3.2.99 \(\int \frac {(e+f x) \csc (c+d x)}{a+a \sin (c+d x)} \, dx\) [199]

Optimal. Leaf size=134 \[ -\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2} \]

[Out]

-2*(f*x+e)*arctanh(exp(I*(d*x+c)))/a/d+(f*x+e)*cot(1/2*c+1/4*Pi+1/2*d*x)/a/d-2*f*ln(sin(1/2*c+1/4*Pi+1/2*d*x))
/a/d^2+I*f*polylog(2,-exp(I*(d*x+c)))/a/d^2-I*f*polylog(2,exp(I*(d*x+c)))/a/d^2

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Rubi [A]
time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4631, 4268, 2317, 2438, 3399, 4269, 3556} \begin {gather*} \frac {i f \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )\right )}{a d^2}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{a d}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(e + f*x)*ArcTanh[E^(I*(c + d*x))])/(a*d) + ((e + f*x)*Cot[c/2 + Pi/4 + (d*x)/2])/(a*d) - (2*f*Log[Sin[c/2
+ Pi/4 + (d*x)/2]])/(a*d^2) + (I*f*PolyLog[2, -E^(I*(c + d*x))])/(a*d^2) - (I*f*PolyLog[2, E^(I*(c + d*x))])/
(a*d^2)

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 3399

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

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 (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \csc (c+d x) \, dx}{a}-\int \frac {e+f x}{a+a \sin (c+d x)} \, dx\\ &=-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {\int (e+f x) \csc ^2\left (\frac {1}{2} \left (c+\frac {\pi }{2}\right )+\frac {d x}{2}\right ) \, dx}{2 a}-\frac {f \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac {f \int \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right ) \, dx}{a d}\\ &=-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {2 f \log \left (\sin \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )\right )}{a d^2}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a 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. \(300\) vs. \(2(134)=268\).
time = 0.62, size = 300, normalized size = 2.24 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (-2 d (e+f x) \sin \left (\frac {1}{2} (c+d x)\right )+f (c+d x) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+d e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )-c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+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 ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{a d^2 (1+\sin (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(-2*d*(e + f*x)*Sin[(c + d*x)/2] + f*(c + d*x)*(Cos[(c + d*x)/2] + Sin[
(c + d*x)/2]) - 2*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + d*e*Log[T
an[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) - c*f*Log[Tan[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Sin[(c
+ d*x)/2]) + 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))]))*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))/(a*d^2*(1 + Sin[c + d*x]))

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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. 244 vs. \(2 (114 ) = 228\).
time = 0.17, size = 245, normalized size = 1.83

method	result	size
risch	\(\frac {2 f x +2 e}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {e \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {e \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}-\frac {f c \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d^{2} a}-\frac {i f \polylog \left (2, {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {i f \polylog \left (2, -{\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}+\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{d^{2} a}+\frac {\ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) f x}{d a}+\frac {\ln \left (1-{\mathrm e}^{i \left (d x +c \right )}\right ) c f}{d^{2} a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) f x}{d a}-\frac {2 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{a \,d^{2}}\)	\(245\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (112) = 224\).
time = 0.40, size = 527, normalized size = 3.93 \begin {gather*} \frac {4 \, d f x \cos \left (d x + c\right ) + 4 i \, d f x \sin \left (d x + c\right ) - 4 \, {\left (f \cos \left (d x + c\right ) + i \, f \sin \left (d x + c\right ) + i \, f\right )} \arctan \left (\cos \left (c\right ) + \sin \left (d x\right ), \cos \left (d x\right ) + \sin \left (c\right )\right ) + 2 \, {\left (-i \, d f x - {\left (d f x + d e\right )} \cos \left (d x + c\right ) - i \, d e + {\left (-i \, d f x - i \, d e\right )} \sin \left (d x + c\right )\right )} \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) + 1\right ) + 2 \, {\left (d \cos \left (d x + c\right ) e + i \, d e \sin \left (d x + c\right ) + i \, d e\right )} \arctan \left (\sin \left (d x + c\right ), \cos \left (d x + c\right ) - 1\right ) - 2 \, {\left (d f x \cos \left (d x + c\right ) + i \, d f x \sin \left (d x + c\right ) + i \, d f x\right )} \arctan \left (\sin \left (d x + c\right ), -\cos \left (d x + c\right ) + 1\right ) + 2 \, {\left (f \cos \left (d x + c\right ) + i \, f \sin \left (d x + c\right ) + i \, f\right )} {\rm Li}_2\left (-e^{\left (i \, d x + i \, c\right )}\right ) - 2 \, {\left (f \cos \left (d x + c\right ) + i \, f \sin \left (d x + c\right ) + i \, f\right )} {\rm Li}_2\left (e^{\left (i \, d x + i \, c\right )}\right ) - 4 i \, d e - {\left (d f x + {\left (-i \, d f x - i \, d e\right )} \cos \left (d x + c\right ) + d e + {\left (d f x + d e\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1\right ) + {\left (d f x - {\left (i \, d f x + i \, d e\right )} \cos \left (d x + c\right ) + d e + {\left (d f x + d e\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right ) + 1\right ) + 2 \, {\left (i \, f \cos \left (d x + c\right ) - f \sin \left (d x + c\right ) - f\right )} \log \left (\cos \left (d x\right )^{2} + \cos \left (c\right )^{2} + 2 \, \cos \left (c\right ) \sin \left (d x\right ) + \sin \left (d x\right )^{2} + 2 \, \cos \left (d x\right ) \sin \left (c\right ) + \sin \left (c\right )^{2}\right )}{-2 i \, a d^{2} \cos \left (d x + c\right ) + 2 \, a d^{2} \sin \left (d x + c\right ) + 2 \, a d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*d*f*x*cos(d*x + c) + 4*I*d*f*x*sin(d*x + c) - 4*(f*cos(d*x + c) + I*f*sin(d*x + c) + I*f)*arctan2(cos(c) +
sin(d*x), cos(d*x) + sin(c)) + 2*(-I*d*f*x - (d*f*x + d*e)*cos(d*x + c) - I*d*e + (-I*d*f*x - I*d*e)*sin(d*x +
c))*arctan2(sin(d*x + c), cos(d*x + c) + 1) + 2*(d*cos(d*x + c)*e + I*d*e*sin(d*x + c) + I*d*e)*arctan2(sin(d
*x + c), cos(d*x + c) - 1) - 2*(d*f*x*cos(d*x + c) + I*d*f*x*sin(d*x + c) + I*d*f*x)*arctan2(sin(d*x + c), -co
s(d*x + c) + 1) + 2*(f*cos(d*x + c) + I*f*sin(d*x + c) + I*f)*dilog(-e^(I*d*x + I*c)) - 2*(f*cos(d*x + c) + I*
f*sin(d*x + c) + I*f)*dilog(e^(I*d*x + I*c)) - 4*I*d*e - (d*f*x + (-I*d*f*x - I*d*e)*cos(d*x + c) + d*e + (d*f
*x + d*e)*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*cos(d*x + c) + 1) + (d*f*x - (I*d*f*x + I*d*e)
*cos(d*x + c) + d*e + (d*f*x + d*e)*sin(d*x + c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*cos(d*x + c) + 1) +
2*(I*f*cos(d*x + c) - f*sin(d*x + c) - f)*log(cos(d*x)^2 + cos(c)^2 + 2*cos(c)*sin(d*x) + sin(d*x)^2 + 2*cos(d
*x)*sin(c) + sin(c)^2))/(-2*I*a*d^2*cos(d*x + c) + 2*a*d^2*sin(d*x + c) + 2*a*d^2)

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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. 626 vs. \(2 (112) = 224\).
time = 0.40, size = 626, normalized size = 4.67 \begin {gather*} \frac {2 \, d f x + 2 \, {\left (d f x + d e\right )} \cos \left (d x + c\right ) + {\left (-i \, f \cos \left (d x + c\right ) - i \, f \sin \left (d x + c\right ) - i \, f\right )} {\rm Li}_2\left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, f \cos \left (d x + c\right ) + i \, f \sin \left (d x + c\right ) + i \, f\right )} {\rm Li}_2\left (\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (-i \, f \cos \left (d x + c\right ) - i \, f \sin \left (d x + c\right ) - i \, f\right )} {\rm Li}_2\left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (i \, f \cos \left (d x + c\right ) + i \, f \sin \left (d x + c\right ) + i \, f\right )} {\rm Li}_2\left (-\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, d e - {\left (d f x + {\left (d f x + d e\right )} \cos \left (d x + c\right ) + d e + {\left (d f x + d e\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) - {\left (d f x + {\left (d f x + d e\right )} \cos \left (d x + c\right ) + d e + {\left (d f x + d e\right )} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) - {\left (c f + {\left (c f - d e\right )} \cos \left (d x + c\right ) - d e + {\left (c f - d e\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2} i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) - {\left (c f + {\left (c f - d e\right )} \cos \left (d x + c\right ) - d e + {\left (c f - d e\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) - \frac {1}{2} i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (d f x + c f + {\left (d f x + c f\right )} \cos \left (d x + c\right ) + {\left (d f x + c f\right )} \sin \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) + {\left (d f x + c f + {\left (d f x + c f\right )} \cos \left (d x + c\right ) + {\left (d f x + c f\right )} \sin \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (f \cos \left (d x + c\right ) + f \sin \left (d x + c\right ) + f\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (d f x + d e\right )} \sin \left (d x + c\right )}{2 \, {\left (a d^{2} \cos \left (d x + c\right ) + a d^{2} \sin \left (d x + c\right ) + a d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {e \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f x \csc {\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(e*csc(c + d*x)/(sin(c + d*x) + 1), x) + Integral(f*x*csc(c + d*x)/(sin(c + d*x) + 1), x))/a

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \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)*(a + a*sin(c + d*x))),x)

[Out]

\text{Hanged}

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