Integrand size = 26, antiderivative size = 158 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {3 e x}{2 a}+\frac {3 f x^2}{4 a}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f \sin ^2(c+d x)}{4 a d^2} \]
3/2*e*x/a+3/4*f*x^2/a+(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
Time = 6.79 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.89 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (-8 d e+6 c d e+4 c f-3 c^2 f+6 d^2 e x-4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )+\cos \left (\frac {1}{2} (c+d x)\right ) \left (8 d (e+f x) \cos (c+d x)-f \cos (2 (c+d x))+2 \left (6 c d e+4 c f-3 c^2 f+6 d^2 e x+4 d f x+3 d^2 f x^2-8 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 f \sin (c+d x)-d (e+f x) \sin (2 (c+d x))\right )\right )\right )}{8 a d^2 (1+\sin (c+d x))} \]
((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]))
Time = 1.17 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, 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 |
\(\displaystyle \frac {\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}}{a}-\frac {\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}}{a}-\frac {\frac {4 f \log \left (-\cos \left (\frac {c}{2}+\frac {d x}{2}-\frac {\pi }{4}\right )\right )}{d^2}-\frac {2 (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {(e+f x)^2}{2 a f}\) |
(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
3.2.93.3.1 Defintions of rubi rules used
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]
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]
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 = 0.59 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {3 f \,x^{2}}{4 a}+\frac {3 e x}{2 a}+\frac {\left (d x f +d e +i f \right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{2}}+\frac {\left (d x f +d e -i f \right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{2}}+\frac {2 i f x}{a d}+\frac {2 i f c}{a \,d^{2}}+\frac {2 f x +2 e}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{2}}-\frac {f \cos \left (2 d x +2 c \right )}{8 a \,d^{2}}-\frac {\left (f x +e \right ) \sin \left (2 d x +2 c \right )}{4 d a}\) | \(187\) |
parallelrisch | \(\frac {16 f \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-32 \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (12 d^{2} x^{2}+24 d x +26\right ) f +24 d^{2} e x +8 d e \right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (\left (12 d^{2} x^{2}-24 d x +26\right ) f +24 d^{2} e x -40 d e \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (6 d x f +6 d e +7 f \right ) \cos \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )+\left (2 d x f +2 d e -f \right ) \cos \left (\frac {5 d x}{2}+\frac {5 c}{2}\right )+\left (6 d x f +6 d e -7 f \right ) \sin \left (\frac {3 d x}{2}+\frac {3 c}{2}\right )-2 \sin \left (\frac {5 d x}{2}+\frac {5 c}{2}\right ) \left (d x f +d e +\frac {1}{2} f \right )}{16 a \,d^{2} \left (\sin \left (\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\) | \(266\) |
default | \(-\frac {-\frac {4 e \left (\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}+2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}-\frac {2 f \,x^{2}+2 f \,x^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 f \,x^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {4 f x}{d}-\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 f x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 f x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {8 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d^{2}}-\frac {4 f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d^{2}}+\frac {4 e \left (\frac {-\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{d}+\frac {4 f \left (\left (d x +c \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )-\frac {3 \left (d x +c \right )^{2}}{4}-\frac {\left (\sin ^{2}\left (d x +c \right )\right )}{4}+\sin \left (d x +c \right )-\cos \left (d x +c \right ) \left (d x +c \right )-c \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\cos \left (d x +c \right ) c +\left (d x +c \right ) c \right )}{d^{2}}}{4 a}\) | \(412\) |
norman | \(\frac {\frac {d e +2 f}{a \,d^{2}}-\frac {2 e \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {5 f \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-3 d e +2 f \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-6 d e +5 f \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-5 d e +3 f \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\frac {\left (-d e +3 f \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}+\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} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {9 f \,x^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 f \,x^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {3 f \,x^{2} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a}+\frac {\left (3 d e +4 f \right ) x}{2 d a}+\frac {\left (3 d e -4 f \right ) x \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (3 d e -2 f \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d a}+\frac {3 \left (3 d e -2 f \right ) x \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {3 \left (3 d e +2 f \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (3 d e +2 f \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (9 d e -4 f \right ) x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}+\frac {\left (9 d e +4 f \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,d^{2}}-\frac {2 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a \,d^{2}}\) | \(589\) |
3/4*f*x^2/a+3/2*e*x/a+1/2*(d*x*f+I*f+d*e)/a/d^2*exp(I*(d*x+c))+1/2*(d*x*f- I*f+d*e)/a/d^2*exp(-I*(d*x+c))+2*I*f/a/d*x+2*I*f/a/d^2*c+2*(f*x+e)/d/a/(ex p(I*(d*x+c))+I)-2*f/a/d^2*ln(exp(I*(d*x+c))+I)-1/8*f/a/d^2*cos(2*d*x+2*c)- 1/4*(f*x+e)/d/a*sin(2*d*x+2*c)
Time = 0.27 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.58 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {6 \, d^{2} f x^{2} + 2 \, {\left (2 \, d f x + 2 \, d e - f\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (4 \, d f x + 4 \, d e + 3 \, f\right )} \cos \left (d x + c\right )^{2} + 8 \, d e + 4 \, {\left (3 \, d^{2} e + 2 \, d f\right )} x + {\left (6 \, d^{2} f x^{2} + 12 \, d e + 12 \, {\left (d^{2} e + d f\right )} x + f\right )} \cos \left (d x + c\right ) - 8 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, d^{2} f x^{2} - 2 \, {\left (2 \, d f x + 2 \, d e + f\right )} \cos \left (d x + c\right )^{2} - 8 \, d e + 4 \, {\left (3 \, d^{2} e - 2 \, d f\right )} x + 4 \, {\left (d f x + d e - 2 \, f\right )} \cos \left (d x + c\right ) - 7 \, f\right )} \sin \left (d x + c\right ) - 7 \, f}{8 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 4653 vs. \(2 (134) = 268\).
Time = 2.15 (sec) , antiderivative size = 4653, normalized size of antiderivative = 29.45 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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/...
Exception generated. \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
Leaf count of result is larger than twice the leaf count of optimal. 7564 vs. \(2 (138) = 276\).
Time = 0.98 (sec) , antiderivative size = 7564, normalized size of antiderivative = 47.87 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Too large to display} \]
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*...
Time = 1.90 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.56 \[ \int \frac {(e+f x) \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx={\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}+\frac {f\,x}{2\,a\,d}\right )-{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-d\,e+f\,1{}\mathrm {i}}{2\,a\,d^2}-\frac {f\,x}{2\,a\,d}\right )+{\mathrm {e}}^{-c\,2{}\mathrm {i}-d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (-2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}-\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,\left (\frac {\left (2\,d\,e+f\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,a\,d^2}+\frac {f\,x\,1{}\mathrm {i}}{8\,a\,d}\right )+\frac {3\,f\,x^2}{4\,a}-\frac {2\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{a\,d^2}+\frac {2\,\left (e+f\,x\right )}{a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {x\,\left (3\,d\,e+f\,4{}\mathrm {i}\right )}{2\,a\,d} \]
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)