Integrand size = 20, antiderivative size = 301 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=-\frac {b^3 f \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b (d e-c f) \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^2}+\frac {b f (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{6 d^2}-\frac {b^4 f \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{12 d^2}+\frac {b^2 (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)}{d^2}-\frac {b^2 f (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {(d e-c f) (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{d^2}+\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d^2}-\frac {b^4 f \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^2 (d e-c f) \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{d^2} \] Output:
-1/12*b^3*f*(d*x+c)^(1/2)*cos(a+b/(d*x+c)^(1/2))/d^2+b*(-c*f+d*e)*(d*x+c)^ (1/2)*cos(a+b/(d*x+c)^(1/2))/d^2+1/6*b*f*(d*x+c)^(3/2)*cos(a+b/(d*x+c)^(1/ 2))/d^2-1/12*b^4*f*Ci(b/(d*x+c)^(1/2))*sin(a)/d^2+b^2*(-c*f+d*e)*Ci(b/(d*x +c)^(1/2))*sin(a)/d^2-1/12*b^2*f*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^2+(-c*f+ d*e)*(d*x+c)*sin(a+b/(d*x+c)^(1/2))/d^2+1/2*f*(d*x+c)^2*sin(a+b/(d*x+c)^(1 /2))/d^2-1/12*b^4*f*cos(a)*Si(b/(d*x+c)^(1/2))/d^2+b^2*(-c*f+d*e)*cos(a)*S i(b/(d*x+c)^(1/2))/d^2
Time = 0.97 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.22 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {e \sqrt {c+d x} \cos \left (\frac {b}{\sqrt {c+d x}}\right ) \left (b \cos (a)+\sqrt {c+d x} \sin (a)\right )}{d}+\frac {f \sqrt {c+d x} \cos \left (\frac {b}{\sqrt {c+d x}}\right ) \left (-b^3 \cos (a)-12 b c \cos (a)+2 b (c+d x) \cos (a)-b^2 \sqrt {c+d x} \sin (a)-12 c \sqrt {c+d x} \sin (a)+6 (c+d x)^{3/2} \sin (a)\right )}{12 d^2}+\frac {e \sqrt {c+d x} \left (\sqrt {c+d x} \cos (a)-b \sin (a)\right ) \sin \left (\frac {b}{\sqrt {c+d x}}\right )}{d}+\frac {f \sqrt {c+d x} \left (-b^2 \sqrt {c+d x} \cos (a)-12 c \sqrt {c+d x} \cos (a)+6 (c+d x)^{3/2} \cos (a)+b^3 \sin (a)+12 b c \sin (a)-2 b (c+d x) \sin (a)\right ) \sin \left (\frac {b}{\sqrt {c+d x}}\right )}{12 d^2}+\frac {b^2 e \left (\operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )}{d}-\frac {b^2 \left (b^2+12 c\right ) f \left (\operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right ) \sin (a)+\cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )\right )}{12 d^2} \] Input:
Integrate[(e + f*x)*Sin[a + b/Sqrt[c + d*x]],x]
Output:
(e*Sqrt[c + d*x]*Cos[b/Sqrt[c + d*x]]*(b*Cos[a] + Sqrt[c + d*x]*Sin[a]))/d + (f*Sqrt[c + d*x]*Cos[b/Sqrt[c + d*x]]*(-(b^3*Cos[a]) - 12*b*c*Cos[a] + 2*b*(c + d*x)*Cos[a] - b^2*Sqrt[c + d*x]*Sin[a] - 12*c*Sqrt[c + d*x]*Sin[a ] + 6*(c + d*x)^(3/2)*Sin[a]))/(12*d^2) + (e*Sqrt[c + d*x]*(Sqrt[c + d*x]* Cos[a] - b*Sin[a])*Sin[b/Sqrt[c + d*x]])/d + (f*Sqrt[c + d*x]*(-(b^2*Sqrt[ c + d*x]*Cos[a]) - 12*c*Sqrt[c + d*x]*Cos[a] + 6*(c + d*x)^(3/2)*Cos[a] + b^3*Sin[a] + 12*b*c*Sin[a] - 2*b*(c + d*x)*Sin[a])*Sin[b/Sqrt[c + d*x]])/( 12*d^2) + (b^2*e*(CosIntegral[b/Sqrt[c + d*x]]*Sin[a] + Cos[a]*SinIntegral [b/Sqrt[c + d*x]]))/d - (b^2*(b^2 + 12*c)*f*(CosIntegral[b/Sqrt[c + d*x]]* Sin[a] + Cos[a]*SinIntegral[b/Sqrt[c + d*x]]))/(12*d^2)
Time = 0.54 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.06, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3912, 2009}
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.
| \(\displaystyle \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx\) |
| \(\Big \downarrow \) 3912 |
| \(\displaystyle -\frac {2 \int \left (\frac {f \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) (c+d x)^{5/2}}{d}+\frac {(d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) (c+d x)^{3/2}}{d}\right )d\frac {1}{\sqrt {c+d x}}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \left (\frac {b^4 f \sin (a) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{24 d}+\frac {b^4 f \cos (a) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{24 d}+\frac {b^3 f \sqrt {c+d x} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{24 d}-\frac {b^2 \sin (a) (d e-c f) \operatorname {CosIntegral}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d}-\frac {b^2 \cos (a) (d e-c f) \text {Si}\left (\frac {b}{\sqrt {c+d x}}\right )}{2 d}+\frac {b^2 f (c+d x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{24 d}-\frac {(c+d x) (d e-c f) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d}-\frac {b \sqrt {c+d x} (d e-c f) \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{2 d}-\frac {f (c+d x)^2 \sin \left (a+\frac {b}{\sqrt {c+d x}}\right )}{4 d}-\frac {b f (c+d x)^{3/2} \cos \left (a+\frac {b}{\sqrt {c+d x}}\right )}{12 d}\right )}{d}\) |
Input:
Int[(e + f*x)*Sin[a + b/Sqrt[c + d*x]],x]
Output:
(-2*((b^3*f*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(24*d) - (b*(d*e - c*f )*Sqrt[c + d*x]*Cos[a + b/Sqrt[c + d*x]])/(2*d) - (b*f*(c + d*x)^(3/2)*Cos [a + b/Sqrt[c + d*x]])/(12*d) + (b^4*f*CosIntegral[b/Sqrt[c + d*x]]*Sin[a] )/(24*d) - (b^2*(d*e - c*f)*CosIntegral[b/Sqrt[c + d*x]]*Sin[a])/(2*d) + ( b^2*f*(c + d*x)*Sin[a + b/Sqrt[c + d*x]])/(24*d) - ((d*e - c*f)*(c + d*x)* Sin[a + b/Sqrt[c + d*x]])/(2*d) - (f*(c + d*x)^2*Sin[a + b/Sqrt[c + d*x]]) /(4*d) + (b^4*f*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(24*d) - (b^2*(d*e - c*f)*Cos[a]*SinIntegral[b/Sqrt[c + d*x]])/(2*d)))/d
Int[((g_.) + (h_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*((e_.) + (f _.)*(x_))^(n_)])^(p_.), x_Symbol] :> Simp[1/(n*f) Subst[Int[ExpandIntegra nd[(a + b*Sin[c + d*x])^p, x^(1/n - 1)*(g - e*(h/f) + h*(x^(1/n)/f))^m, x], x], x, (e + f*x)^n], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IGtQ[p , 0] && IntegerQ[1/n]
Time = 2.92 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.98
| method | result | size |
| derivativedivides | \(-\frac {2 b^{2} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+f \,b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )\right )}{d^{2}}\) | \(295\) |
| default | \(-\frac {2 b^{2} \left (-c f \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+d e \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{2 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{2}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{2}\right )+f \,b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )\right )}{d^{2}}\) | \(295\) |
| parts | \(\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) x^{2} f +\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) x e +\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) c f x}{d}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) c e}{d}+\frac {b \cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}\, f x}{d}+\frac {b \cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}\, e}{d}+\frac {b^{2} \operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right ) f x}{d}+\frac {b^{2} \operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right ) e}{d}+\frac {b^{2} \operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right ) f x}{d}+\frac {b^{2} \operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right ) e}{d}+\frac {b^{2} f \left (\frac {2 \sin \left (a \right ) b^{2} \left (-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\cos \left (\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{4 b^{2}}+\frac {\sin \left (\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{4 b}-\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right )}{4}\right )}{d}+\frac {2 \cos \left (a \right ) b^{2} \left (-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{2 b^{2}}-\frac {\sin \left (\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{4 b^{2}}-\frac {\cos \left (\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{4 b}-\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right )}{4}\right )}{d}+\frac {2 b^{2} \left (-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{6 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{6 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{6}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{6}\right )}{d}+\frac {2 b^{2} \left (-\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{2}}{4 b^{4}}-\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )^{\frac {3}{2}}}{12 b^{3}}+\frac {\sin \left (a +\frac {b}{\sqrt {d x +c}}\right ) \left (d x +c \right )}{24 b^{2}}+\frac {\cos \left (a +\frac {b}{\sqrt {d x +c}}\right ) \sqrt {d x +c}}{24 b}+\frac {\operatorname {Si}\left (\frac {b}{\sqrt {d x +c}}\right ) \cos \left (a \right )}{24}+\frac {\operatorname {Ci}\left (\frac {b}{\sqrt {d x +c}}\right ) \sin \left (a \right )}{24}\right )}{d}\right )}{d}\) | \(621\) |
Input:
int((f*x+e)*sin(a+b/(d*x+c)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-2/d^2*b^2*(-c*f*(-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))+d*e*(-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))+f*b^2*(-1/4*sin(a+b/(d*x+c)^(1/2))/b^4*(d*x+c)^2-1/12*c os(a+b/(d*x+c)^(1/2))/b^3*(d*x+c)^(3/2)+1/24*sin(a+b/(d*x+c)^(1/2))/b^2*(d *x+c)+1/24*cos(a+b/(d*x+c)^(1/2))/b*(d*x+c)^(1/2)+1/24*Si(b/(d*x+c)^(1/2)) *cos(a)+1/24*Ci(b/(d*x+c)^(1/2))*sin(a)))
Time = 0.10 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.67 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {{\left (12 \, b^{2} d e - {\left (b^{4} + 12 \, b^{2} c\right )} f\right )} \operatorname {Ci}\left (\frac {b}{\sqrt {d x + c}}\right ) \sin \left (a\right ) + {\left (12 \, b^{2} d e - {\left (b^{4} + 12 \, b^{2} c\right )} f\right )} \cos \left (a\right ) \operatorname {Si}\left (\frac {b}{\sqrt {d x + c}}\right ) + {\left (2 \, b d f x + 12 \, b d e - {\left (b^{3} + 10 \, b c\right )} f\right )} \sqrt {d x + c} \cos \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right ) + {\left (6 \, d^{2} f x^{2} + 12 \, c d e - {\left (b^{2} c + 6 \, c^{2}\right )} f - {\left (b^{2} d f - 12 \, d^{2} e\right )} x\right )} \sin \left (\frac {a d x + a c + \sqrt {d x + c} b}{d x + c}\right )}{12 \, d^{2}} \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^(1/2)),x, algorithm="fricas")
Output:
1/12*((12*b^2*d*e - (b^4 + 12*b^2*c)*f)*cos_integral(b/sqrt(d*x + c))*sin( a) + (12*b^2*d*e - (b^4 + 12*b^2*c)*f)*cos(a)*sin_integral(b/sqrt(d*x + c) ) + (2*b*d*f*x + 12*b*d*e - (b^3 + 10*b*c)*f)*sqrt(d*x + c)*cos((a*d*x + a *c + sqrt(d*x + c)*b)/(d*x + c)) + (6*d^2*f*x^2 + 12*c*d*e - (b^2*c + 6*c^ 2)*f - (b^2*d*f - 12*d^2*e)*x)*sin((a*d*x + a*c + sqrt(d*x + c)*b)/(d*x + c)))/d^2
\[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \left (e + f x\right ) \sin {\left (a + \frac {b}{\sqrt {c + d x}} \right )}\, dx \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)**(1/2)),x)
Output:
Integral((e + f*x)*sin(a + b/sqrt(c + d*x)), x)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.35 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\frac {12 \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} e - \frac {12 \, {\left ({\left ({\left (-i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) + {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{2} + 2 \, \sqrt {d x + c} b \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) + 2 \, {\left (d x + c\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} c f}{d} + \frac {{\left ({\left ({\left (i \, {\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) - i \, {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \cos \left (a\right ) - {\left ({\rm Ei}\left (\frac {i \, b}{\sqrt {d x + c}}\right ) + {\rm Ei}\left (-\frac {i \, b}{\sqrt {d x + c}}\right )\right )} \sin \left (a\right )\right )} b^{4} - 2 \, {\left (\sqrt {d x + c} b^{3} - 2 \, {\left (d x + c\right )}^{\frac {3}{2}} b\right )} \cos \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right ) - 2 \, {\left ({\left (d x + c\right )} b^{2} - 6 \, {\left (d x + c\right )}^{2}\right )} \sin \left (\frac {\sqrt {d x + c} a + b}{\sqrt {d x + c}}\right )\right )} f}{d}}{24 \, d} \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^(1/2)),x, algorithm="maxima")
Output:
1/24*(12*(((-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)))*e - 12*(((-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)))*c*f/d + (((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^4 - 2*(sqrt(d*x + c)*b^3 - 2*(d*x + c)^(3/2)*b)*cos((sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 2*((d*x + c)*b^2 - 6*(d*x + c)^2)*sin((sqrt(d*x + c)*a + b)/sqrt(d*x + c)))*f/d)/d
Leaf count of result is larger than twice the leaf count of optimal. 2158 vs. \(2 (263) = 526\).
Time = 0.47 (sec) , antiderivative size = 2158, normalized size of antiderivative = 7.17 \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*sin(a+b/(d*x+c)^(1/2)),x, algorithm="giac")
Output:
1/12*(12*(a^2*b^3*cos_integral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*s in(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)/sqrt(d*x + c))/sqrt(d*x + c) + (s qrt(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)))*e/((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) - (a^4*b^5*cos_integral(-a + (sqrt(d*x + c)*a + b )/sqrt(d*x + c))*sin(a) - a^4*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c)) - 4*(sqrt(d*x + c)*a + b)*a^3*b^5*cos_integral(-a + ( sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/sqrt(d*x + c) + 4*(sqrt(d*x + c )*a + b)*a^3*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x + c))/sqrt(d*x + c) + 6*(sqrt(d*x + c)*a + b)^2*a^2*b^5*cos_integral(-a + (s qrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a)/(d*x + c) + 12*a^4*b^3*c*cos_int egral(-a + (sqrt(d*x + c)*a + b)/sqrt(d*x + c))*sin(a) - 6*(sqrt(d*x + c)* a + b)^2*a^2*b^5*cos(a)*sin_integral(a - (sqrt(d*x + c)*a + b)/sqrt(d*x...
Timed out. \[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\int \sin \left (a+\frac {b}{\sqrt {c+d\,x}}\right )\,\left (e+f\,x\right ) \,d x \] Input:
int(sin(a + b/(c + d*x)^(1/2))*(e + f*x),x)
Output:
int(sin(a + b/(c + d*x)^(1/2))*(e + f*x), x)
\[ \int (e+f x) \sin \left (a+\frac {b}{\sqrt {c+d x}}\right ) \, dx=\left (\int \sin \left (\frac {\sqrt {d x +c}\, a +b}{\sqrt {d x +c}}\right )d x \right ) e +\left (\int \sin \left (\frac {\sqrt {d x +c}\, a +b}{\sqrt {d x +c}}\right ) x d x \right ) f \] Input:
int((f*x+e)*sin(a+b/(d*x+c)^(1/2)),x)
Output:
int(sin((sqrt(c + d*x)*a + b)/sqrt(c + d*x)),x)*e + int(sin((sqrt(c + d*x) *a + b)/sqrt(c + d*x))*x,x)*f