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

3.2.99.1 Optimal result

Integrand size = 14, antiderivative size = 94 \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {b \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{d}+\frac {(c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {b^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d} \]

output

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
 

3.2.99.2 Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {b \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )+b^2 \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+c \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )+d x \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )+b^2 \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d} \]

input

Integrate[Sin[a + b/Sqrt[c + d*x]],x]
 

output

(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
 

3.2.99.3 Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3842, 3042, 3778, 3042, 3778, 25, 3042, 3784, 3042, 3780, 3783}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx\)

\(\Big \downarrow \) 3842

\(\displaystyle -\frac {2 \int (c+d x)^{3/2} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \int (c+d x)^{3/2} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}}{d}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \int (c+d x) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \int (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt {c+d x}}-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (b \int -\sqrt {c+d x} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \int \sqrt {c+d x} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \int \sqrt {c+d x} \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3784

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt {c+d x} \cos \left (\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}+\cos (a) \int \sqrt {c+d x} \sin \left (\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}\right )-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt {c+d x} \sin \left (\frac {b}{\sqrt {c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt {c+d x}}+\cos (a) \int \sqrt {c+d x} \sin \left (\frac {b}{\sqrt {c+d x}}\right )d\frac {1}{\sqrt {c+d x}}\right )-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \left (\sin (a) \int \sqrt {c+d x} \sin \left (\frac {b}{\sqrt {c+d x}}+\frac {\pi }{2}\right )d\frac {1}{\sqrt {c+d x}}+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

\(\Big \downarrow \) 3783

\(\displaystyle -\frac {2 \left (\frac {1}{2} b \left (-b \left (\sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )-\sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )-\frac {1}{2} (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )\right )}{d}\)

input

Int[Sin[a + b/Sqrt[c + d*x]],x]
 

output

(-2*(-1/2*((c + d*x)*Sin[a + b/Sqrt[c + d*x]]) + (b*(-(Sqrt[c + d*x]*Cos[a 
 + b/Sqrt[c + d*x]]) - b*(CosIntegral[b/Sqrt[c + d*x]]*Sin[a] + Cos[a]*Sin 
Integral[b/Sqrt[c + d*x]])))/2))/d
 

3.2.99.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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] - Simp[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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 3783

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte 
gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - 
c*f, 0]
 

rule 3784

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* 
e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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 3842

Int[((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)])^(p_.), x_S 
ymbol] :> Simp[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] && Intege 
rQ[1/n]
 

3.2.99.4 Maple [A] (verified)

Time = 0.78 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.89

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

input

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

output

-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))
 

3.2.99.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11 \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {b^{2} \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + b^{2} \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right ) + \sqrt {d x + c} b \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) + {\left (d x + c\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{d} \]

input

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

output

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

3.2.99.6 Sympy [F]

\[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}\, dx \]

input

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

output

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

3.2.99.7 Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.32 \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {{\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{2 \, d} \]

input

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

output

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
 

3.2.99.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (84) = 168\).

Time = 0.35 (sec) , antiderivative size = 413, normalized size of antiderivative = 4.39 \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {a^{2} b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) - a^{2} b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) - \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right )}{\sqrt {d x + c}} + \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{\sqrt {d x + c}} + \frac {{\left (\sqrt {d x + c} a + b\right )}^{2} b^{3} \operatorname {Ci}\left (-a + \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) \sin \left (a\right )}{d x + c} - \frac {{\left (\sqrt {d x + c} a + b\right )}^{2} b^{3} \cos \left (a\right ) \operatorname {Si}\left (a - \frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{d x + c} - a b^{3} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + \frac {{\left (\sqrt {d x + c} a + b\right )} b^{3} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{\sqrt {d x + c}} + b^{3} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )}{{\left (a^{2} - \frac {2 \, {\left (\sqrt {d x + c} a + b\right )} a}{\sqrt {d x + c}} + \frac {{\left (\sqrt {d x + c} a + b\right )}^{2}}{d x + c}\right )} b d} \]

input

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

output

(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*(s 
qrt(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_int 
egral(a - (sqrt(d*x + c)*a + b)/sqrt(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))*si 
n(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)/sqr 
t(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)
 

3.2.99.9 Mupad [F(-1)]

Timed out. \[ \int \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right ) \,d x \]

input

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

output

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