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

Optimal. Leaf size=211 \[ -\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}+\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d} \]

[Out]

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

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Rubi [A]  time = 0.528379, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4588, 3321, 2264, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}+\frac{\text{PolyLog}\left (2,-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4588

Int[(((f_.) + (g_.)*(x_))^(m_.)*Sec[(d_.) + (e_.)*(x_)]^2)/((b_) + (c_.)*Tan[(d_.) + (e_.)*(x_)]^2), x_Symbol]
:> Dist[2, Int[(f + g*x)^m/(b + c + (b - c)*Cos[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && IGtQ
[m, 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+b \tan ^2(c+d x)} \, dx &=2 \int \frac{x}{a+b+(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) 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} \sqrt{b}+2 (a+b)+2 (a-b) e^{i (2 c+2 d x)}} \, dx}{\sqrt{a} \sqrt{b}}-\frac{(2 (a-b)) \int \frac{e^{i (2 c+2 d x)} x}{4 \sqrt{a} \sqrt{b}+2 (a+b)+2 (a-b) e^{i (2 c+2 d x)}} \, dx}{\sqrt{a} \sqrt{b}}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i \int \log \left (1+\frac{2 (a-b) e^{i (2 c+2 d x)}}{-4 \sqrt{a} \sqrt{b}+2 (a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{b} d}-\frac{i \int \log \left (1+\frac{2 (a-b) e^{i (2 c+2 d x)}}{4 \sqrt{a} \sqrt{b}+2 (a+b)}\right ) \, dx}{2 \sqrt{a} \sqrt{b} d}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a-b) x}{-4 \sqrt{a} \sqrt{b}+2 (a+b)}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt{a} \sqrt{b} d^2}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a-b) x}{4 \sqrt{a} \sqrt{b}+2 (a+b)}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt{a} \sqrt{b} d^2}\\ &=-\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}+\frac{i x \log \left (1+\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{2 \sqrt{a} \sqrt{b} d}-\frac{\text{Li}_2\left (-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}-\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}+\frac{\text{Li}_2\left (-\frac{(a-b) e^{2 i (c+d x)}}{\left (\sqrt{a}+\sqrt{b}\right )^2}\right )}{4 \sqrt{a} \sqrt{b} d^2}\\ \end{align*}

Mathematica [B]  time = 6.53475, size = 512, normalized size = 2.43 \[ \frac{x \left (-\sqrt{a} \text{PolyLog}\left (2,\frac{\sqrt{b} (1-i \tan (c+d x))}{\sqrt{b}+i \sqrt{-a}}\right )-\sqrt{a} \text{PolyLog}\left (2,\frac{\sqrt{b} (1+i \tan (c+d x))}{\sqrt{b}+i \sqrt{-a}}\right )+\sqrt{a} \text{PolyLog}\left (2,-\frac{\sqrt{b} (\tan (c+d x)-i)}{\sqrt{-a}+i \sqrt{b}}\right )+\sqrt{a} \text{PolyLog}\left (2,\frac{\sqrt{b} (\tan (c+d x)+i)}{\sqrt{-a}+i \sqrt{b}}\right )+4 i \sqrt{-a} c \tan ^{-1}\left (\frac{\sqrt{b} \tan (c+d x)}{\sqrt{a}}\right )-\sqrt{a} \log (1+i \tan (c+d x)) \log \left (\frac{\sqrt{-a}-\sqrt{b} \tan (c+d x)}{\sqrt{-a}-i \sqrt{b}}\right )+\sqrt{a} \log (1-i \tan (c+d x)) \log \left (\frac{\sqrt{-a}-\sqrt{b} \tan (c+d x)}{\sqrt{-a}+i \sqrt{b}}\right )-\sqrt{a} \log (1-i \tan (c+d x)) \log \left (\frac{\sqrt{-a}+\sqrt{b} \tan (c+d x)}{\sqrt{-a}-i \sqrt{b}}\right )+\sqrt{a} \log (1+i \tan (c+d x)) \log \left (\frac{\sqrt{-a}+\sqrt{b} \tan (c+d x)}{\sqrt{-a}+i \sqrt{b}}\right )\right )}{2 \sqrt{-a^2} \sqrt{b} d (\log (1-i \tan (c+d x))-\log (1+i \tan (c+d x))+2 i c)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.157, size = 1003, normalized size = 4.8 \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+b*tan(d*x+c)^2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 7.23159, size = 8015, normalized size = 37.99 \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+b*tan(d*x+c)^2),x, algorithm="fricas")

[Out]

1/16*(-4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a +
b)/(a - b)) + 2*cos(d*x + c) + 2*I*sin(d*x + c)) + 4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt(-(2*
(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) + 4*I*(a - b)*sqr
t(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) - 2*cos(d*
x + c) + 2*I*sin(d*x + c)) - 4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt(-(2*(a - b)*sqrt(a*b/(a^2
- 2*a*b + b^2)) + a + b)/(a - b)) - 2*cos(d*x + c) - 2*I*sin(d*x + c)) + 4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b
^2))*c*log(2*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) + 2*cos(d*x + c) + 2*I*sin(d*x +
c)) - 4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)
/(a - b)) + 2*cos(d*x + c) - 2*I*sin(d*x + c)) - 4*I*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt((2*(a
- b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) - 2*cos(d*x + c) + 2*I*sin(d*x + c)) + 4*I*(a - b)*sqrt(a
*b/(a^2 - 2*a*b + b^2))*c*log(2*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) - 2*cos(d*x +
c) - 2*I*sin(d*x + c)) + 4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dilog(1/2*((2*(a + b)*cos(d*x + c) + (2*I*a +
2*I*b)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqr
t(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) - 2*a + 2*b)/(a - b) + 1) + 4*(a - b)*sqrt(a*b/(
a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b)*cos(d*x + c) - (2*I*a + 2*I*b)*sin(d*x + c) - 4*((a - b)*cos(d*x +
c) + (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))
+ a + b)/(a - b)) + 2*a - 2*b)/(a - b) + 1) + 4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dilog(1/2*((2*(a + b)*co
s(d*x + c) + (-2*I*a - 2*I*b)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2
- 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) - 2*a + 2*b)/(a - b) + 1) +
4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b)*cos(d*x + c) - (-2*I*a - 2*I*b)*sin(d*x + c) -
4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/
(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) + 2*a - 2*b)/(a - b) + 1) - 4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dil
og(1/2*((2*(a + b)*cos(d*x + c) + (2*I*a + 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x
+ c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) - 2*a +
2*b)/(a - b) + 1) - 4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b)*cos(d*x + c) - (2*I*a + 2*I
*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) - (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*
(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) + 2*a - 2*b)/(a - b) + 1) - 4*(a - b)*sqrt(a*b/(a^2 -
2*a*b + b^2))*dilog(1/2*((2*(a + b)*cos(d*x + c) + (-2*I*a - 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (
-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b
)/(a - b)) - 2*a + 2*b)/(a - b) + 1) - 4*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))*dilog(-1/2*((2*(a + b)*cos(d*x
+ c) - (-2*I*a - 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*
a*b + b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) + 2*a - 2*b)/(a - b) + 1) + 4*(I*
(a - b)*d*x + I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a + b)*cos(d*x + c) + (2*I*a + 2*I*b)*s
in(d*x + c) - 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a
- b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*
sqrt(a*b/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b)*cos(d*x + c) - (2*I*a + 2*I*b)*sin(d*x + c) - 4*((a - b)*cos
(d*x + c) + (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b +
b^2)) + a + b)/(a - b)) + 2*a - 2*b)/(a - b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2)
)*log(-1/2*((2*(a + b)*cos(d*x + c) + (-2*I*a - 2*I*b)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) - (I*a - I*b)*si
n(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) -
2*a + 2*b)/(a - b)) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b)*cos(d*
x + c) - (-2*I*a - 2*I*b)*sin(d*x + c) - 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2
*a*b + b^2)))*sqrt(-(2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) + a + b)/(a - b)) + 2*a - 2*b)/(a - b)) + 4*(-I*(
a - b)*d*x - I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2))*log(-1/2*((2*(a + b)*cos(d*x + c) + (2*I*a + 2*I*b)*si
n(d*x + c) + 4*((a - b)*cos(d*x + c) + (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b
)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) - 2*a + 2*b)/(a - b)) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt
(a*b/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b)*cos(d*x + c) - (2*I*a + 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x
+ c) - (I*a - I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2))
- a - b)/(a - b)) + 2*a - 2*b)/(a - b)) + 4*(I*(a - b)*d*x + I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2))*log(-
1/2*((2*(a + b)*cos(d*x + c) + (-2*I*a - 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) + (-I*a + I*b)*sin(d*x
+ c))*sqrt(a*b/(a^2 - 2*a*b + b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) - 2*a + 2
*b)/(a - b)) + 4*(-I*(a - b)*d*x - I*(a - b)*c)*sqrt(a*b/(a^2 - 2*a*b + b^2))*log(1/2*((2*(a + b)*cos(d*x + c)
- (-2*I*a - 2*I*b)*sin(d*x + c) + 4*((a - b)*cos(d*x + c) - (-I*a + I*b)*sin(d*x + c))*sqrt(a*b/(a^2 - 2*a*b
+ b^2)))*sqrt((2*(a - b)*sqrt(a*b/(a^2 - 2*a*b + b^2)) - a - b)/(a - b)) + 2*a - 2*b)/(a - b)))/(a*b*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 )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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