text {Si}(d x)}{a^3}-\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2 b}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2 b}+\frac {\sin (c+d x)}{a^2 (a+b x)}+\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 a b^2}+\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 a b^2}+\frac {\sin (c+d x)}{2 a (a+b x)^2}+\frac {d \cos (c+d x)}{2 a b (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x)^3),x]

[Out]

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

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

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Mathematica [A]
time = 0.62, size = 449, normalized size = 1.72 \begin {gather*} \frac {a^3 b d \cos (c+d x)+a^2 b^2 d x \cos (c+d x)+2 b^2 (a+b x)^2 \text {Ci}(d x) \sin (c)+(a+b x)^2 \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (-2 a b d \cos \left (c-\frac {a d}{b}\right )+\left (-2 b^2+a^2 d^2\right ) \sin \left (c-\frac {a d}{b}\right )\right )+3 a^2 b^2 \sin (c+d x)+2 a b^3 x \sin (c+d x)+2 a^2 b^2 \cos (c) \text {Si}(d x)+4 a b^3 x \cos (c) \text {Si}(d x)+2 b^4 x^2 \cos (c) \text {Si}(d x)-2 a^2 b^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a^4 d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-4 a b^3 x \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d^2 x \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-2 b^4 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+a^2 b^2 d^2 x^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a^3 b d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+4 a^2 b^2 d x \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+2 a b^3 d x^2 \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{2 a^3 b^2 (a+b x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.14, size = 359, normalized size = 1.38

method	result	size
derivativedivides	\(-\frac {d b \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{3}}-\frac {d^{2} b \left (-\frac {\sin \left (d x +c \right )}{2 \left (d a -c b +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}-\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{a}-\frac {b \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a^{3}}\)	\(359\)
default	\(-\frac {d b \left (-\frac {\sin \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}+\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}+\frac {\sinIntegral \left (d x \right ) \cos \left (c \right )+\cosineIntegral \left (d x \right ) \sin \left (c \right )}{a^{3}}-\frac {d^{2} b \left (-\frac {\sin \left (d x +c \right )}{2 \left (d a -c b +b \left (d x +c \right )\right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (d a -c b +b \left (d x +c \right )\right ) b}-\frac {\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right )}{a}-\frac {b \left (\frac {\sinIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\cosineIntegral \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a^{3}}\)	\(359\)
risch	\(\frac {d \,{\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{2} b}+\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right )}{2 a^{3}}+\frac {i {\mathrm e}^{i c} \expIntegral \left (1, -i d x \right )}{2 a^{3}}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d x \right )}{2 a^{3}}+\frac {{\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right ) d}{2 a^{2} b}+\frac {i d^{2} {\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{4 b^{2} a}-\frac {i {\mathrm e}^{-\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, -i d x -i c -\frac {i a d -i b c}{b}\right )}{2 a^{3}}-\frac {{\mathrm e}^{-i c} \pi \,\mathrm {csgn}\left (d x \right )}{2 a^{3}}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d x \right )}{a^{3}}-\frac {i {\mathrm e}^{\frac {i \left (d a -c b \right )}{b}} \expIntegral \left (1, i d x +i c +\frac {i \left (d a -c b \right )}{b}\right ) d^{2}}{4 a \,b^{2}}-\frac {b^{2} \cos \left (d x +c \right ) d^{3} x^{3}}{2 a \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {3 b \cos \left (d x +c \right ) d^{3} x^{2}}{2 \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {3 a \cos \left (d x +c \right ) d^{3} x}{2 \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {a^{2} \cos \left (d x +c \right ) d^{3}}{2 b \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {\sin \left (d x +c \right ) b^{3} d^{2} x^{3}}{a^{2} \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {7 \sin \left (d x +c \right ) b^{2} d^{2} x^{2}}{2 a \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {4 \sin \left (d x +c \right ) b \,d^{2} x}{\left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}-\frac {3 a \sin \left (d x +c \right ) d^{2}}{2 \left (b x +a \right )^{2} \left (-d^{2} x^{2} b^{2}-2 a b \,d^{2} x -d^{2} a^{2}\right )}\)	\(798\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(sin(d*x + c)/((b*x + a)^3*x), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 532 vs. \(2 (259) = 518\).
time = 0.40, size = 532, normalized size = 2.04 \begin {gather*} \frac {4 \, {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \cos \left (c\right ) \operatorname {Si}\left (d x\right ) + 2 \, {\left (a^{2} b^{2} d x + a^{3} b d\right )} \cos \left (d x + c\right ) - 2 \, {\left ({\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + 2 \, {\left (2 \, a b^{3} x + 3 \, a^{2} b^{2}\right )} \sin \left (d x + c\right ) + 2 \, {\left ({\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \operatorname {Ci}\left (d x\right ) + {\left (b^{4} x^{2} + 2 \, a b^{3} x + a^{2} b^{2}\right )} \operatorname {Ci}\left (-d x\right )\right )} \sin \left (c\right ) - {\left ({\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{4} d^{2} - 2 \, a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} - 2 \, b^{4}\right )} x^{2} + 2 \, {\left (a^{3} b d^{2} - 2 \, a b^{3}\right )} x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 4 \, {\left (a b^{3} d x^{2} + 2 \, a^{2} b^{2} d x + a^{3} b d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (a^{3} b^{4} x^{2} + 2 \, a^{4} b^{3} x + a^{5} b^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.62, size = 17806, normalized size = 68.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(a^2*b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b
^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^2*
x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^2*x^2*real_part(c
os_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*b^2*d^2*x^2*real_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a^2*b^2*d^2*x^2*real_part(cos_integral(d*x + a
*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a^3*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a^3*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2
*tan(1/2*a*d/b)^2 - 2*a*b^3*d*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d
/b)^2 - 2*a*b^3*d*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a
^3*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b^2*d^2*x^2*imag_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d
/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2
+ 4*a^2*b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*b^2
*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*b^2*d^2*x^2*si
n_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b^3*d*x^2*imag_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a*b^3*d*x^2*imag_part(cos_integral(-d*x - a*d/b))*t
an(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a^3*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*
tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a^3*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*
tan(1/2*a*d/b) + 8*a*b^3*d*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - a^2*
b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^2*b^2*d^2*x^2*imag_part(c
os_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*t
an(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b^3*d*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*
tan(1/2*a*d/b)^2 + 4*a*b^3*d*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b
)^2 - 4*a^3*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a^3*b*
d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 8*a*b^3*d*x^2*sin_int
egral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + a^2*b^2*d^2*x^2*imag_part(cos_integral(d*x
+ a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + 2*a^2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^4*d^2*ima
g_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^4*x^2*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b^4*x^2*imag_part(cos_integral(d*x))*tan(1/
2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^4*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*
c)^2*tan(1/2*a*d/b)^2 + 2*b^4*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*
d/b)^2 + 2*b^4*x^2*imag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*a^2*b^2*d*x*
real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*a^2*b^2*d*x*real_part(co
s_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^4*x^2*sin_integral(d*x)*tan(1/2*d
*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^4*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(
1/2*a*d/b)^2 - 4*b^4*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*b^
2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a^2*b^2*d^2*x^2*real_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a^3*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*
x)^2*tan(1/2*c)^2 + 2*a^3*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b^3*
d*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b^3*d*x^2*real_part(cos_integral(
-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a^3*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/
2*c)^2 - 2*a^2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a^2*b^2*d^2*
x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a^3*b*d^2*x*imag_part(cos_integral
(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x)^3),x)

[Out]

int(sin(c + d*x)/(x*(a + b*x)^3), x)

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