Integrand size = 24, antiderivative size = 76 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {(e+f x)^2}{2 a f}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2} \] Output:
1/2*(f*x+e)^2/a/f+(f*x+e)*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d-2*f*ln(sin(1/2*c+1 /4*Pi+1/2*d*x))/a/d^2
Leaf count is larger than twice the leaf count of optimal. \(199\) vs. \(2(76)=152\).
Time = 0.91 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.62 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 d f x \cos \left (c+\frac {d x}{2}\right )+\cos \left (\frac {d x}{2}\right ) \left (d^2 x (2 e+f x)-4 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-4 d e \sin \left (\frac {d x}{2}\right )-2 d f x \sin \left (\frac {d x}{2}\right )+2 d^2 e x \sin \left (c+\frac {d x}{2}\right )+d^2 f x^2 \sin \left (c+\frac {d x}{2}\right )-4 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sin \left (c+\frac {d x}{2}\right )}{2 a d^2 \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:
Integrate[((e + f*x)*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
(2*d*f*x*Cos[c + (d*x)/2] + Cos[(d*x)/2]*(d^2*x*(2*e + f*x) - 4*f*Log[Cos[ (c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*d*e*Sin[(d*x)/2] - 2*d*f*x*Sin[(d*x) /2] + 2*d^2*e*x*Sin[c + (d*x)/2] + d^2*f*x^2*Sin[c + (d*x)/2] - 4*f*Log[Co s[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[c + (d*x)/2])/(2*a*d^2*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
Time = 0.47 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5026, 17, 3042, 3799, 3042, 4672, 3042, 25, 3956}
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.
| \(\displaystyle \int \frac {(e+f x) \sin (c+d x)}{a \sin (c+d x)+a} \, dx\) |
| \(\Big \downarrow \) 5026 |
| \(\displaystyle \frac {\int (e+f x)dx}{a}-\int \frac {e+f x}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\int \frac {e+f x}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\int \frac {e+f x}{\sin (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {\int (e+f x) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {\frac {2 f \int \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {\frac {2 f \int -\tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {-\frac {2 f \int \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {(e+f x)^2}{2 a f}-\frac {\frac {4 f \log \left (-\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\pi }{4}\right )\right )}{d^2}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}\) |
Input:
Int[((e + f*x)*Sin[c + d*x])/(a + a*Sin[c + d*x]),x]
Output:
(e + f*x)^2/(2*a*f) - ((-2*(e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/d + (4*f*L og[-Cos[c/2 - Pi/4 + (d*x)/2]])/d^2)/(2*a)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_. )*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b Int[(e + f*x)^m*Sin[c + d*x]^(n - 1), x], x] - Simp[a/b Int[(e + f*x)^m*(Sin[c + d*x]^(n - 1)/(a + b*Sin[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] & & IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 0.84 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\frac {f \,x^{2}}{2 a}+\frac {e x}{a}+\frac {2 i f x}{d a}+\frac {2 i f c}{d^{2} a}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{2} a}\) | \(88\) |
| parallelrisch | \(\frac {f \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-2 f \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (\frac {f x}{2}+e \right ) \left (d x -2\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+x \left (\left (\frac {f x}{2}+e \right ) d +f \right )\right ) d}{d^{2} a \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(109\) |
| norman | \(\frac {-\frac {2 e \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}-\frac {2 e \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d a}+\frac {\left (d e +f \right ) x}{a d}+\frac {\left (d e -f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {\left (d e -f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{a d}+\frac {\left (d e +f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{a d}+\frac {f \,x^{2}}{2 a}+\frac {f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+\frac {f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a}+\frac {f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d^{2} a}-\frac {2 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d^{2} a}\) | \(267\) |
Input:
int((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/2*f/a*x^2+1/a*e*x+2*I*f/d/a*x+2*I*f/d^2/a*c+2*(f*x+e)/d/a/(exp(I*(d*x+c) )+I)-2*f/d^2/a*ln(exp(I*(d*x+c))+I)
Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (62) = 124\).
Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.99 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {d^{2} f x^{2} + 2 \, d e + 2 \, {\left (d^{2} e + d f\right )} x + {\left (d^{2} f x^{2} + 2 \, d e + 2 \, {\left (d^{2} e + d f\right )} x\right )} \cos \left (d x + c\right ) - 2 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (d^{2} f x^{2} - 2 \, d e + 2 \, {\left (d^{2} e - d f\right )} x\right )} \sin \left (d x + c\right )}{2 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \] Input:
integrate((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")
Output:
1/2*(d^2*f*x^2 + 2*d*e + 2*(d^2*e + d*f)*x + (d^2*f*x^2 + 2*d*e + 2*(d^2*e + d*f)*x)*cos(d*x + c) - 2*(f*cos(d*x + c) + f*sin(d*x + c) + f)*log(sin( d*x + c) + 1) + (d^2*f*x^2 - 2*d*e + 2*(d^2*e - d*f)*x)*sin(d*x + c))/(a*d ^2*cos(d*x + c) + a*d^2*sin(d*x + c) + a*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (56) = 112\).
Time = 0.67 (sec) , antiderivative size = 456, normalized size of antiderivative = 6.00 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\begin {cases} \frac {2 d^{2} e x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d^{2} e x}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {d^{2} f x^{2}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {4 d e}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {2 d f x \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 d f x}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {4 f \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} - \frac {4 f \log {\left (\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 f \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} + \frac {2 f \log {\left (\tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1 \right )}}{2 a d^{2} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 2 a d^{2}} & \text {for}\: d \neq 0 \\\frac {\left (e x + \frac {f x^{2}}{2}\right ) \sin {\left (c \right )}}{a \sin {\left (c \right )} + a} & \text {otherwise} \end {cases} \] Input:
integrate((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
Piecewise((2*d**2*e*x*tan(c/2 + d*x/2)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d* *2) + 2*d**2*e*x/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) + d**2*f*x**2*tan( c/2 + d*x/2)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) + d**2*f*x**2/(2*a*d** 2*tan(c/2 + d*x/2) + 2*a*d**2) + 4*d*e/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d* *2) - 2*d*f*x*tan(c/2 + d*x/2)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) + 2* d*f*x/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) - 4*f*log(tan(c/2 + d*x/2) + 1)*tan(c/2 + d*x/2)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) - 4*f*log(tan(c /2 + d*x/2) + 1)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) + 2*f*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2) + 2*f*log(tan(c/2 + d*x/2)**2 + 1)/(2*a*d**2*tan(c/2 + d*x/2) + 2*a*d**2), Ne(d, 0)), ((e*x + f*x**2/2)*sin(c)/(a*sin(c) + a), True))
Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (62) = 124\).
Time = 0.11 (sec) , antiderivative size = 273, normalized size of antiderivative = 3.59 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {4 \, c f {\left (\frac {1}{a d + \frac {a d \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}} + \frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a d}\right )} - 4 \, e {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}}\right )} - \frac {{\left ({\left (d x + c\right )}^{2} \cos \left (d x + c\right )^{2} + {\left (d x + c\right )}^{2} \sin \left (d x + c\right )^{2} + 2 \, {\left (d x + c\right )}^{2} \sin \left (d x + c\right ) + {\left (d x + c\right )}^{2} + 4 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 2 \, {\left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right )\right )} f}{a d \cos \left (d x + c\right )^{2} + a d \sin \left (d x + c\right )^{2} + 2 \, a d \sin \left (d x + c\right ) + a d}}{2 \, d} \] Input:
integrate((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/2*(4*c*f*(1/(a*d + a*d*sin(d*x + c)/(cos(d*x + c) + 1)) + arctan(sin(d* x + c)/(cos(d*x + c) + 1))/(a*d)) - 4*e*(arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a + 1/(a + a*sin(d*x + c)/(cos(d*x + c) + 1))) - ((d*x + c)^2*cos(d *x + c)^2 + (d*x + c)^2*sin(d*x + c)^2 + 2*(d*x + c)^2*sin(d*x + c) + (d*x + c)^2 + 4*(d*x + c)*cos(d*x + c) - 2*(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1))*f/(a*d*cos(d*x + c)^2 + a*d*sin(d*x + c)^2 + 2*a*d*sin(d*x + c) + a*d ))/d
Leaf count of result is larger than twice the leaf count of optimal. 656 vs. \(2 (62) = 124\).
Time = 0.27 (sec) , antiderivative size = 656, normalized size of antiderivative = 8.63 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
1/2*(d^2*f*x^2*tan(1/2*d*x)*tan(1/2*c) - d^2*f*x^2*tan(1/2*d*x) - d^2*f*x^ 2*tan(1/2*c) + 2*d^2*e*x*tan(1/2*d*x)*tan(1/2*c) - d^2*f*x^2 - 2*d^2*e*x*t an(1/2*d*x) - 2*d^2*e*x*tan(1/2*c) + 2*d*f*x*tan(1/2*d*x)*tan(1/2*c) - 2*d ^2*e*x + 2*d*f*x*tan(1/2*d*x) + 2*d*f*x*tan(1/2*c) + 2*d*e*tan(1/2*d*x)*ta n(1/2*c) - 2*f*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1 /2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*ta n(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x) ^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x)*tan(1/2*c) - 2*d*f*x + 2*d*e*tan(1/2* d*x) + 2*f*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c ) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/ 2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2*d*x) + 2*d*e*tan(1/2*c) + 2*f*log(2*(tan(1/2*d *x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*tan(1/2*c) - 2*tan(1/2*d*x)*tan(1/2* c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/ (tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 1))*tan(1/2 *c) - 2*d*e + 2*f*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*ta n(1/2*c) - 2*tan(1/2*d*x)*tan(1/2*c)^2 + tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2 *tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*d*x)^2*tan(1/2*c)^2 + tan(1/2*d *x)^2 + tan(1/2*c)^2 + 1)))/(a*d^2*tan(1/2*d*x)*tan(1/2*c) - a*d^2*tan(1/2 *d*x) - a*d^2*tan(1/2*c) - a*d^2)
Time = 35.50 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {f\,x^2}{2\,a}-\frac {2\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{a\,d^2}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {x\,\left (d\,e+f\,2{}\mathrm {i}\right )}{a\,d} \] Input:
int((sin(c + d*x)*(e + f*x))/(a + a*sin(c + d*x)),x)
Output:
(f*x^2)/(2*a) - (2*f*log(exp(c*1i)*exp(d*x*1i) + 1i))/(a*d^2) + (2*(e + f* x))/(a*d*(exp(c*1i + d*x*1i) + 1i)) + (x*(f*2i + d*e))/(a*d)
Time = 0.17 (sec) , antiderivative size = 184, normalized size of antiderivative = 2.42 \[ \int \frac {(e+f x) \sin (c+d x)}{a+a \sin (c+d x)} \, dx=\frac {2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) f +2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) f -4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) f -4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) f +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d^{2} e x +\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d^{2} f \,x^{2}-4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d e -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) d f x +2 d^{2} e x +d^{2} f \,x^{2}+2 d f x}{2 a \,d^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \] Input:
int((f*x+e)*sin(d*x+c)/(a+a*sin(d*x+c)),x)
Output:
(2*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)*f + 2*log(tan((c + d*x)/2 )**2 + 1)*f - 4*log(tan((c + d*x)/2) + 1)*tan((c + d*x)/2)*f - 4*log(tan(( c + d*x)/2) + 1)*f + 2*tan((c + d*x)/2)*d**2*e*x + tan((c + d*x)/2)*d**2*f *x**2 - 4*tan((c + d*x)/2)*d*e - 2*tan((c + d*x)/2)*d*f*x + 2*d**2*e*x + d **2*f*x**2 + 2*d*f*x)/(2*a*d**2*(tan((c + d*x)/2) + 1))