3.3.33 \(\int \frac {(e+f x)^2 \csc (c+d x)}{a+b \sin (c+d x)} \, dx\) [233]

Optimal. Leaf size=528 \[ -\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {2 i f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {2 b f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {2 b f (e+f x) \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {2 i b f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {2 i b f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3} \]

[Out]

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

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Rubi [A]
time = 0.63, antiderivative size = 528, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4631, 4268, 2611, 2320, 6724, 3404, 2296, 2221} \begin {gather*} \frac {2 i b f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^3 \sqrt {a^2-b^2}}-\frac {2 i b f^2 \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d^3 \sqrt {a^2-b^2}}+\frac {2 b f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^2 \sqrt {a^2-b^2}}-\frac {2 b f (e+f x) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d^2 \sqrt {a^2-b^2}}-\frac {2 f^2 \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {2 f^2 \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {2 i f (e+f x) \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {2 i f (e+f x) \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {i b (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {i b (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d \sqrt {a^2-b^2}}-\frac {2 (e+f x)^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2296

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 2320

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 2611

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 3404

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 4268

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

Rule 4631

Int[(Csc[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Csc[c + d*x]^n, x], x] - Dist[b/a, Int[(e + f*x)^m*(Csc[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]

Rule 6724

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]

Rubi steps

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

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Mathematica [A]
time = 2.49, size = 982, normalized size = 1.86 \begin {gather*} \frac {-2 d^2 e^2 \tanh ^{-1}\left (e^{i (c+d x)}\right )+2 d^2 e f x \log \left (1-e^{i (c+d x)}\right )+d^2 f^2 x^2 \log \left (1-e^{i (c+d x)}\right )-2 d^2 e f x \log \left (1+e^{i (c+d x)}\right )-d^2 f^2 x^2 \log \left (1+e^{i (c+d x)}\right )+2 i d f (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right )-2 i d f (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right )-2 f^2 \text {Li}_3\left (-e^{i (c+d x)}\right )+2 f^2 \text {Li}_3\left (e^{i (c+d x)}\right )+\frac {b \left (2 \sqrt {a^2-b^2} d e^{i c} f (e+f x) \text {Li}_2\left (\frac {i b e^{i (2 c+d x)}}{a e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+i \left (2 d^2 e^2 \sqrt {\left (a^2-b^2\right ) e^{2 i c}} \tanh ^{-1}\left (\frac {-a+i b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right )+2 \sqrt {a^2-b^2} d^2 e e^{i c} f x \log \left (1-\frac {i b e^{i (2 c+d x)}}{a e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+\sqrt {a^2-b^2} d^2 e^{i c} f^2 x^2 \log \left (1-\frac {i b e^{i (2 c+d x)}}{a e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-2 \sqrt {a^2-b^2} d^2 e e^{i c} f x \log \left (1-\frac {i b e^{i (2 c+d x)}}{a e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-\sqrt {a^2-b^2} d^2 e^{i c} f^2 x^2 \log \left (1-\frac {i b e^{i (2 c+d x)}}{a e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 i \sqrt {a^2-b^2} d e^{i c} f (e+f x) \text {Li}_2\left (\frac {i b e^{i (2 c+d x)}}{a e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )+2 \sqrt {a^2-b^2} e^{i c} f^2 \text {Li}_3\left (\frac {i b e^{i (2 c+d x)}}{a e^{i c}-\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )-2 \sqrt {a^2-b^2} e^{i c} f^2 \text {Li}_3\left (\frac {i b e^{i (2 c+d x)}}{a e^{i c}+\sqrt {\left (a^2-b^2\right ) e^{2 i c}}}\right )\right )\right )}{\sqrt {a^2-b^2} \sqrt {\left (a^2-b^2\right ) e^{2 i c}}}}{a d^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d^2*e^2*ArcTanh[E^(I*(c + d*x))] + 2*d^2*e*f*x*Log[1 - E^(I*(c + d*x))] + d^2*f^2*x^2*Log[1 - E^(I*(c + d*
x))] - 2*d^2*e*f*x*Log[1 + E^(I*(c + d*x))] - d^2*f^2*x^2*Log[1 + E^(I*(c + d*x))] + (2*I)*d*f*(e + f*x)*PolyL
og[2, -E^(I*(c + d*x))] - (2*I)*d*f*(e + f*x)*PolyLog[2, E^(I*(c + d*x))] - 2*f^2*PolyLog[3, -E^(I*(c + d*x))]
+ 2*f^2*PolyLog[3, E^(I*(c + d*x))] + (b*(2*Sqrt[a^2 - b^2]*d*E^(I*c)*f*(e + f*x)*PolyLog[2, (I*b*E^(I*(2*c +
d*x)))/(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + I*(2*d^2*e^2*Sqrt[(a^2 - b^2)*E^((2*I)*c)]*ArcTanh[(-a
+ I*b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 2*Sqrt[a^2 - b^2]*d^2*e*E^(I*c)*f*x*Log[1 - (I*b*E^(I*(2*c + d*x)))/
(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + Sqrt[a^2 - b^2]*d^2*E^(I*c)*f^2*x^2*Log[1 - (I*b*E^(I*(2*c + d*
x)))/(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*Sqrt[a^2 - b^2]*d^2*e*E^(I*c)*f*x*Log[1 - (I*b*E^(I*(2*c
+ d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - Sqrt[a^2 - b^2]*d^2*E^(I*c)*f^2*x^2*Log[1 - (I*b*E^(I
*(2*c + d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + (2*I)*Sqrt[a^2 - b^2]*d*E^(I*c)*f*(e + f*x)*Poly
Log[2, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])] + 2*Sqrt[a^2 - b^2]*E^(I*c)*f^2*Po
lyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) - Sqrt[(a^2 - b^2)*E^((2*I)*c)])] - 2*Sqrt[a^2 - b^2]*E^(I*c)*f^2*
PolyLog[3, (I*b*E^(I*(2*c + d*x)))/(a*E^(I*c) + Sqrt[(a^2 - b^2)*E^((2*I)*c)])])))/(Sqrt[a^2 - b^2]*Sqrt[(a^2
- b^2)*E^((2*I)*c)]))/(a*d^3)

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{2} \csc \left (d x +c \right )}{a +b \sin \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^2*csc(d*x+c)/(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

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2426 vs. \(2 (461) = 922\).
time = 0.61, size = 2426, normalized size = 4.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b^2*f^2*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(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) - 2*b^2*f^2*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(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) + 2*b^2*f^2*sqrt(-(a^2 - b^2)/b^
2)*polylog(3, -(-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) - 2*b^2*f^2*sqrt(-(a^2 - b^2)/b^2)*polylog(3, -(-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) - 2*(a^2 - b^2)*f^2*polylog(3, cos(d*x + c) + I*sin(d*x + c)) - 2*
(a^2 - b^2)*f^2*polylog(3, cos(d*x + c) - I*sin(d*x + c)) + 2*(a^2 - b^2)*f^2*polylog(3, -cos(d*x + c) + I*sin
(d*x + c)) + 2*(a^2 - b^2)*f^2*polylog(3, -cos(d*x + c) - I*sin(d*x + c)) - 2*(-I*b^2*d*f^2*x - I*b^2*d*f*e)*s
qrt(-(a^2 - b^2)/b^2)*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) - 2*(I*b^2*d*f^2*x + I*b^2*d*f*e)*sqrt(-(a^2 - b^2)/b^2)*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) - 2*(I*b^2*d*f^2*x + I*
b^2*d*f*e)*sqrt(-(a^2 - b^2)/b^2)*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) - 2*(-I*b^2*d*f^2*x - I*b^2*d*f*e)*sqrt(-(a^2 - b^2)/b^2)*dilog((-I*a*c
os(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) + (b^2*c
^2*f^2 - 2*b^2*c*d*f*e + b^2*d^2*e^2)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*cos(d*x + c) + 2*I*b*sin(d*x + c) + 2*b*s
qrt(-(a^2 - b^2)/b^2) + 2*I*a) + (b^2*c^2*f^2 - 2*b^2*c*d*f*e + b^2*d^2*e^2)*sqrt(-(a^2 - b^2)/b^2)*log(2*b*co
s(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) - (b^2*c^2*f^2 - 2*b^2*c*d*f*e + b^2*d^2
*e^2)*sqrt(-(a^2 - b^2)/b^2)*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)
- (b^2*c^2*f^2 - 2*b^2*c*d*f*e + b^2*d^2*e^2)*sqrt(-(a^2 - b^2)/b^2)*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) - (b^2*d^2*f^2*x^2 - b^2*c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*e)*sqrt
(-(a^2 - b^2)/b^2)*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) + (b^2*d^2*f^2*x^2 - b^2*c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*e)*sqrt(-(a^2 - b^2)/b^2)*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) - (b
^2*d^2*f^2*x^2 - b^2*c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*e)*sqrt(-(a^2 - b^2)/b^2)*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) + (b^2*d^2*f^2*x^2 - b^2*
c^2*f^2 + 2*(b^2*d^2*f*x + b^2*c*d*f)*e)*sqrt(-(a^2 - b^2)/b^2)*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) - 2*(-I*(a^2 - b^2)*d*f^2*x - I*(a^2 - b^2)*d*
f*e)*dilog(cos(d*x + c) + I*sin(d*x + c)) - 2*(I*(a^2 - b^2)*d*f^2*x + I*(a^2 - b^2)*d*f*e)*dilog(cos(d*x + c)
- I*sin(d*x + c)) - 2*(-I*(a^2 - b^2)*d*f^2*x - I*(a^2 - b^2)*d*f*e)*dilog(-cos(d*x + c) + I*sin(d*x + c)) -
2*(I*(a^2 - b^2)*d*f^2*x + I*(a^2 - b^2)*d*f*e)*dilog(-cos(d*x + c) - I*sin(d*x + c)) + ((a^2 - b^2)*d^2*f^2*x
^2 + 2*(a^2 - b^2)*d^2*f*x*e + (a^2 - b^2)*d^2*e^2)*log(cos(d*x + c) + I*sin(d*x + c) + 1) + ((a^2 - b^2)*d^2*
f^2*x^2 + 2*(a^2 - b^2)*d^2*f*x*e + (a^2 - b^2)*d^2*e^2)*log(cos(d*x + c) - I*sin(d*x + c) + 1) - ((a^2 - b^2)
*c^2*f^2 - 2*(a^2 - b^2)*c*d*f*e + (a^2 - b^2)*d^2*e^2)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2) - ((
a^2 - b^2)*c^2*f^2 - 2*(a^2 - b^2)*c*d*f*e + (a^2 - b^2)*d^2*e^2)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) +
1/2) - ((a^2 - b^2)*d^2*f^2*x^2 - (a^2 - b^2)*c^2*f^2 + 2*((a^2 - b^2)*d^2*f*x + (a^2 - b^2)*c*d*f)*e)*log(-c
os(d*x + c) + I*sin(d*x + c) + 1) - ((a^2 - b^2)*d^2*f^2*x^2 - (a^2 - b^2)*c^2*f^2 + 2*((a^2 - b^2)*d^2*f*x +
(a^2 - b^2)*c*d*f)*e)*log(-cos(d*x + c) - I*sin(d*x + c) + 1))/((a^3 - a*b^2)*d^3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e + f x\right )^{2} \csc {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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