3.169 \(\int \frac{(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx\)

Optimal. Leaf size=671 \[ -\frac{2 a d (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac{2 a d (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac{2 i d^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}+\frac{2 i d^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^3 \left (a^2-b^2\right )}-\frac{2 i a d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}+\frac{2 i a d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^2 \left (a^2-b^2\right )}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac{b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{i (c+d x)^2}{f \left (a^2-b^2\right )} \]

[Out]

(I*(c + d*x)^2)/((a^2 - b^2)*f) - (2*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 -
b^2)*f^2) - (I*a*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (2
*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^2*Log[1
- (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + ((2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f
*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[
a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + ((2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((
a^2 - b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)
*f^2) - ((2*I)*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + ((2*I)
*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (b*(c + d*x)^2*Cos[e
+ f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.20514, antiderivative size = 671, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3324, 3323, 2264, 2190, 2531, 2282, 6589, 4519, 2279, 2391} \[ -\frac{2 a d (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac{2 a d (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^2 \left (a^2-b^2\right )^{3/2}}+\frac{2 i d^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )}+\frac{2 i d^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^3 \left (a^2-b^2\right )}-\frac{2 i a d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}+\frac{2 i a d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^3 \left (a^2-b^2\right )^{3/2}}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f^2 \left (a^2-b^2\right )}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f^2 \left (a^2-b^2\right )}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )}{f \left (a^2-b^2\right )^{3/2}}+\frac{b (c+d x)^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{i (c+d x)^2}{f \left (a^2-b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*(c + d*x)^2)/((a^2 - b^2)*f) - (2*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 -
b^2)*f^2) - (I*a*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) - (2
*d*(c + d*x)*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^2) + (I*a*(c + d*x)^2*Log[1
- (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f) + ((2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f
*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a - Sqrt[
a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^2) + ((2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((
a^2 - b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)
*f^2) - ((2*I)*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + ((2*I)
*a*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])])/((a^2 - b^2)^(3/2)*f^3) + (b*(c + d*x)^2*Cos[e
+ f*x])/((a^2 - b^2)*f*(a + b*Sin[e + f*x]))

Rule 3324

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

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 4519

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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{(c+d x)^2}{(a+b \sin (e+f x))^2} \, dx &=\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{a \int \frac{(c+d x)^2}{a+b \sin (e+f x)} \, dx}{a^2-b^2}-\frac{(2 b d) \int \frac{(c+d x) \cos (e+f x)}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{(2 a) \int \frac{e^{i (e+f x)} (c+d x)^2}{i b+2 a e^{i (e+f x)}-i b e^{2 i (e+f x)}} \, dx}{a^2-b^2}-\frac{(2 b d) \int \frac{e^{i (e+f x)} (c+d x)}{a-\sqrt{a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}-\frac{(2 b d) \int \frac{e^{i (e+f x)} (c+d x)}{a+\sqrt{a^2-b^2}-i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right ) f}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{(2 i a b) \int \frac{e^{i (e+f x)} (c+d x)^2}{2 a-2 \sqrt{a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{(2 i a b) \int \frac{e^{i (e+f x)} (c+d x)^2}{2 a+2 \sqrt{a^2-b^2}-2 i b e^{i (e+f x)}} \, dx}{\left (a^2-b^2\right )^{3/2}}+\frac{\left (2 d^2\right ) \int \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}+\frac{\left (2 d^2\right ) \int \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right ) f^2}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (2 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right ) f^3}-\frac{\left (2 i d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right ) f^3}+\frac{(2 i a d) \int (c+d x) \log \left (1-\frac{2 i b e^{i (e+f x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}-\frac{(2 i a d) \int (c+d x) \log \left (1-\frac{2 i b e^{i (e+f x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left (2 a d^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (e+f x)}}{2 a-2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{\left (2 a d^2\right ) \int \text{Li}_2\left (\frac{2 i b e^{i (e+f x)}}{2 a+2 \sqrt{a^2-b^2}}\right ) \, dx}{\left (a^2-b^2\right )^{3/2} f^2}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (2 i a d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a-\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{\left (2 i a d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i b x}{a+\sqrt{a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{\left (a^2-b^2\right )^{3/2} f^3}\\ &=\frac{i (c+d x)^2}{\left (a^2-b^2\right ) f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}-\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}-\frac{2 d (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^2}+\frac{i a (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}-\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}+\frac{2 i d^2 \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right ) f^3}+\frac{2 a d (c+d x) \text{Li}_2\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^2}-\frac{2 i a d^2 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{2 i a d^2 \text{Li}_3\left (\frac{i b e^{i (e+f x)}}{a+\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} f^3}+\frac{b (c+d x)^2 \cos (e+f x)}{\left (a^2-b^2\right ) f (a+b \sin (e+f x))}\\ \end{align*}

Mathematica [A]  time = 1.67465, size = 530, normalized size = 0.79 \[ \frac{-\frac{i a \left (-2 i d f (c+d x) \text{PolyLog}\left (2,-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )+2 i d f (c+d x) \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+2 d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{a-\sqrt{a^2-b^2}}\right )-2 d^2 \text{PolyLog}\left (3,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+f^2 (c+d x)^2 \log \left (1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )-f^2 (c+d x)^2 \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )\right )}{\sqrt{a^2-b^2}}+2 i d^2 \text{PolyLog}\left (2,-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )+2 i d^2 \text{PolyLog}\left (2,\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )-2 d f (c+d x) \log \left (1+\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}-a}\right )-2 d f (c+d x) \log \left (1-\frac{i b e^{i (e+f x)}}{\sqrt{a^2-b^2}+a}\right )+\frac{b f^2 (c+d x)^2 \cos (e+f x)}{a+b \sin (e+f x)}+i f^2 (c+d x)^2}{f^3 \left (a^2-b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*f^2*(c + d*x)^2 - 2*d*f*(c + d*x)*Log[1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - 2*d*f*(c + d*x)*L
og[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + (2*I)*d^2*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt
[a^2 - b^2])] + (2*I)*d^2*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] - (I*a*(f^2*(c + d*x)^2*Log[
1 + (I*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] - f^2*(c + d*x)^2*Log[1 - (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^
2 - b^2])] - (2*I)*d*f*(c + d*x)*PolyLog[2, ((-I)*b*E^(I*(e + f*x)))/(-a + Sqrt[a^2 - b^2])] + (2*I)*d*f*(c +
d*x)*PolyLog[2, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])] + 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a - Sqr
t[a^2 - b^2])] - 2*d^2*PolyLog[3, (I*b*E^(I*(e + f*x)))/(a + Sqrt[a^2 - b^2])]))/Sqrt[a^2 - b^2] + (b*f^2*(c +
d*x)^2*Cos[e + f*x])/(a + b*Sin[e + f*x]))/((a^2 - b^2)*f^3)

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Maple [F]  time = 1.273, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx+c \right ) ^{2}}{ \left ( a+b\sin \left ( fx+e \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [C]  time = 4.36677, size = 7017, normalized size = 10.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*cos(f*x + e) - 2*a*s
in(f*x + e) + 2*(b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) - 4*(a*b^2*d^2*sin(f*x + e) + a
^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(2*I*a*cos(f*x + e) - 2*a*sin(f*x + e) - 2*(b*cos(f*x + e) + I
*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) + 4*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*po
lylog(3, 1/2*(-2*I*a*cos(f*x + e) - 2*a*sin(f*x + e) + 2*(b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)
/b^2))/b) - 4*(a*b^2*d^2*sin(f*x + e) + a^2*b*d^2)*sqrt(-(a^2 - b^2)/b^2)*polylog(3, 1/2*(-2*I*a*cos(f*x + e)
- 2*a*sin(f*x + e) - 2*(b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))/b) - 4*((a^2*b - b^3)*d^2*f
^2*x^2 + 2*(a^2*b - b^3)*c*d*f^2*x + (a^2*b - b^3)*c^2*f^2)*cos(f*x + e) - (-4*I*(a^2*b - b^3)*d^2*sin(f*x + e
) - 4*I*(a^3 - a*b^2)*d^2 + 2*(2*I*a^2*b*d^2*f*x + 2*I*a^2*b*c*d*f + (2*I*a*b^2*d^2*f*x + 2*I*a*b^2*c*d*f)*sin
(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(f*x + e) + 2*a*sin(f*x + e) + 2*(b*cos(f*x + e) - I*b
*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - (-4*I*(a^2*b - b^3)*d^2*sin(f*x + e) - 4*I*(a^3 - a*b^2)
*d^2 + 2*(-2*I*a^2*b*d^2*f*x - 2*I*a^2*b*c*d*f + (-2*I*a*b^2*d^2*f*x - 2*I*a*b^2*c*d*f)*sin(f*x + e))*sqrt(-(a
^2 - b^2)/b^2))*dilog(-1/2*(2*I*a*cos(f*x + e) + 2*a*sin(f*x + e) - 2*(b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt
(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - (4*I*(a^2*b - b^3)*d^2*sin(f*x + e) + 4*I*(a^3 - a*b^2)*d^2 + 2*(-2*I*a^2*b
*d^2*f*x - 2*I*a^2*b*c*d*f + (-2*I*a*b^2*d^2*f*x - 2*I*a*b^2*c*d*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*dilo
g(-1/2*(-2*I*a*cos(f*x + e) + 2*a*sin(f*x + e) + 2*(b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2)
+ 2*b)/b + 1) - (4*I*(a^2*b - b^3)*d^2*sin(f*x + e) + 4*I*(a^3 - a*b^2)*d^2 + 2*(2*I*a^2*b*d^2*f*x + 2*I*a^2*b
*c*d*f + (2*I*a*b^2*d^2*f*x + 2*I*a*b^2*c*d*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*dilog(-1/2*(-2*I*a*cos(f*
x + e) + 2*a*sin(f*x + e) - 2*(b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b + 1) - 2*(2*
(a^3 - a*b^2)*d^2*e - 2*(a^3 - a*b^2)*c*d*f + 2*((a^2*b - b^3)*d^2*e - (a^2*b - b^3)*c*d*f)*sin(f*x + e) + (a^
2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sin(f*x + e)
)*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - 2*
(2*(a^3 - a*b^2)*d^2*e - 2*(a^3 - a*b^2)*c*d*f + 2*((a^2*b - b^3)*d^2*e - (a^2*b - b^3)*c*d*f)*sin(f*x + e) +
(a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sin(f*x +
e))*sqrt(-(a^2 - b^2)/b^2))*log(2*b*cos(f*x + e) - 2*I*b*sin(f*x + e) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) -
2*(2*(a^3 - a*b^2)*d^2*e - 2*(a^3 - a*b^2)*c*d*f + 2*((a^2*b - b^3)*d^2*e - (a^2*b - b^3)*c*d*f)*sin(f*x + e)
- (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*sin(f*
x + e))*sqrt(-(a^2 - b^2)/b^2))*log(-2*b*cos(f*x + e) + 2*I*b*sin(f*x + e) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*
a) - 2*(2*(a^3 - a*b^2)*d^2*e - 2*(a^3 - a*b^2)*c*d*f + 2*((a^2*b - b^3)*d^2*e - (a^2*b - b^3)*c*d*f)*sin(f*x
+ e) - (a^2*b*d^2*e^2 - 2*a^2*b*c*d*e*f + a^2*b*c^2*f^2 + (a*b^2*d^2*e^2 - 2*a*b^2*c*d*e*f + a*b^2*c^2*f^2)*si
n(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*log(-2*b*cos(f*x + e) - 2*I*b*sin(f*x + e) + 2*b*sqrt(-(a^2 - b^2)/b^2) -
2*I*a) + 2*(2*(a^3 - a*b^2)*d^2*f*x + 2*(a^3 - a*b^2)*d^2*e + 2*((a^2*b - b^3)*d^2*f*x + (a^2*b - b^3)*d^2*e)*
sin(f*x + e) - (a^2*b*d^2*f^2*x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^2*x^2 +
2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos
(f*x + e) + 2*a*sin(f*x + e) + 2*(b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 2*(2*(
a^3 - a*b^2)*d^2*f*x + 2*(a^3 - a*b^2)*d^2*e + 2*((a^2*b - b^3)*d^2*f*x + (a^2*b - b^3)*d^2*e)*sin(f*x + e) +
(a^2*b*d^2*f^2*x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^
2*x - a*b^2*d^2*e^2 + 2*a*b^2*c*d*e*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(2*I*a*cos(f*x + e) + 2*a
*sin(f*x + e) - 2*(b*cos(f*x + e) - I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 2*(2*(a^3 - a*b^2)*d^
2*f*x + 2*(a^3 - a*b^2)*d^2*e + 2*((a^2*b - b^3)*d^2*f*x + (a^2*b - b^3)*d^2*e)*sin(f*x + e) - (a^2*b*d^2*f^2*
x^2 + 2*a^2*b*c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2
*e^2 + 2*a*b^2*c*d*e*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(-2*I*a*cos(f*x + e) + 2*a*sin(f*x + e)
+ 2*(b*cos(f*x + e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b) + 2*(2*(a^3 - a*b^2)*d^2*f*x + 2*(a^3
- a*b^2)*d^2*e + 2*((a^2*b - b^3)*d^2*f*x + (a^2*b - b^3)*d^2*e)*sin(f*x + e) + (a^2*b*d^2*f^2*x^2 + 2*a^2*b*
c*d*f^2*x - a^2*b*d^2*e^2 + 2*a^2*b*c*d*e*f + (a*b^2*d^2*f^2*x^2 + 2*a*b^2*c*d*f^2*x - a*b^2*d^2*e^2 + 2*a*b^2
*c*d*e*f)*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2))*log(1/2*(-2*I*a*cos(f*x + e) + 2*a*sin(f*x + e) - 2*(b*cos(f*x
+ e) + I*b*sin(f*x + e))*sqrt(-(a^2 - b^2)/b^2) + 2*b)/b))/((a^4*b - 2*a^2*b^3 + b^5)*f^3*sin(f*x + e) + (a^5
- 2*a^3*b^2 + a*b^4)*f^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d x + c\right )}^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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