\(\int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [300]

Optimal result

Integrand size = 26, antiderivative size = 298 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {a e x}{b^2}+\frac {a f x^2}{2 b^2}+\frac {(e+f x) \cos (c+d x)}{b d}+\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d}+\frac {\sqrt {a^2-b^2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {\sqrt {a^2-b^2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {f \sin (c+d x)}{b d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 298, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4621, 3377, 2717, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {f \sqrt {a^2-b^2} \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d^2}-\frac {f \sqrt {a^2-b^2} \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^2 d^2}+\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^2 d}-\frac {i \sqrt {a^2-b^2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b^2 d}+\frac {a e x}{b^2}+\frac {a f x^2}{2 b^2}-\frac {f \sin (c+d x)}{b d^2}+\frac {(e+f x) \cos (c+d x)}{b d} \]

[In]

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

[Out]

(a*e*x)/b^2 + (a*f*x^2)/(2*b^2) + ((e + f*x)*Cos[c + d*x])/(b*d) + (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E
^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b^2*d) - (I*Sqrt[a^2 - b^2]*(e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(
a + Sqrt[a^2 - b^2])])/(b^2*d) + (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(
b^2*d^2) - (Sqrt[a^2 - b^2]*f*PolyLog[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b^2*d^2) - (f*Sin[c +
d*x])/(b*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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 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 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]

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(780\) vs. \(2(298)=596\).

Time = 6.41 (sec) , antiderivative size = 780, normalized size of antiderivative = 2.62 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-a (c+d x) (c f-d (2 e+f x))+2 b d (e+f x) \cos (c+d x)+\frac {2 \left (-a^2+b^2\right ) d (e+f x) \left (\frac {2 (d e-c f) \arctan \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 \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 )}{\sqrt {-a^2+b^2}}-\frac {i f \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 )}{\sqrt {-a^2+b^2}}-\frac {i f \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 )}{\sqrt {-a^2+b^2}}+\frac {i f \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 )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\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 )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\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 )}{\sqrt {-a^2+b^2}}+\frac {i f \operatorname {PolyLog}\left (2,\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}-\frac {i f \operatorname {PolyLog}\left (2,\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\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 )}-2 b f \sin (c+d x)}{2 b^2 d^2} \]

[In]

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

[Out]

(-(a*(c + d*x)*(c*f - d*(2*e + f*x))) + 2*b*d*(e + f*x)*Cos[c + d*x] + (2*(-a^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])])/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])])/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])])/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])])/Sqrt[-a^2 + b^2] - (I*f*PolyLog[2, (a*(1 - I*Tan[(c + d*x)/2]))/(a + I*(b + Sq
rt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] + (I*f*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*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 +
b^2] - (I*f*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]]) - 2*b*f*Sin[c + d*x])/(2*b^2*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. 1122 vs. \(2 (266 ) = 532\).

Time = 0.64 (sec) , antiderivative size = 1123, normalized size of antiderivative = 3.77

method	result	size
risch	\(\text {Expression too large to display}\)	\(1123\)

[In]

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

[Out]

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

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. 1034 vs. \(2 (263) = 526\).

Time = 0.41 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.47 \[ \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

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

[In]

integrate((f*x+e)*cos(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

\text{Hanged}