\(\int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [339]

Optimal result

Integrand size = 32, antiderivative size = 386 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2} \]

[Out]

1/2*I*b*(f*x+e)^2/a^2/f+1/2*I*(a^2-b^2)*(f*x+e)^2/a^2/b/f-f*arctanh(cos(d*x+c))/a/d^2-(f*x+e)*csc(d*x+c)/a/d-b
*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/a^2/d-(a^2-b^2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d-(
a^2-b^2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d+1/2*I*b*f*polylog(2,exp(2*I*(d*x+c)))/a^
2/d^2+I*(a^2-b^2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/d^2+I*(a^2-b^2)*f*polylog(2,I*b*ex
p(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b/d^2

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used = {4639, 4493, 3377, 2718, 4495, 3855, 4489, 2715, 8, 3798, 2221, 2317, 2438, 4621, 4615} \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i f \left (a^2-b^2\right ) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i f \left (a^2-b^2\right ) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b d}+\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a^2 b f}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d} \]

[In]

Int[((e + f*x)*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

((I/2)*b*(e + f*x)^2)/(a^2*f) + ((I/2)*(a^2 - b^2)*(e + f*x)^2)/(a^2*b*f) - (f*ArcTanh[Cos[c + d*x]])/(a*d^2)
- ((e + f*x)*Csc[c + d*x])/(a*d) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])]
)/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d) - (b*(e +
f*x)*Log[1 - E^((2*I)*(c + d*x))])/(a^2*d) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2
- b^2])])/(a^2*b*d^2) + (I*(a^2 - b^2)*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(a^2*b*d^2)
+ ((I/2)*b*f*PolyLog[2, E^((2*I)*(c + d*x))])/(a^2*d^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2715

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

Rule 2718

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

Rule 3377

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

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3855

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

Rule 4489

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[(c + d
*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))), x] - Dist[d*(m/(b*(n + 1))), Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n +
1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 4493

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4495

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[
(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x] + Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4615

Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x])
/; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]

Rule 4621

Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol
] :> Dist[a/b^2, Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Dist[1/b, Int[(e + f*x)^m*Cos[c + d*x]^(n -
2)*Sin[c + d*x], x], x] - Dist[(a^2 - b^2)/b^2, Int[(e + f*x)^m*(Cos[c + d*x]^(n - 2)/(a + b*Sin[c + d*x])), x
], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4639

Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin
[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Dist[b/a
, Int[(e + f*x)^m*Cos[c + d*x]^(p + 1)*(Cot[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] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (e+f x) \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a} \\ & = -\frac {\int (e+f x) \cos (c+d x) \, dx}{a}+\frac {\int (e+f x) \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos ^2(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2} \\ & = -\frac {(e+f x) \csc (c+d x)}{a d}-\frac {(e+f x) \sin (c+d x)}{a d}+\frac {\int (e+f x) \cos (c+d x) \, dx}{a}-\frac {b \int (e+f x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx+\frac {f \int \csc (c+d x) \, dx}{a d}+\frac {f \int \sin (c+d x) \, dx}{a d} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)}{1-e^{2 i (c+d x)}} \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx-\left (1-\frac {b^2}{a^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx-\frac {f \int \sin (c+d x) \, dx}{a d} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {(b f) \int \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {\left (i \left (1-\frac {b^2}{a^2}\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b d^2}-\frac {\left (i \left (1-\frac {b^2}{a^2}\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b d^2} \\ & = \frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i \left (1-\frac {b^2}{a^2}\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i \left (1-\frac {b^2}{a^2}\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(915\) vs. \(2(386)=772\).

Time = 7.38 (sec) , antiderivative size = 915, normalized size of antiderivative = 2.37 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {4 a b d (e+f x) \cot \left (\frac {1}{2} (c+d x)\right )+8 a^2 d e \log \left (1+\frac {b \sin (c+d x)}{a}\right )-8 b^2 d e \log \left (1+\frac {b \sin (c+d x)}{a}\right )-8 a^2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )+8 b^2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )-8 a b f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+8 b^2 d e (\log (\cos (c+d x))+\log (\tan (c+d x)))-8 b^2 c f (\log (\cos (c+d x))+\log (\tan (c+d x)))+a^2 f \left (i (-2 c+\pi -2 d x)^2-32 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \cot \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}{\sqrt {a^2-b^2}}\right )-4 \left (-2 c+\pi -2 d x+4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )-4 \left (-2 c+\pi -2 d x-4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+4 (-2 c+\pi -2 d x) \log (a+b \sin (c+d x))+8 (c+d x) \log (a+b \sin (c+d x))+8 i \left (\operatorname {PolyLog}\left (2,\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )\right )\right )-b^2 f \left (i (-2 c+\pi -2 d x)^2-32 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \cot \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}{\sqrt {a^2-b^2}}\right )-4 \left (-2 c+\pi -2 d x+4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )-4 \left (-2 c+\pi -2 d x-4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+4 (-2 c+\pi -2 d x) \log (a+b \sin (c+d x))+8 (c+d x) \log (a+b \sin (c+d x))+8 i \left (\operatorname {PolyLog}\left (2,\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )\right )\right )+8 b^2 f \left ((c+d x) \log \left (1-e^{2 i (c+d x)}\right )-\frac {1}{2} i \left ((c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )\right )\right )+4 a b d (e+f x) \tan \left (\frac {1}{2} (c+d x)\right )}{8 a^2 b d^2} \]

[In]

Integrate[((e + f*x)*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]

[Out]

-1/8*(4*a*b*d*(e + f*x)*Cot[(c + d*x)/2] + 8*a^2*d*e*Log[1 + (b*Sin[c + d*x])/a] - 8*b^2*d*e*Log[1 + (b*Sin[c
+ d*x])/a] - 8*a^2*c*f*Log[1 + (b*Sin[c + d*x])/a] + 8*b^2*c*f*Log[1 + (b*Sin[c + d*x])/a] - 8*a*b*f*Log[Tan[(
c + d*x)/2]] + 8*b^2*d*e*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]) - 8*b^2*c*f*(Log[Cos[c + d*x]] + Log[Tan[c +
d*x]]) + a^2*f*(I*(-2*c + Pi - 2*d*x)^2 - (32*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Cot[(2*c + Pi
+ 2*d*x)/4])/Sqrt[a^2 - b^2]] - 4*(-2*c + Pi - 2*d*x + 4*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 - (I*(-a + Sq
rt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] - 4*(-2*c + Pi - 2*d*x - 4*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + (I*(a
+ Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + 4*(-2*c + Pi - 2*d*x)*Log[a + b*Sin[c + d*x]] + 8*(c + d*x)*Log[a
+ b*Sin[c + d*x]] + (8*I)*(PolyLog[2, (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + PolyLog[2, ((-I)*(a +
Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))])) - b^2*f*(I*(-2*c + Pi - 2*d*x)^2 - (32*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt
[2]]*ArcTan[((a - b)*Cot[(2*c + Pi + 2*d*x)/4])/Sqrt[a^2 - b^2]] - 4*(-2*c + Pi - 2*d*x + 4*ArcSin[Sqrt[(a + b
)/b]/Sqrt[2]])*Log[1 - (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] - 4*(-2*c + Pi - 2*d*x - 4*ArcSin[Sqrt[
(a + b)/b]/Sqrt[2]])*Log[1 + (I*(a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + 4*(-2*c + Pi - 2*d*x)*Log[a + b*
Sin[c + d*x]] + 8*(c + d*x)*Log[a + b*Sin[c + d*x]] + (8*I)*(PolyLog[2, (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c
+ d*x)))] + PolyLog[2, ((-I)*(a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))])) + 8*b^2*f*((c + d*x)*Log[1 - E^((2*
I)*(c + d*x))] - (I/2)*((c + d*x)^2 + PolyLog[2, E^((2*I)*(c + d*x))])) + 4*a*b*d*(e + f*x)*Tan[(c + d*x)/2])/
(a^2*b*d^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1704 vs. \(2 (351 ) = 702\).

Time = 0.79 (sec) , antiderivative size = 1705, normalized size of antiderivative = 4.42

method	result	size
risch	\(\text {Expression too large to display}\)	\(1705\)

[In]

int((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/a^2*b/d*f*ln(exp(I*(d*x+c))+1)*x+1/a^2*b/d^2*f*c*ln(exp(I*(d*x+c))-1)+2/b/d*e*ln(exp(I*(d*x+c)))+1/a/d^2*f*
ln(exp(I*(d*x+c))-1)-1/b/d*e*ln(I*b*exp(2*I*(d*x+c))-I*b-2*a*exp(I*(d*x+c)))-1/a/d^2*f*ln(exp(I*(d*x+c))+1)+1/
2*I/b*f*x^2+a^2/b/d^2*f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*c-I*a^
2/b/d^2*f/(-a^2+b^2)*dilog((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))-I*a^2/b/d^2*f/(-a
^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))-I/a^2*b^3/d^2*f/(-a^2+b^2)*dilog
((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))-I/a^2*b^3/d^2*f/(-a^2+b^2)*dilog((I*a+b*exp
(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))-I/b*e*x-1/a^2*b/d^2*f*c*ln(I*b*exp(2*I*(d*x+c))-I*b-2*a*
exp(I*(d*x+c)))+I/a^2*b/d^2*f*dilog(exp(I*(d*x+c))+1)-2*b/d*f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^
(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*x-2*b/d*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2
)^(1/2)))*x-2*b/d^2*f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*c-2*b/d^
2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c-I/a^2*b/d^2*f*dilog(exp(I*
(d*x+c)))+2*I/b/d*f*c*x+2*I*b/d^2*f/(-a^2+b^2)*dilog((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)
^(1/2)))+2*I*b/d^2*f/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+a^2/b/d*
f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*x+a^2/b/d*f/(-a^2+b^2)*ln((I
*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x+1/a^2*b^3/d^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*
x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c+1/a^2*b^3/d^2*f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b
^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))*c+1/a^2*b^3/d*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a
+(-a^2+b^2)^(1/2)))*x+1/a^2*b^3/d*f/(-a^2+b^2)*ln((-I*a-b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1
/2)))*x+a^2/b/d^2*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c-2*I*(f*x+e
)*exp(I*(d*x+c))/d/a/(exp(2*I*(d*x+c))-1)+1/b/d^2*f*c*ln(I*b*exp(2*I*(d*x+c))-I*b-2*a*exp(I*(d*x+c)))+I/b/d^2*
f*c^2-2/b/d^2*f*c*ln(exp(I*(d*x+c)))-1/a^2*b/d*e*ln(exp(I*(d*x+c))+1)-1/a^2*b/d*e*ln(exp(I*(d*x+c))-1)+1/a^2*b
/d*e*ln(I*b*exp(2*I*(d*x+c))-I*b-2*a*exp(I*(d*x+c)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1419 vs. \(2 (343) = 686\).

Time = 0.48 (sec) , antiderivative size = 1419, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*a*b*d*f*x - I*b^2*f*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) + I*b^2*f*dilog(cos(d*x + c) - I
*sin(d*x + c))*sin(d*x + c) + I*b^2*f*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) - I*b^2*f*dilog(-cos(
d*x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*a*b*d*e - I*(a^2 - b^2)*f*dilog((I*a*cos(d*x + c) - a*sin(d*x + c)
+ (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) - I*(a^2 - b^2)*f*dilog
((I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*s
in(d*x + c) + I*(a^2 - b^2)*f*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*
sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + I*(a^2 - b^2)*f*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) -
(b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1)*sin(d*x + c) + ((a^2 - b^2)*d*e - (a^2
- b^2)*c*f)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + ((a
^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*
a)*sin(d*x + c) + ((a^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(-2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(
a^2 - b^2)/b^2) + 2*I*a)*sin(d*x + c) + ((a^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(-2*b*cos(d*x + c) - 2*I*b*sin(
d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a)*sin(d*x + c) + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*log(-(I*a*
cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c
) + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(
d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*log(-(-I*a*cos(d
*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + (
(a^2 - b^2)*d*f*x + (a^2 - b^2)*c*f)*log(-(-I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) - I*b*sin(d*x
+ c))*sqrt(-(a^2 - b^2)/b^2) - b)/b)*sin(d*x + c) + (b^2*d*f*x + b^2*d*e + a*b*f)*log(cos(d*x + c) + I*sin(d*x
+ c) + 1)*sin(d*x + c) + (b^2*d*f*x + b^2*d*e + a*b*f)*log(cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d*x + c) +
(b^2*d*e - (b^2*c + a*b)*f)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) + (b^2*d*e - (b^2*c
+ a*b)*f)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) + (b^2*d*f*x + b^2*c*f)*log(-cos(d*x
+ c) + I*sin(d*x + c) + 1)*sin(d*x + c) + (b^2*d*f*x + b^2*c*f)*log(-cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d
*x + c))/(a^2*b*d^2*sin(d*x + c))

Sympy [F]

\[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]

[In]

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)

[Out]

Integral((e + f*x)*cos(c + d*x)*cot(c + d*x)**2/(a + b*sin(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)*cos(d*x + c)*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]

[In]

int((cos(c + d*x)*cot(c + d*x)^2*(e + f*x))/(a + b*sin(c + d*x)),x)

[Out]

\text{Hanged}