\(\int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [193]

Optimal result

Integrand size = 26, antiderivative size = 155 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 (e+f x)^2}{4 a f}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2} \] Output:

3/4*(f*x+e)^2/a/f+(f*x+e)*cos(d*x+c)/a/d+(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-f*sin(d*x+c)/a/d^2-1/2*(f*x+e 
)*cos(d*x+c)*sin(d*x+c)/a/d+1/4*f*sin(d*x+c)^2/a/d^2
 

Mathematica [A] (verified)

Time = 8.75 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.92 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (-8 d e+6 c d e+4 c f-3 c^2 f+6 d^2 e x-4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (6 c d e+4 c f-3 c^2 f+6 d^2 e x+4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )\right )}{8 a d^2 (1+\sin (c+d x))} \] Input:

Integrate[((e + f*x)*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
 

Output:

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(Sin[(c + d*x)/2]*(8*d*(e + f*x)*Co 
s[c + d*x] - f*Cos[2*(c + d*x)] + 2*(-8*d*e + 6*c*d*e + 4*c*f - 3*c^2*f + 
6*d^2*e*x - 4*d*f*x + 3*d^2*f*x^2 - 8*f*Log[Cos[(c + d*x)/2] + Sin[(c + d* 
x)/2]] - 4*f*Sin[c + d*x] - d*(e + f*x)*Sin[2*(c + d*x)])) + Cos[(c + d*x) 
/2]*(8*d*(e + f*x)*Cos[c + d*x] - f*Cos[2*(c + d*x)] + 2*(6*c*d*e + 4*c*f 
- 3*c^2*f + 6*d^2*e*x + 4*d*f*x + 3*d^2*f*x^2 - 8*f*Log[Cos[(c + d*x)/2] + 
 Sin[(c + d*x)/2]] - 4*f*Sin[c + d*x] - d*(e + f*x)*Sin[2*(c + d*x)]))))/( 
8*a*d^2*(1 + Sin[c + d*x]))
 

Rubi [A] (verified)

Time = 1.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.12, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {5026, 3042, 3791, 17, 5026, 3042, 3777, 3042, 3117, 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.

\(\displaystyle \int \frac {(e+f x) \sin ^3(c+d x)}{a \sin (c+d x)+a} \, dx\)

\(\Big \downarrow \) 5026

\(\displaystyle \frac {\int (e+f x) \sin ^2(c+d x)dx}{a}-\int \frac {(e+f x) \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x) \sin (c+d x)^2dx}{a}-\int \frac {(e+f x) \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {\frac {1}{2} \int (e+f x)dx+\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}}{a}-\int \frac {(e+f x) \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\int \frac {(e+f x) \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 5026

\(\displaystyle -\frac {\int (e+f x) \sin (c+d x)dx}{a}+\int \frac {(e+f x) \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (e+f x) \sin (c+d x)dx}{a}+\int \frac {(e+f x) \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \int \frac {(e+f x) \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {f \int \cos (c+d x)dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e+f x) \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}\)

\(\Big \downarrow \) 3117

\(\displaystyle \int \frac {(e+f x) \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 5026

\(\displaystyle -\int \frac {e+f x}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)dx}{a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\int \frac {e+f x}{\sin (c+d x) a+a}dx+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {e+f x}{\sin (c+d x) a+a}dx+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 3799

\(\displaystyle -\frac {\int (e+f x) \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (e+f x) \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {2 f \int \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {2 f \int -\tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {2 f \int \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}+\frac {(e+f x)^2}{2 a f}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}-\frac {\frac {4 f \log \left (-\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\pi }{4}\right )\right )}{d^2}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {(e+f x)^2}{2 a f}\)

Input:

Int[((e + f*x)*Sin[c + d*x]^3)/(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) - (-(((e + f*x)*Cos[c + d*x])/d 
) + (f*Sin[c + d*x])/d^2)/a + ((e + f*x)^2/(4*f) - ((e + f*x)*Cos[c + d*x] 
*Sin[c + d*x])/(2*d) + (f*Sin[c + d*x]^2)/(4*d^2))/a
 

Defintions of rubi rules used

rule 17

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]
 

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 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; 
 FreeQ[{c, d}, x]
 

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3799

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])
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4672

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]
 

rule 5026

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]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 2.07 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21

method	result	size
risch	\(\frac {3 f \,x^{2}}{4 a}+\frac {3 e x}{2 a}+\frac {\left (d x f +d e +i f \right ) {\mathrm e}^{i \left (d x +c \right )}}{2 d^{2} a}+\frac {\left (d x f +d e -i f \right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 d^{2} a}+\frac {2 i f x}{d a}+\frac {2 i f c}{d^{2} a}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d^{2} a}-\frac {f \cos \left (2 d x +2 c \right )}{8 d^{2} a}-\frac {\left (f x +e \right ) \sin \left (2 d x +2 c \right )}{4 d a}\)	\(187\)
parallelrisch	\(\frac {16 \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) f \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-32 f \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (12 x^{2} d^{2}+24 d x +26\right ) f +24 d^{2} e x +8 d e \right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (12 x^{2} d^{2}-24 d x +26\right ) f +24 d^{2} e x -40 d e \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (6 d x f +6 d e +7 f \right ) \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (2 d x f +2 d e -f \right ) \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (6 d x f +6 d e -7 f \right ) \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-2 \left (d x f +d e +\frac {1}{2} f \right ) \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )}{16 \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) d^{2} a}\)	\(266\)
default	\(-\frac {-\frac {4 e \left (\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}-\frac {2 f \,x^{2}+2 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\frac {4 f x}{d}-\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}-\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}+\frac {8 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d^{2}}-\frac {4 f \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d^{2}}+\frac {4 e \left (\frac {-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}+\frac {4 f \left (\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )-\left (d x +c \right ) \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {\left (d x +c \right )^{2}}{4}-\frac {\sin \left (d x +c \right )^{2}}{4}+\cos \left (d x +c \right ) c +c \left (-\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{2}}}{4 a}\)	\(405\)
norman	\(\frac {\frac {d e +2 f}{d^{2} a}-\frac {2 e \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {5 f \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d^{2} a}+\frac {\left (-3 d e +2 f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{d^{2} a}+\frac {\left (-6 d e +5 f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d^{2} a}+\frac {\left (-5 d e +3 f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d^{2} a}+\frac {\left (-d e +3 f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d^{2} a}+\frac {3 f \,x^{2}}{4 a}+\frac {3 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a}+\frac {9 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4 a}+\frac {9 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 a}+\frac {9 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 a}+\frac {9 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 a}+\frac {3 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 a}+\frac {3 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a}+\frac {\left (3 d e +4 f \right ) x}{2 a d}+\frac {\left (3 d e -4 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 a d}+\frac {\left (3 d e -2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a d}+\frac {3 \left (3 d e -2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{2 a d}+\frac {3 \left (3 d e +2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a d}+\frac {\left (3 d e +2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a d}+\frac {\left (9 d e -4 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a d}+\frac {\left (9 d e +4 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \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}\)	\(589\)

Input:

int((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

3/4*f/a*x^2+3/2/a*e*x+1/2*(d*x*f+I*f+d*e)/d^2/a*exp(I*(d*x+c))+1/2*(d*x*f- 
I*f+d*e)/d^2/a*exp(-I*(d*x+c))+2*I*f/d/a*x+2*I*f/d^2/a*c+2*(f*x+e)/d/a/(ex 
p(I*(d*x+c))+I)-2*f/d^2/a*ln(exp(I*(d*x+c))+I)-1/8*f/d^2/a*cos(2*d*x+2*c)- 
1/4*(f*x+e)/d/a*sin(2*d*x+2*c)
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.61 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \, d^{2} f x^{2} + 2 \, {\left (2 \, d f x + 2 \, d e - f\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (4 \, d f x + 4 \, d e + 3 \, f\right )} \cos \left (d x + c\right )^{2} + 8 \, d e + 4 \, {\left (3 \, d^{2} e + 2 \, d f\right )} x + {\left (6 \, d^{2} f x^{2} + 12 \, d e + 12 \, {\left (d^{2} e + d f\right )} x + f\right )} \cos \left (d x + c\right ) - 8 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, d^{2} f x^{2} - 2 \, {\left (2 \, d f x + 2 \, d e + f\right )} \cos \left (d x + c\right )^{2} - 8 \, d e + 4 \, {\left (3 \, d^{2} e - 2 \, d f\right )} x + 4 \, {\left (d f x + d e - 2 \, f\right )} \cos \left (d x + c\right ) - 7 \, f\right )} \sin \left (d x + c\right ) - 7 \, f}{8 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \] Input:

integrate((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="fricas")
 

Output:

1/8*(6*d^2*f*x^2 + 2*(2*d*f*x + 2*d*e - f)*cos(d*x + c)^3 + 2*(4*d*f*x + 4 
*d*e + 3*f)*cos(d*x + c)^2 + 8*d*e + 4*(3*d^2*e + 2*d*f)*x + (6*d^2*f*x^2 
+ 12*d*e + 12*(d^2*e + d*f)*x + f)*cos(d*x + c) - 8*(f*cos(d*x + c) + f*si 
n(d*x + c) + f)*log(sin(d*x + c) + 1) + (6*d^2*f*x^2 - 2*(2*d*f*x + 2*d*e 
+ f)*cos(d*x + c)^2 - 8*d*e + 4*(3*d^2*e - 2*d*f)*x + 4*(d*f*x + d*e - 2*f 
)*cos(d*x + c) - 7*f)*sin(d*x + c) - 7*f)/(a*d^2*cos(d*x + c) + a*d^2*sin( 
d*x + c) + a*d^2)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4653 vs. \(2 (129) = 258\).

Time = 2.24 (sec) , antiderivative size = 4653, normalized size of antiderivative = 30.02 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*sin(d*x+c)**3/(a+a*sin(d*x+c)),x)
 

Output:

Piecewise((6*d**2*e*x*tan(c/2 + d*x/2)**5/(4*a*d**2*tan(c/2 + d*x/2)**5 + 
4*a*d**2*tan(c/2 + d*x/2)**4 + 8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan 
(c/2 + d*x/2)**2 + 4*a*d**2*tan(c/2 + d*x/2) + 4*a*d**2) + 6*d**2*e*x*tan( 
c/2 + d*x/2)**4/(4*a*d**2*tan(c/2 + d*x/2)**5 + 4*a*d**2*tan(c/2 + d*x/2)* 
*4 + 8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan(c/2 + d*x/2)**2 + 4*a*d** 
2*tan(c/2 + d*x/2) + 4*a*d**2) + 12*d**2*e*x*tan(c/2 + d*x/2)**3/(4*a*d**2 
*tan(c/2 + d*x/2)**5 + 4*a*d**2*tan(c/2 + d*x/2)**4 + 8*a*d**2*tan(c/2 + d 
*x/2)**3 + 8*a*d**2*tan(c/2 + d*x/2)**2 + 4*a*d**2*tan(c/2 + d*x/2) + 4*a* 
d**2) + 12*d**2*e*x*tan(c/2 + d*x/2)**2/(4*a*d**2*tan(c/2 + d*x/2)**5 + 4* 
a*d**2*tan(c/2 + d*x/2)**4 + 8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan(c 
/2 + d*x/2)**2 + 4*a*d**2*tan(c/2 + d*x/2) + 4*a*d**2) + 6*d**2*e*x*tan(c/ 
2 + d*x/2)/(4*a*d**2*tan(c/2 + d*x/2)**5 + 4*a*d**2*tan(c/2 + d*x/2)**4 + 
8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan(c/2 + d*x/2)**2 + 4*a*d**2*tan 
(c/2 + d*x/2) + 4*a*d**2) + 6*d**2*e*x/(4*a*d**2*tan(c/2 + d*x/2)**5 + 4*a 
*d**2*tan(c/2 + d*x/2)**4 + 8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan(c/ 
2 + d*x/2)**2 + 4*a*d**2*tan(c/2 + d*x/2) + 4*a*d**2) + 3*d**2*f*x**2*tan( 
c/2 + d*x/2)**5/(4*a*d**2*tan(c/2 + d*x/2)**5 + 4*a*d**2*tan(c/2 + d*x/2)* 
*4 + 8*a*d**2*tan(c/2 + d*x/2)**3 + 8*a*d**2*tan(c/2 + d*x/2)**2 + 4*a*d** 
2*tan(c/2 + d*x/2) + 4*a*d**2) + 3*d**2*f*x**2*tan(c/2 + d*x/2)**4/(4*a*d* 
*2*tan(c/2 + d*x/2)**5 + 4*a*d**2*tan(c/2 + d*x/2)**4 + 8*a*d**2*tan(c/...
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7564 vs. \(2 (137) = 274\).

Time = 0.91 (sec) , antiderivative size = 7564, normalized size of antiderivative = 48.80 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x, algorithm="giac")
 

Output:

1/8*(6*d^2*f*x^2*tan(1/2*d*x)^5*tan(1/2*c)^5 - 6*d^2*f*x^2*tan(1/2*d*x)^5* 
tan(1/2*c)^4 - 6*d^2*f*x^2*tan(1/2*d*x)^4*tan(1/2*c)^5 + 12*d^2*e*x*tan(1/ 
2*d*x)^5*tan(1/2*c)^5 + 12*d^2*f*x^2*tan(1/2*d*x)^5*tan(1/2*c)^3 - 6*d^2*f 
*x^2*tan(1/2*d*x)^4*tan(1/2*c)^4 - 12*d^2*e*x*tan(1/2*d*x)^5*tan(1/2*c)^4 
+ 12*d^2*f*x^2*tan(1/2*d*x)^3*tan(1/2*c)^5 - 12*d^2*e*x*tan(1/2*d*x)^4*tan 
(1/2*c)^5 + 16*d*f*x*tan(1/2*d*x)^5*tan(1/2*c)^5 - 12*d^2*f*x^2*tan(1/2*d* 
x)^5*tan(1/2*c)^2 - 12*d^2*f*x^2*tan(1/2*d*x)^4*tan(1/2*c)^3 + 24*d^2*e*x* 
tan(1/2*d*x)^5*tan(1/2*c)^3 - 12*d^2*f*x^2*tan(1/2*d*x)^3*tan(1/2*c)^4 - 1 
2*d^2*e*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 8*d*f*x*tan(1/2*d*x)^5*tan(1/2*c)^ 
4 - 12*d^2*f*x^2*tan(1/2*d*x)^2*tan(1/2*c)^5 + 24*d^2*e*x*tan(1/2*d*x)^3*t 
an(1/2*c)^5 + 8*d*f*x*tan(1/2*d*x)^4*tan(1/2*c)^5 + 16*d*e*tan(1/2*d*x)^5* 
tan(1/2*c)^5 - 8*f*log(2*(tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^2*t 
an(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)^5*tan(1/2*c)^5 + 6*d^2*f*x^2*tan( 
1/2*d*x)^5*tan(1/2*c) - 12*d^2*f*x^2*tan(1/2*d*x)^4*tan(1/2*c)^2 - 24*d^2* 
e*x*tan(1/2*d*x)^5*tan(1/2*c)^2 + 24*d^2*f*x^2*tan(1/2*d*x)^3*tan(1/2*c)^3 
 - 24*d^2*e*x*tan(1/2*d*x)^4*tan(1/2*c)^3 + 8*d*f*x*tan(1/2*d*x)^5*tan(1/2 
*c)^3 - 12*d^2*f*x^2*tan(1/2*d*x)^2*tan(1/2*c)^4 - 24*d^2*e*x*tan(1/2*d*x) 
^3*tan(1/2*c)^4 - 64*d*f*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 8*d*e*tan(1/2*...
 

Mupad [B] (verification not implemented)

Time = 41.17 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.59 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx={\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )-{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right )+{\mathrm {e}}^{-c\,2{}\mathrm {i}-d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+\frac {3\,f\,x^2}{4\,a}-\frac {2\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{a\,d^2}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {x\,\left (3\,d\,e+f\,4{}\mathrm {i}\right )}{2\,a\,d} \] Input:

int((sin(c + d*x)^3*(e + f*x))/(a + a*sin(c + d*x)),x)
 

Output:

exp(c*1i + d*x*1i)*((f*1i + d*e)/(2*a*d^2) + (f*x)/(2*a*d)) - exp(- c*1i - 
 d*x*1i)*((f*1i - d*e)/(2*a*d^2) - (f*x)/(2*a*d)) + exp(- c*2i - d*x*2i)*( 
((f*1i - 2*d*e)*1i)/(16*a*d^2) - (f*x*1i)/(8*a*d)) + exp(c*2i + d*x*2i)*(( 
(f*1i + 2*d*e)*1i)/(16*a*d^2) + (f*x*1i)/(8*a*d)) + (3*f*x^2)/(4*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*4i + 3*d*e))/(2*a*d)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 515, normalized size of antiderivative = 3.32 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {-8 f +4 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) f -8 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) f -4 \cos \left (d x +c \right ) \sin \left (d x +c \right ) f +4 \cos \left (d x +c \right ) c f -4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) \sin \left (d x +c \right ) f +8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right ) f +2 \sin \left (d x +c \right )^{3} d e -4 \sin \left (d x +c \right )^{2} d e -4 \sin \left (d x +c \right ) c f -2 \sin \left (d x +c \right ) d e -6 c d e -4 \sin \left (d x +c \right ) f -\sin \left (d x +c \right )^{3} f +8 \cos \left (d x +c \right ) f +3 \sin \left (d x +c \right )^{2} f +\cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} f +2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d e -2 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d e +6 \cos \left (d x +c \right ) c d e +6 \cos \left (d x +c \right ) d^{2} e x +3 \cos \left (d x +c \right ) d^{2} f \,x^{2}+2 \sin \left (d x +c \right )^{3} d f x -4 \sin \left (d x +c \right )^{2} d f x -6 \sin \left (d x +c \right ) c d e -6 \sin \left (d x +c \right ) d^{2} e x -3 \sin \left (d x +c \right ) d^{2} f \,x^{2}-6 \sin \left (d x +c \right ) d f x -12 \cos \left (d x +c \right ) d e +12 d e -6 d^{2} e x +2 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} d f x -2 \cos \left (d x +c \right ) \sin \left (d x +c \right ) d f x -3 d^{2} f \,x^{2}-4 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) f +8 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) f -8 \cos \left (d x +c \right ) d f x -4 c f +8 d f x}{4 a \,d^{2} \left (\cos \left (d x +c \right )-\sin \left (d x +c \right )-1\right )} \] Input:

int((f*x+e)*sin(d*x+c)^3/(a+a*sin(d*x+c)),x)
 

Output:

(4*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*f - 8*cos(c + d*x)*log(tan((c 
 + d*x)/2) + 1)*f + 2*cos(c + d*x)*sin(c + d*x)**2*d*e + 2*cos(c + d*x)*si 
n(c + d*x)**2*d*f*x + cos(c + d*x)*sin(c + d*x)**2*f - 2*cos(c + d*x)*sin( 
c + d*x)*d*e - 2*cos(c + d*x)*sin(c + d*x)*d*f*x - 4*cos(c + d*x)*sin(c + 
d*x)*f + 6*cos(c + d*x)*c*d*e + 4*cos(c + d*x)*c*f + 6*cos(c + d*x)*d**2*e 
*x + 3*cos(c + d*x)*d**2*f*x**2 - 12*cos(c + d*x)*d*e - 8*cos(c + d*x)*d*f 
*x + 8*cos(c + d*x)*f - 4*log(tan((c + d*x)/2)**2 + 1)*sin(c + d*x)*f - 4* 
log(tan((c + d*x)/2)**2 + 1)*f + 8*log(tan((c + d*x)/2) + 1)*sin(c + d*x)* 
f + 8*log(tan((c + d*x)/2) + 1)*f + 2*sin(c + d*x)**3*d*e + 2*sin(c + d*x) 
**3*d*f*x - sin(c + d*x)**3*f - 4*sin(c + d*x)**2*d*e - 4*sin(c + d*x)**2* 
d*f*x + 3*sin(c + d*x)**2*f - 6*sin(c + d*x)*c*d*e - 4*sin(c + d*x)*c*f - 
6*sin(c + d*x)*d**2*e*x - 3*sin(c + d*x)*d**2*f*x**2 - 2*sin(c + d*x)*d*e 
- 6*sin(c + d*x)*d*f*x - 4*sin(c + d*x)*f - 6*c*d*e - 4*c*f - 6*d**2*e*x - 
 3*d**2*f*x**2 + 12*d*e + 8*d*f*x - 8*f)/(4*a*d**2*(cos(c + d*x) - sin(c + 
 d*x) - 1))