3.3.38 \(\int \frac {(e+f x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [238]

Optimal. Leaf size=370 \[ \frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}-\frac {(e+f x) \cot (c+d x)}{a d}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {i b f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {Li}_2\left (e^{i (c+d x)}\right )}{a^2 d^2}-\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2}+\frac {b^2 f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 \sqrt {a^2-b^2} d^2} \]

[Out]

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

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Rubi [A]
time = 0.41, antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4631, 4269, 3556, 4268, 2317, 2438, 3404, 2296, 2221} \begin {gather*} -\frac {b^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d^2 \sqrt {a^2-b^2}}+\frac {b^2 f \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d^2 \sqrt {a^2-b^2}}-\frac {i b f \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a^2 d^2}+\frac {i b f \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a^2 d^2}-\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {i b^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 d \sqrt {a^2-b^2}}+\frac {2 b (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a^2 d}+\frac {f \log (\sin (c+d x))}{a d^2}-\frac {(e+f x) \cot (c+d x)}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 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 3556

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

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 4269

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

Rule 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]

Rubi steps

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(822\) vs. \(2(370)=740\).
time = 7.61, size = 822, normalized size = 2.22 \begin {gather*} \frac {-a d (e+f x) \cot \left (\frac {1}{2} (c+d x)\right )+2 a f \log (\sin (c+d x))-2 b d e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )+2 b c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )-2 b f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )+\frac {2 b^2 d (e+f x) \left (\frac {2 (d e-c f) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {-b+\sqrt {-a^2+b^2}-a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )\right )}{\sqrt {-a^2+b^2}}-\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}\right )}{d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}+a d (e+f x) \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a*d*(e + f*x)*Cot[(c + d*x)/2]) + 2*a*f*Log[Sin[c + d*x]] - 2*b*d*e*Log[Tan[(c + d*x)/2]] + 2*b*c*f*Log[Tan
[(c + d*x)/2]] - 2*b*f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + d*x))]) + I*(PolyLog[2, -E^(I*
(c + d*x))] - PolyLog[2, E^(I*(c + d*x))])) + (2*b^2*d*(e + f*x)*((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2
])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c
+ d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 +
b^2]))]))/Sqrt[-a^2 + b^2] + (I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])
/(I*a + b + Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]))/Sqr
t[-a^2 + b^2] + (I*f*(Log[1 - I*Tan[(c + d*x)/2]]*Log[(-b + Sqrt[-a^2 + b^2] - a*Tan[(c + d*x)/2])/(I*a - b +
Sqrt[-a^2 + b^2])] + PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])]))/Sqrt[-a^2 + b^2] -
(I*f*(Log[1 + I*Tan[(c + d*x)/2]]*Log[(b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])
] + PolyLog[2, (a + I*a*Tan[(c + d*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))]))/Sqrt[-a^2 + b^2]))/(d*e - c*f + I
*f*Log[1 - I*Tan[(c + d*x)/2]] - I*f*Log[1 + I*Tan[(c + d*x)/2]]) + a*d*(e + f*x)*Tan[(c + d*x)/2])/(2*a^2*d^2
)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (332 ) = 664\).
time = 0.16, size = 766, normalized size = 2.07

method	result	size
risch	\(\frac {i b^{2} f \dilog \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {b^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {-a^{2}+b^{2}}}+\frac {b^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {b f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) x}{a^{2} d}-\frac {i b f \dilog \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d^{2}}-\frac {b^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) x}{a^{2} d \sqrt {-a^{2}+b^{2}}}-\frac {b^{2} f \ln \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}+\sqrt {-a^{2}+b^{2}}}{i a +\sqrt {-a^{2}+b^{2}}}\right ) c}{a^{2} d^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}-\frac {b e \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d}+\frac {b e \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{2} d}+\frac {b f c \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a^{2} d^{2}}-\frac {2 i \left (f x +e \right )}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}-\frac {i b^{2} f \dilog \left (\frac {i a +b \,{\mathrm e}^{i \left (d x +c \right )}-\sqrt {-a^{2}+b^{2}}}{i a -\sqrt {-a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 i b^{2} f c \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a^{2} d^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 i b^{2} e \arctan \left (\frac {2 i b \,{\mathrm e}^{i \left (d x +c \right )}-2 a}{2 \sqrt {-a^{2}+b^{2}}}\right )}{a^{2} d \sqrt {-a^{2}+b^{2}}}-\frac {i b f \dilog \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a^{2} d^{2}}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a \,d^{2}}\)	\(766\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)*csc(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

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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. 1695 vs. \(2 (324) = 648\).
time = 0.63, size = 1695, normalized size = 4.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(I*b^3*f*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)*sin(d*x + c) - I*b^3*f*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)*sin(d*x + c) - I*b^3
*f*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)*sin(d*x + c) + I*b^3*f*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)*sin(d*x + c) + I*(a^2*b - b^
3)*f*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) - I*(a^2*b - b^3)*f*dilog(cos(d*x + c) - I*sin(d*x + c)
)*sin(d*x + c) + I*(a^2*b - b^3)*f*dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) - I*(a^2*b - b^3)*f*dilo
g(-cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) - (b^3*c*f - b^3*d*e)*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)*sin(d*x + c) - (b^3*c*f - b^3*d*e)*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)*sin(d*x + c) + (b^3*
c*f - b^3*d*e)*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)*sin(d*x + c) + (b^3*c*f - b^3*d*e)*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)*sin(d*x + c) - (b^3*d*f*x + b^3*c*f)*sqrt(-(a^2 - b^2)/b^2)*log(-(I*a*co
s(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^3*d*f*x + b^3*c*f)*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)*sin(d*x + c) - (b^3*d*f*x + b^3*c*f)*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)*sin(
d*x + c) + (b^3*d*f*x + b^3*c*f)*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)*sin(d*x + c) + ((a^2*b - b^3)*d*f*x + (a^2*b - b^3)*d*
e + (a^3 - a*b^2)*f)*log(cos(d*x + c) + I*sin(d*x + c) + 1)*sin(d*x + c) + ((a^2*b - b^3)*d*f*x + (a^2*b - b^3
)*d*e + (a^3 - a*b^2)*f)*log(cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d*x + c) - ((a^2*b - b^3)*d*e - (a^3 - a*b
^2 + (a^2*b - b^3)*c)*f)*log(-1/2*cos(d*x + c) + 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) - ((a^2*b - b^3)*d*e -
(a^3 - a*b^2 + (a^2*b - b^3)*c)*f)*log(-1/2*cos(d*x + c) - 1/2*I*sin(d*x + c) + 1/2)*sin(d*x + c) - ((a^2*b -
b^3)*d*f*x + (a^2*b - b^3)*c*f)*log(-cos(d*x + c) + I*sin(d*x + c) + 1)*sin(d*x + c) - ((a^2*b - b^3)*d*f*x +
(a^2*b - b^3)*c*f)*log(-cos(d*x + c) - I*sin(d*x + c) + 1)*sin(d*x + c) - 2*((a^3 - a*b^2)*d*f*x + (a^3 - a*b
^2)*d*e)*cos(d*x + c))/((a^4 - a^2*b^2)*d^2*sin(d*x + c))

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

[Out]

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

[Out]

\text{Hanged}

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