3.163 \(\int \frac{x \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx\)

Optimal. Leaf size=267 \[ -\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right )}{4 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt{a+c} \sqrt{b+c}+c\right )+a+b}\right )}{4 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right )}{2 d \sqrt{a+c} \sqrt{b+c}}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt{a+c} \sqrt{b+c}+c\right )+a+b}\right )}{2 d \sqrt{a+c} \sqrt{b+c}} \]

[Out]

((-I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(Sqrt[a + c]*Sqrt[
b + c]*d) + ((I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(Sqrt[a
+ c]*Sqrt[b + c]*d) - PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))]/
(4*Sqrt[a + c]*Sqrt[b + c]*d^2) + PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[
b + c])))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2)

________________________________________________________________________________________

Rubi [A]  time = 0.718662, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4589, 3321, 2264, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right )}{4 d^2 \sqrt{a+c} \sqrt{b+c}}+\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt{a+c} \sqrt{b+c}+c\right )+a+b}\right )}{4 d^2 \sqrt{a+c} \sqrt{b+c}}-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{-2 \sqrt{a+c} \sqrt{b+c}+a+b+2 c}\right )}{2 d \sqrt{a+c} \sqrt{b+c}}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt{a+c} \sqrt{b+c}+c\right )+a+b}\right )}{2 d \sqrt{a+c} \sqrt{b+c}} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]

[Out]

((-I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(Sqrt[a + c]*Sqrt[
b + c]*d) + ((I/2)*x*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(Sqrt[a
+ c]*Sqrt[b + c]*d) - PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c]))]/
(4*Sqrt[a + c]*Sqrt[b + c]*d^2) + PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[
b + c])))]/(4*Sqrt[a + c]*Sqrt[b + c]*d^2)

Rule 4589

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

Rule 3321

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c
+ d*x)^m*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(
2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && 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 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{x \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx &=2 \int \frac{x}{a+b+2 c+(a-b) \cos (2 c+2 d x)} \, dx\\ &=4 \int \frac{e^{i (2 c+2 d x)} x}{a-b+2 (a+b+2 c) e^{i (2 c+2 d x)}+(a-b) e^{2 i (2 c+2 d x)}} \, dx\\ &=\frac{(2 (a-b)) \int \frac{e^{i (2 c+2 d x)} x}{-4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)+2 (a-b) e^{i (2 c+2 d x)}} \, dx}{\sqrt{a+c} \sqrt{b+c}}-\frac{(2 (a-b)) \int \frac{e^{i (2 c+2 d x)} x}{4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)+2 (a-b) e^{i (2 c+2 d x)}} \, dx}{\sqrt{a+c} \sqrt{b+c}}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt{a+c} \sqrt{b+c}\right )}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}+\frac{i \int \log \left (1+\frac{2 (a-b) e^{i (2 c+2 d x)}}{-4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)}\right ) \, dx}{2 \sqrt{a+c} \sqrt{b+c} d}-\frac{i \int \log \left (1+\frac{2 (a-b) e^{i (2 c+2 d x)}}{4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)}\right ) \, dx}{2 \sqrt{a+c} \sqrt{b+c} d}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt{a+c} \sqrt{b+c}\right )}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a-b) x}{-4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt{a+c} \sqrt{b+c} d^2}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a-b) x}{4 \sqrt{a+c} \sqrt{b+c}+2 (a+b+2 c)}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt{a+c} \sqrt{b+c} d^2}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt{a+c} \sqrt{b+c}\right )}\right )}{2 \sqrt{a+c} \sqrt{b+c} d}-\frac{\text{Li}_2\left (-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt{a+c} \sqrt{b+c}}\right )}{4 \sqrt{a+c} \sqrt{b+c} d^2}+\frac{\text{Li}_2\left (-\frac{(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt{a+c} \sqrt{b+c}\right )}\right )}{4 \sqrt{a+c} \sqrt{b+c} d^2}\\ \end{align*}

Mathematica [B]  time = 4.34411, size = 751, normalized size = 2.81 \[ \frac{x \left (\sqrt{a+c}-\sqrt{-b-c} \tan (c+d x)\right ) \left (\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right ) \left (i \sqrt{b+c} \text{PolyLog}\left (2,\frac{\sqrt{-b-c} (1-i \tan (c+d x))}{\sqrt{-b-c}-i \sqrt{a+c}}\right )-i \sqrt{b+c} \text{PolyLog}\left (2,\frac{\sqrt{-b-c} (1-i \tan (c+d x))}{\sqrt{-b-c}+i \sqrt{a+c}}\right )+i \sqrt{b+c} \text{PolyLog}\left (2,\frac{\sqrt{-b-c} (1+i \tan (c+d x))}{\sqrt{-b-c}-i \sqrt{a+c}}\right )-i \sqrt{b+c} \text{PolyLog}\left (2,\frac{\sqrt{-b-c} (1+i \tan (c+d x))}{\sqrt{-b-c}+i \sqrt{a+c}}\right )+4 c \sqrt{-b-c} \tan ^{-1}\left (\frac{\sqrt{b+c} \tan (c+d x)}{\sqrt{a+c}}\right )-i \sqrt{b+c} \log (1+i \tan (c+d x)) \log \left (\frac{i \left (\sqrt{a+c}-\sqrt{-b-c} \tan (c+d x)\right )}{\sqrt{-b-c}+i \sqrt{a+c}}\right )+i \sqrt{b+c} \log (1-i \tan (c+d x)) \log \left (\frac{i \left (\sqrt{-b-c} \tan (c+d x)-\sqrt{a+c}\right )}{\sqrt{-b-c}-i \sqrt{a+c}}\right )+i \sqrt{b+c} \log (1+i \tan (c+d x)) \log \left (-\frac{i \left (\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right )}{\sqrt{-b-c}-i \sqrt{a+c}}\right )-i \sqrt{b+c} \log (1-i \tan (c+d x)) \log \left (\frac{i \left (\sqrt{a+c}+\sqrt{-b-c} \tan (c+d x)\right )}{\sqrt{-b-c}+i \sqrt{a+c}}\right )\right )}{2 d \sqrt{a+c} \sqrt{-(b+c)^2} (-i \log (1-i \tan (c+d x))+i \log (1+i \tan (c+d x))+2 c) \left (a+b \tan ^2(c+d x)+c \sec ^2(c+d x)\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]

[Out]

(x*(4*Sqrt[-b - c]*c*ArcTan[(Sqrt[b + c]*Tan[c + d*x])/Sqrt[a + c]] - I*Sqrt[b + c]*Log[1 + I*Tan[c + d*x]]*Lo
g[(I*(Sqrt[a + c] - Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])] + I*Sqrt[b + c]*Log[1 - I*Tan[
c + d*x]]*Log[(I*(-Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] + I*Sqrt[b + c]*L
og[1 + I*Tan[c + d*x]]*Log[((-I)*(Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] -
I*Sqrt[b + c]*Log[1 - I*Tan[c + d*x]]*Log[(I*(Sqrt[a + c] + Sqrt[-b - c]*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt
[a + c])] + I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 - I*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] - I*S
qrt[b + c]*PolyLog[2, (Sqrt[-b - c]*(1 - I*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])] + I*Sqrt[b + c]*Poly
Log[2, (Sqrt[-b - c]*(1 + I*Tan[c + d*x]))/(Sqrt[-b - c] - I*Sqrt[a + c])] - I*Sqrt[b + c]*PolyLog[2, (Sqrt[-b
- c]*(1 + I*Tan[c + d*x]))/(Sqrt[-b - c] + I*Sqrt[a + c])])*(Sqrt[a + c] - Sqrt[-b - c]*Tan[c + d*x])*(Sqrt[a
+ c] + Sqrt[-b - c]*Tan[c + d*x]))/(2*Sqrt[a + c]*Sqrt[-(b + c)^2]*d*(2*c - I*Log[1 - I*Tan[c + d*x]] + I*Log
[1 + I*Tan[c + d*x]])*(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2))

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Maple [B]  time = 0.201, size = 1670, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x)

[Out]

-1/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*a*c*x-1/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2
)-a-b-2*c)*b*c*x-I/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((
a+c)*(b+c))^(1/2)-a-b-2*c))*c^2-1/2/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*a*c^2-1/2/d^2/((a
+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*c^2-1/4/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-
b-2*c)*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*a-1/4/d^2/((a+c)*(b+c))^(1/2)/(-2*((
a+c)*(b+c))^(1/2)-a-b-2*c)*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*b-1/2/d^2/((a+c)
*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*
c))*c-2/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c^2*x-I/d^2/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln
(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*c-1/2*I/d^2/((a+c)*(b+c))^(1/2)*ln(1-(a-b)*exp(2*I
*(d*x+c))/(2*((a+c)*(b+c))^(1/2)-a-b-2*c))*c-I/d^2*c/(a*b+a*c+b*c+c^2)^(1/2)*arctanh(1/4*(2*(a-b)*exp(2*I*(d*x
+c))+2*a+2*b+4*c)/(a*b+a*c+b*c+c^2)^(1/2))-I/d/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-
2*((a+c)*(b+c))^(1/2)-a-b-2*c))*x-1/2*I/d/((a+c)*(b+c))^(1/2)*ln(1-(a-b)*exp(2*I*(d*x+c))/(2*((a+c)*(b+c))^(1/
2)-a-b-2*c))*x-1/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c*x^2-1/2/((a+c)*(b+c))^(1/2)/(-2*((a+c)
*(b+c))^(1/2)-a-b-2*c)*a*x^2-1/2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*x^2-2/d/(-2*((a+c)*(b+
c))^(1/2)-a-b-2*c)*c*x-1/2/d^2/((a+c)*(b+c))^(1/2)*c^2-1/d^2/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c^2-1/2/d^2/(-2*
((a+c)*(b+c))^(1/2)-a-b-2*c)*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))-1/4/d^2/((a+c)
*(b+c))^(1/2)*polylog(2,(a-b)*exp(2*I*(d*x+c))/(2*((a+c)*(b+c))^(1/2)-a-b-2*c))-1/2/((a+c)*(b+c))^(1/2)*x^2-1/
(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*x^2-1/2*I/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)
*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*a*c-1/2*I/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-
a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*b*c-1/2*I/d/((a+c)*(b+c))^(1/2)/(-2*((a
+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*a*x-1/2*I/d/((a+c)*(b+
c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*b*x-I
/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-
b-2*c))*c*x-1/d/((a+c)*(b+c))^(1/2)*c*x-1/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c^3

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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(x*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.02899, size = 10267, normalized size = 38.45 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/16*(-4*I*(a - b)*c*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt(-(2*(a - b)*sqrt((a*b + (a +
b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) + 4*I*(a - b)*c
*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 -
2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) + 4*I*(a - b)*c*sqrt((a*b + (a + b)
*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a +
b + 2*c)/(a - b)) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) - 4*I*(a - b)*c*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a
*b + b^2))*log(2*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) -
2*cos(d*x + c) - 2*I*sin(d*x + c)) + 4*I*(a - b)*c*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqr
t((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) + 2*cos(d*x + c) + 2*I*
sin(d*x + c)) - 4*I*(a - b)*c*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt((2*(a - b)*sqrt((a*
b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) - 4*I*(
a - b)*c*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/
(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) + 4*I*(a - b)*c*sqrt((a*b +
(a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(2*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))
- a - b - 2*c)/(a - b)) - 2*cos(d*x + c) - 2*I*sin(d*x + c)) + 4*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2
*a*b + b^2))*dilog(1/2*((2*(a + b + 2*c)*cos(d*x + c) + (2*I*a + 2*I*b + 4*I*c)*sin(d*x + c) - 4*((a - b)*cos(
d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt
((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) - 2*a + 2*b)/(a - b) + 1) + 4*(a - b)*sq
rt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b + 2*c)*cos(d*x + c) - (2*I*a + 2*I*b + 4
*I*c)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) + (-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 -
2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*
a - 2*b)/(a - b) + 1) + 4*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*dilog(1/2*((2*(a + b + 2*c
)*cos(d*x + c) + (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (I*a - I*b)*sin(d*x + c))*s
qrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b +
b^2)) + a + b + 2*c)/(a - b)) - 2*a + 2*b)/(a - b) + 1) + 4*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b
+ b^2))*dilog(-1/2*((2*(a + b + 2*c)*cos(d*x + c) - (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) - 4*((a - b)*cos(d*
x + c) + (I*a - I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt((a
*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*a - 2*b)/(a - b) + 1) - 4*(a - b)*sqrt(
(a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*dilog(1/2*((2*(a + b + 2*c)*cos(d*x + c) + (2*I*a + 2*I*b + 4*I*c
)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b
+ b^2)))*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) - 2*a + 2*
b)/(a - b) + 1) - 4*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b + 2*c)*cos
(d*x + c) - (2*I*a + 2*I*b + 4*I*c)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) - (I*a - I*b)*sin(d*x + c))*sqrt((a
*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))
- a - b - 2*c)/(a - b)) + 2*a - 2*b)/(a - b) + 1) - 4*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)
)*dilog(1/2*((2*(a + b + 2*c)*cos(d*x + c) + (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) +
(-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt((a*b + (a
+ b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) - 2*a + 2*b)/(a - b) + 1) - 4*(a - b)*sqrt((a*b + (
a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b + 2*c)*cos(d*x + c) - (-2*I*a - 2*I*b - 4*I*c)*sin(
d*x + c) + 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^
2)))*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) + 2*a - 2*b)/(a
- b) + 1) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a
+ b + 2*c)*cos(d*x + c) + (2*I*a + 2*I*b + 4*I*c)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d
*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2
- 2*a*b + b^2)) + a + b + 2*c)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt((a*b +
(a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b + 2*c)*cos(d*x + c) - (2*I*a + 2*I*b + 4*I*c)*sin(d*x
+ c) - 4*((a - b)*cos(d*x + c) + (-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))
)*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*a - 2*b)/(a -
b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a + b
+ 2*c)*cos(d*x + c) + (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (I*a - I*b)*sin(d*x +
c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*
a*b + b^2)) + a + b + 2*c)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt((a*b + (a + b
)*c + c^2)/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b + 2*c)*cos(d*x + c) - (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c)
- 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt
(-(2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) + a + b + 2*c)/(a - b)) + 2*a - 2*b)/(a - b)) +
4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a + b + 2*c)
*cos(d*x + c) + (2*I*a + 2*I*b + 4*I*c)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqr
t((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^
2)) - a - b - 2*c)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^
2)/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b + 2*c)*cos(d*x + c) - (2*I*a + 2*I*b + 4*I*c)*sin(d*x + c) + 4*((a
- b)*cos(d*x + c) - (I*a - I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a -
b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) + 2*a - 2*b)/(a - b)) + 4*(I*(a -
b)*d*x + I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a + b + 2*c)*cos(d*x +
c) + (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (-I*a + I*b)*sin(d*x + c))*sqrt((a*b +
(a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a -
b - 2*c)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt((a*b + (a + b)*c + c^2)/(a^2
- 2*a*b + b^2))*log(1/2*((2*(a + b + 2*c)*cos(d*x + c) - (-2*I*a - 2*I*b - 4*I*c)*sin(d*x + c) + 4*((a - b)*co
s(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqr
t((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2)) - a - b - 2*c)/(a - b)) + 2*a - 2*b)/(a - b)))/((a*b + (a + b)*
c + c^2)*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sec ^{2}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )} + c \sec ^{2}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sec(d*x+c)**2/(a+c*sec(d*x+c)**2+b*tan(d*x+c)**2),x)

[Out]

Integral(x*sec(c + d*x)**2/(a + b*tan(c + d*x)**2 + c*sec(c + d*x)**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="giac")

[Out]

integrate(x*sec(d*x + c)^2/(c*sec(d*x + c)^2 + b*tan(d*x + c)^2 + a), x)