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

Optimal. Leaf size=31 \[ a \log (x)+\frac {1}{2} b \text {Ci}\left (d x^2\right ) \sin (c)+\frac {1}{2} b \cos (c) \text {Si}\left (d x^2\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {14, 3458, 3457, 3456} \begin {gather*} a \log (x)+\frac {1}{2} b \sin (c) \text {CosIntegral}\left (d x^2\right )+\frac {1}{2} b \cos (c) \text {Si}\left (d x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sin[c + d*x^2])/x,x]

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 3456

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3457

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3458

Int[Sin[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Sin[c], Int[Cos[d*x^n]/x, x], x] + Dist[Cos[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 29, normalized size = 0.94 \begin {gather*} a \log (x)+\frac {1}{2} b \left (\text {Ci}\left (d x^2\right ) \sin (c)+\cos (c) \text {Si}\left (d x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sin[c + d*x^2])/x,x]

[Out]

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

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Maple [A]
time = 0.04, size = 29, normalized size = 0.94

method	result	size
default	\(a \ln \left (x \right )+b \left (\frac {\cos \left (c \right ) \sinIntegral \left (d \,x^{2}\right )}{2}+\frac {\sin \left (c \right ) \cosineIntegral \left (d \,x^{2}\right )}{2}\right )\)	\(29\)
risch	\(a \ln \left (x \right )-\frac {{\mathrm e}^{-i c} \mathrm {csgn}\left (d \,x^{2}\right ) \pi b}{4}+\frac {{\mathrm e}^{-i c} \sinIntegral \left (d \,x^{2}\right ) b}{2}-\frac {i {\mathrm e}^{-i c} \expIntegral \left (1, -i d \,x^{2}\right ) b}{4}+\frac {i b \,{\mathrm e}^{i c} \expIntegral \left (1, -i d \,x^{2}\right )}{4}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.36, size = 50, normalized size = 1.61 \begin {gather*} -\frac {1}{4} \, {\left ({\left (i \, {\rm Ei}\left (i \, d x^{2}\right ) - i \, {\rm Ei}\left (-i \, d x^{2}\right )\right )} \cos \left (c\right ) - {\left ({\rm Ei}\left (i \, d x^{2}\right ) + {\rm Ei}\left (-i \, d x^{2}\right )\right )} \sin \left (c\right )\right )} b + a \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((I*Ei(I*d*x^2) - I*Ei(-I*d*x^2))*cos(c) - (Ei(I*d*x^2) + Ei(-I*d*x^2))*sin(c))*b + a*log(x)

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Fricas [A]
time = 0.36, size = 38, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, b \cos \left (c\right ) \operatorname {Si}\left (d x^{2}\right ) + a \log \left (x\right ) + \frac {1}{4} \, {\left (b \operatorname {Ci}\left (d x^{2}\right ) + b \operatorname {Ci}\left (-d x^{2}\right )\right )} \sin \left (c\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 4.92, size = 32, normalized size = 1.03 \begin {gather*} \frac {1}{2} \, b \operatorname {Ci}\left (d x^{2}\right ) \sin \left (c\right ) + \frac {1}{2} \, b \cos \left (c\right ) \operatorname {Si}\left (d x^{2}\right ) + \frac {1}{2} \, a \log \left (d x^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*cos_integral(d*x^2)*sin(c) + 1/2*b*cos(c)*sin_integral(d*x^2) + 1/2*a*log(d*x^2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} a\,\ln \left (x\right )+\frac {b\,\sin \left (c\right )\,\mathrm {cosint}\left (d\,x^2\right )}{2}+\frac {b\,\cos \left (c\right )\,\mathrm {sinint}\left (d\,x^2\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(c + d*x^2))/x,x)

[Out]

a*log(x) + (b*sin(c)*cosint(d*x^2))/2 + (b*cos(c)*sinint(d*x^2))/2

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