3.164 \(\int \frac {x^2 \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx\)
Optimal. Leaf size=407 \[ -\frac {i \text {Li}_3\left (-\frac {(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt {a+c} \sqrt {b+c}}\right )}{4 d^3 \sqrt {a+c} \sqrt {b+c}}+\frac {i \text {Li}_3\left (-\frac {(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt {a+c} \sqrt {b+c}\right )}\right )}{4 d^3 \sqrt {a+c} \sqrt {b+c}}-\frac {x \text {Li}_2\left (-\frac {(a-b) e^{2 i (c+d x)}}{a+b+2 c-2 \sqrt {a+c} \sqrt {b+c}}\right )}{2 d^2 \sqrt {a+c} \sqrt {b+c}}+\frac {x \text {Li}_2\left (-\frac {(a-b) e^{2 i (c+d x)}}{a+b+2 \left (c+\sqrt {a+c} \sqrt {b+c}\right )}\right )}{2 d^2 \sqrt {a+c} \sqrt {b+c}}-\frac {i x^2 \log \left (1+\frac {(a-b) e^{2 i (c+d x)}}{-2 \sqrt {a+c} \sqrt {b+c}+a+b+2 c}\right )}{2 d \sqrt {a+c} \sqrt {b+c}}+\frac {i x^2 \log \left (1+\frac {(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt {a+c} \sqrt {b+c}+c\right )+a+b}\right )}{2 d \sqrt {a+c} \sqrt {b+c}} \]
[Out]
-1/2*I*x^2*ln(1+(a-b)*exp(2*I*(d*x+c))/(a+b+2*c-2*(a+c)^(1/2)*(b+c)^(1/2)))/d/(a+c)^(1/2)/(b+c)^(1/2)+1/2*I*x^
2*ln(1+(a-b)*exp(2*I*(d*x+c))/(a+b+2*c+2*(a+c)^(1/2)*(b+c)^(1/2)))/d/(a+c)^(1/2)/(b+c)^(1/2)-1/2*x*polylog(2,-
(a-b)*exp(2*I*(d*x+c))/(a+b+2*c-2*(a+c)^(1/2)*(b+c)^(1/2)))/d^2/(a+c)^(1/2)/(b+c)^(1/2)+1/2*x*polylog(2,-(a-b)
*exp(2*I*(d*x+c))/(a+b+2*c+2*(a+c)^(1/2)*(b+c)^(1/2)))/d^2/(a+c)^(1/2)/(b+c)^(1/2)-1/4*I*polylog(3,-(a-b)*exp(
2*I*(d*x+c))/(a+b+2*c-2*(a+c)^(1/2)*(b+c)^(1/2)))/d^3/(a+c)^(1/2)/(b+c)^(1/2)+1/4*I*polylog(3,-(a-b)*exp(2*I*(
d*x+c))/(a+b+2*c+2*(a+c)^(1/2)*(b+c)^(1/2)))/d^3/(a+c)^(1/2)/(b+c)^(1/2)
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Rubi [A] time = 1.08, antiderivative size = 407, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used
= {4589, 3321, 2264, 2190, 2531, 2282, 6589} \[ -\frac {x \text {PolyLog}\left (2,-\frac {(a-b) e^{2 i (c+d x)}}{-2 \sqrt {a+c} \sqrt {b+c}+a+b+2 c}\right )}{2 d^2 \sqrt {a+c} \sqrt {b+c}}+\frac {x \text {PolyLog}\left (2,-\frac {(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt {a+c} \sqrt {b+c}+c\right )+a+b}\right )}{2 d^2 \sqrt {a+c} \sqrt {b+c}}-\frac {i \text {PolyLog}\left (3,-\frac {(a-b) e^{2 i (c+d x)}}{-2 \sqrt {a+c} \sqrt {b+c}+a+b+2 c}\right )}{4 d^3 \sqrt {a+c} \sqrt {b+c}}+\frac {i \text {PolyLog}\left (3,-\frac {(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt {a+c} \sqrt {b+c}+c\right )+a+b}\right )}{4 d^3 \sqrt {a+c} \sqrt {b+c}}-\frac {i x^2 \log \left (1+\frac {(a-b) e^{2 i (c+d x)}}{-2 \sqrt {a+c} \sqrt {b+c}+a+b+2 c}\right )}{2 d \sqrt {a+c} \sqrt {b+c}}+\frac {i x^2 \log \left (1+\frac {(a-b) e^{2 i (c+d x)}}{2 \left (\sqrt {a+c} \sqrt {b+c}+c\right )+a+b}\right )}{2 d \sqrt {a+c} \sqrt {b+c}} \]
Antiderivative was successfully verified.
[In]
Int[(x^2*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]
[Out]
((-I/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b + c])])/(Sqrt[a + c]*Sqr
t[b + c]*d) + ((I/2)*x^2*Log[1 + ((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a + c]*Sqrt[b + c]))])/(Sq
rt[a + c]*Sqrt[b + c]*d) - (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*c - 2*Sqrt[a + c]*Sqrt[b +
c]))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) + (x*PolyLog[2, -(((a - b)*E^((2*I)*(c + d*x)))/(a + b + 2*(c + Sqrt[a
+ c]*Sqrt[b + c])))])/(2*Sqrt[a + c]*Sqrt[b + c]*d^2) - ((I/4)*PolyLog[3, -(((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^3) + ((I/4)*PolyLog[3, -(((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^3)
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 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 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[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 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 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rubi steps
\begin {align*} \int \frac {x^2 \sec ^2(c+d x)}{a+c \sec ^2(c+d x)+b \tan ^2(c+d x)} \, dx &=2 \int \frac {x^2}{a+b+2 c+(a-b) \cos (2 c+2 d x)} \, dx\\ &=4 \int \frac {e^{i (2 c+2 d x)} x^2}{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^2}{-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^2}{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^2 \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^2 \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 x \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}{\sqrt {a+c} \sqrt {b+c} d}-\frac {i \int x \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}{\sqrt {a+c} \sqrt {b+c} d}\\ &=-\frac {i x^2 \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^2 \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 {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}+\frac {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}+\frac {\int \text {Li}_2\left (-\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^2}-\frac {\int \text {Li}_2\left (-\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^2}\\ &=-\frac {i x^2 \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^2 \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 {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}+\frac {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}-\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(-a+b) x}{a+b+2 c-2 \sqrt {a+c} \sqrt {b+c}}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt {a+c} \sqrt {b+c} d^3}+\frac {i \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {(-a+b) x}{a+b+2 \left (c+\sqrt {a+c} \sqrt {b+c}\right )}\right )}{x} \, dx,x,e^{i (2 c+2 d x)}\right )}{4 \sqrt {a+c} \sqrt {b+c} d^3}\\ &=-\frac {i x^2 \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^2 \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 {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}+\frac {x \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 )}{2 \sqrt {a+c} \sqrt {b+c} d^2}-\frac {i \text {Li}_3\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^3}+\frac {i \text {Li}_3\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^3}\\ \end {align*}
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Mathematica [A] time = 2.21, size = 499, normalized size = 1.23 \[ -\frac {i e^{2 i c} \left (2 d^2 x^2 \log \left (1+\frac {(a-b) e^{2 i (2 c+d x)}}{-2 \sqrt {e^{4 i c} (a+c) (b+c)}+a e^{2 i c}+b e^{2 i c}+2 c e^{2 i c}}\right )-2 d^2 x^2 \log \left (1+\frac {(a-b) e^{2 i (2 c+d x)}}{2 \sqrt {e^{4 i c} (a+c) (b+c)}+a e^{2 i c}+b e^{2 i c}+2 c e^{2 i c}}\right )-2 i d x \text {Li}_2\left (\frac {(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}-2 \sqrt {(a+c) (b+c) e^{4 i c}}}\right )+2 i d x \text {Li}_2\left (\frac {(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}+2 \sqrt {(a+c) (b+c) e^{4 i c}}}\right )+\text {Li}_3\left (\frac {(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}-2 \sqrt {(a+c) (b+c) e^{4 i c}}}\right )-\text {Li}_3\left (\frac {(b-a) e^{2 i (2 c+d x)}}{e^{2 i c} a+b e^{2 i c}+2 c e^{2 i c}+2 \sqrt {(a+c) (b+c) e^{4 i c}}}\right )\right )}{4 d^3 \sqrt {e^{4 i c} (a+c) (b+c)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(x^2*Sec[c + d*x]^2)/(a + c*Sec[c + d*x]^2 + b*Tan[c + d*x]^2),x]
[Out]
((-1/4*I)*E^((2*I)*c)*(2*d^2*x^2*Log[1 + ((a - b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*
E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - 2*d^2*x^2*Log[1 + ((a - b)*E^((2*I)*(2*c + d*x)))/(a*E^(
(2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - (2*I)*d*x*PolyLog[2, ((-a
+ b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I
)*c)])] + (2*I)*d*x*PolyLog[2, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*
c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] + PolyLog[3, ((-a + b)*E^((2*I)*(2*c + d*x)))/(a*E^((2*I)*c) + b*E^
((2*I)*c) + 2*c*E^((2*I)*c) - 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])] - PolyLog[3, ((-a + b)*E^((2*I)*(2*c + d*x
)))/(a*E^((2*I)*c) + b*E^((2*I)*c) + 2*c*E^((2*I)*c) + 2*Sqrt[(a + c)*(b + c)*E^((4*I)*c)])]))/(d^3*Sqrt[(a +
c)*(b + c)*E^((4*I)*c)])
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fricas [C] time = 1.28, size = 5772, normalized size = 14.18 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="fricas")
[Out]
1/16*(8*(a - b)*d*x*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) + 8*(a - b)*d*x*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*d
ilog(-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) + 8*(a - b)*d*x*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) + 8*(a - b)*d*x*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) - 8*(a - b)*d*x*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) - 8*(a - b)*d*x*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) - 8*(a - b)*d*x*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) - 8*(a - b)*d*x*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)*
c^2*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^2*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^2*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^2*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^2*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^2*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^2*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^2*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)*d^2*x^2 - I*(a - b)*c^2)*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^2*x^2 + I*(a - b)*c^2)*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^2*x^2 + I*(a - b)*c^2)*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^2*x^2 - I*(a - b)*c^2)*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^2*x^2 + I*(a - b)*c^2)
*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^2*x^2 - I*(a - b)*c^2)*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^2*x^2 - I*(a - b)*c^2
)*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^2*x^2 + I*(a - b)*c^2)*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*(2*I*a - 2*I*b)*sqrt((a*b +
(a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3, -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))/(a - b)) +
4*(-2*I*a + 2*I*b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3, 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))/(a - b)) + 4*(-2*I*a + 2*I*b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3
, -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))/(a - b)) + 4*(2*I*a - 2*I*b)*sqrt((a*b + (a + b)*c + c^2)
/(a^2 - 2*a*b + b^2))*polylog(3, 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))/(a - b)) + 4*(-2*I*a + 2*I*
b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3, -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
))/(a - b)) + 4*(2*I*a - 2*I*b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3, 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))/(a - b)) + 4*(2*I*a - 2*I*b)*sqrt((a*b + (a + b)*c + c^2)/(a^2 - 2*a*b + b^2))*
polylog(3, -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))/(a - b)) + 4*(-2*I*a + 2*I*b)*sqrt((a*b + (a + b
)*c + c^2)/(a^2 - 2*a*b + b^2))*polylog(3, 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))/(a - b)))/((a*b +
(a + b)*c + c^2)*d^3)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sec \left (d x + c\right )^{2}}{c \sec \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="giac")
[Out]
integrate(x^2*sec(d*x + c)^2/(c*sec(d*x + c)^2 + b*tan(d*x + c)^2 + a), x)
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maple [B] time = 0.55, size = 2061, normalized size = 5.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x)
[Out]
-I/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2
)-a-b-2*c))*x^2-1/2*I/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2
*((a+c)*(b+c))^(1/2)-a-b-2*c))*x^2-1/2*I/d/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*a*ln(1-(a-b)*e
xp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*x^2+1/2*I/d^3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-
b-2*c)*b*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*I/d^3/((a+c)*(b+c))^(1/2)/(-2*(
(a+c)*(b+c))^(1/2)-a-b-2*c)*a*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)*b*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-
2*c))*x-1/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c*polylog(2,(a-b)*exp(2*I*(d*x+c))/(-2*((a+
c)*(b+c))^(1/2)-a-b-2*c))*x-1/2/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*a*polylog(2,(a-b)*exp
(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))*x+1/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*a
*c^2*x+1/d^2/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*c^2*x-1/4*I/d^3/((a+c)*(b+c))^(1/2)/(-2*((
a+c)*(b+c))^(1/2)-a-b-2*c)*a*polylog(3,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c))-1/4*I/d^3/((a+
c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*polylog(3,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/2)-a-
b-2*c))+2/d^2*c^3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*x+2/3/d^3/((a+c)*(b+c))^(1/2)/(-2*((a+c
)*(b+c))^(1/2)-a-b-2*c)*a*c^3+2/3/d^3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*c^3+I/d^3*c^2/(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^2-1/2*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^2+1/2*I/d^3*c^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))+I/d^3*c^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))-1/3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*
a*x^3-1/3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*b*x^3-2/3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))
^(1/2)-a-b-2*c)*c*x^3-1/2/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))*x-1/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))*x+4/3/d^3*c^4/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)+1/d^2*c^2/((a+c)*(b+c))^(1/2)*x+2/d^2*c
^2/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*x-1/4*I/d^3/((a+c)*(b+c))^(1/2)*polylog(3,(a-b)*exp(2*I*(d*x+c))/(2*((a+c)
*(b+c))^(1/2)-a-b-2*c))-1/2*I/d^3/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*polylog(3,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)
*(b+c))^(1/2)-a-b-2*c))+2/3/d^3*c^3/((a+c)*(b+c))^(1/2)+4/3/d^3*c^3/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)-1/2*I/d^3
/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c*polylog(3,(a-b)*exp(2*I*(d*x+c))/(-2*((a+c)*(b+c))^(1/
2)-a-b-2*c))+I/d^3/((a+c)*(b+c))^(1/2)/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*c^3*ln(1-(a-b)*exp(2*I*(d*x+c))/(-2*((
a+c)*(b+c))^(1/2)-a-b-2*c))-1/3/((a+c)*(b+c))^(1/2)*x^3-2/3/(-2*((a+c)*(b+c))^(1/2)-a-b-2*c)*x^3
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sec \left (d x + c\right )^{2}}{c \sec \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*sec(d*x+c)^2/(a+c*sec(d*x+c)^2+b*tan(d*x+c)^2),x, algorithm="maxima")
[Out]
integrate(x^2*sec(d*x + c)^2/(c*sec(d*x + c)^2 + b*tan(d*x + c)^2 + a), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\cos \left (c+d\,x\right )}^2\,\left (a+\frac {c}{{\cos \left (c+d\,x\right )}^2}+b\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2/(cos(c + d*x)^2*(a + c/cos(c + d*x)^2 + b*tan(c + d*x)^2)),x)
[Out]
int(x^2/(cos(c + d*x)^2*(a + c/cos(c + d*x)^2 + b*tan(c + d*x)^2)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \sec ^{2}{\left (c + d x \right )}}{a + b \tan ^{2}{\left (c + d x \right )} + c \sec ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*sec(d*x+c)**2/(a+c*sec(d*x+c)**2+b*tan(d*x+c)**2),x)
[Out]
Integral(x**2*sec(c + d*x)**2/(a + b*tan(c + d*x)**2 + c*sec(c + d*x)**2), x)
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