3.2.99 \(\int \sin (a+\frac {b}{\sqrt {c+d x}}) \, dx\) [199]

Optimal. Leaf size=94 \[ \frac {b \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b^2 \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3442, 3378, 3384, 3380, 3383} \begin {gather*} \frac {b^2 \sin (a) \text {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b/Sqrt[c + d*x]],x]

[Out]

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

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 3442

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_Symbol] :> Dist[1/(n*f), Subst[Int[x
^(1/n - 1)*(a + b*Sin[c + d*x])^p, x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p, 0] && In
tegerQ[1/n]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 99, normalized size = 1.05 \begin {gather*} \frac {b \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )+b^2 \text {Ci}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+c \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )+d x \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )+b^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b/Sqrt[c + d*x]],x]

[Out]

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

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Maple [A]
time = 0.02, size = 84, normalized size = 0.89

method	result	size
derivativedivides	\(-\frac {2 b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )}{d}\)	\(84\)
default	\(-\frac {2 b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\sinIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\cosineIntegral \left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )}{d}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b/(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.39, size = 124, normalized size = 1.32 \begin {gather*} \frac {{\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.41, size = 125, normalized size = 1.33 \begin {gather*} \frac {b^{2} \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + b^{2} \operatorname {Ci}\left (-\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + 2 \, b^{2} \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right ) + 2 \, \sqrt {d x + c} b \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/2)),x, algorithm="fricas")

[Out]

1/2*(b^2*cos_integral(b/sqrt(d*x + c))*sin(a) + b^2*cos_integral(-b/sqrt(d*x + c))*sin(a) + 2*b^2*cos(a)*sin_i
ntegral(b/sqrt(d*x + c)) + 2*sqrt(d*x + c)*b*cos((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)) + 2*(d*x + c)*sin(
(a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)**(1/2)),x)

[Out]

Integral(sin(a + b/sqrt(c + d*x)), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (84) = 168\).
time = 4.29, size = 413, normalized size = 4.39 \begin {gather*} \frac {a^{2} b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) - a^{2} b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) - \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right )}{\sqrt {d x + c}} + \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{\sqrt {d x + c}} + \frac {{\left (\sqrt {d x + c} a + b\right )}^{2} b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right )}{d x + c} - \frac {{\left (\sqrt {d x + c} a + b\right )}^{2} b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{d x + c} - a b^{3} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + \frac {{\left (\sqrt {d x + c} a + b\right )} b^{3} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{\sqrt {d x + c}} + b^{3} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{{\left (a^{2} - \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a}{\sqrt {d x + c}} + \frac {{\left (\sqrt {d x + c} a + b\right )}^{2}}{d x + c}\right )} b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b/(d*x+c)^(1/2)),x, algorithm="giac")

[Out]

(a^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - a^2*b^3*cos(a)*sin_integral(a - (sqrt
(d*x + c)*a + b)/sqrt(d*x + c)) - 2*(sqrt(d*x + c)*a + b)*a*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d
*x + c))*sin(a)/sqrt(d*x + c) + 2*(sqrt(d*x + c)*a + b)*a*b^3*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sq
rt(d*x + c))/sqrt(d*x + c) + (sqrt(d*x + c)*a + b)^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c)
)*sin(a)/(d*x + c) - (sqrt(d*x + c)*a + b)^2*b^3*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/
(d*x + c) - a*b^3*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) + (sqrt(d*x + c)*a + b)*b^3*cos((sqrt(d*x + c)*a +
b)/sqrt(d*x + c))/sqrt(d*x + c) + b^3*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))/((a^2 - 2*(sqrt(d*x + c)*a + b
)*a/sqrt(d*x + c) + (sqrt(d*x + c)*a + b)^2/(d*x + c))*b*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b/(c + d*x)^(1/2)),x)

[Out]

int(sin(a + b/(c + d*x)^(1/2)), x)

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