\(\int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx\) [39]

Optimal result

Integrand size = 20, antiderivative size = 1523 \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {6 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {6 i b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}+\frac {12 i b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 i b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^3}-\frac {12 i b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 b^3 d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}-\frac {12 b d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}-\frac {6 b^3 d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^4}+\frac {12 b d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]

[Out]

-6*I*b^2*d^2*(d*x+c)*polylog(2,-a*exp(I*(f*x+e))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^3-6*I*b^2*d^2*(d*x+c)*
polylog(2,-a*exp(I*(f*x+e))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^3-6*I*b^3*d^2*(d*x+c)*polylog(3,-a*exp(I*(f
*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^3-12*I*b*d^2*(d*x+c)*polylog(3,-a*exp(I*(f*x+e))/(b+(-a^2+
b^2)^(1/2)))/a^2/f^3/(-a^2+b^2)^(1/2)+b^2*(d*x+c)^3*sin(f*x+e)/a/(a^2-b^2)/f/(b+a*cos(f*x+e))+1/4*(d*x+c)^4/a^
2/d+3*b^2*d*(d*x+c)^2*ln(1+a*exp(I*(f*x+e))/(b-I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^2+3*b^2*d*(d*x+c)^2*ln(1+a*
exp(I*(f*x+e))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^2-3*b^3*d*(d*x+c)^2*polylog(2,-a*exp(I*(f*x+e))/(b-(-a^2
+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^2+3*b^3*d*(d*x+c)^2*polylog(2,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2
/(-a^2+b^2)^(3/2)/f^2+6*b*d*(d*x+c)^2*polylog(2,-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/f^2/(-a^2+b^2)^(1/
2)-6*b*d*(d*x+c)^2*polylog(2,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/f^2/(-a^2+b^2)^(1/2)-I*b^3*(d*x+c)^3*
ln(1+a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f-2*I*b*(d*x+c)^3*ln(1+a*exp(I*(f*x+e))/(b+(-
a^2+b^2)^(1/2)))/a^2/f/(-a^2+b^2)^(1/2)+2*I*b*(d*x+c)^3*ln(1+a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/f/(-a^
2+b^2)^(1/2)+I*b^3*(d*x+c)^3*ln(1+a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f+6*I*b^3*d^2*(d
*x+c)*polylog(3,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^3+12*I*b*d^2*(d*x+c)*polylog(3,
-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/f^3/(-a^2+b^2)^(1/2)+6*b^2*d^3*polylog(3,-a*exp(I*(f*x+e))/(b-I*(a
^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f^4+6*b^2*d^3*polylog(3,-a*exp(I*(f*x+e))/(b+I*(a^2-b^2)^(1/2)))/a^2/(a^2-b^2)/f
^4+6*b^3*d^3*polylog(4,-a*exp(I*(f*x+e))/(b-(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^4-6*b^3*d^3*polylog(4,-a
*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/(-a^2+b^2)^(3/2)/f^4-12*b*d^3*polylog(4,-a*exp(I*(f*x+e))/(b-(-a^2+b
^2)^(1/2)))/a^2/f^4/(-a^2+b^2)^(1/2)+12*b*d^3*polylog(4,-a*exp(I*(f*x+e))/(b+(-a^2+b^2)^(1/2)))/a^2/f^4/(-a^2+
b^2)^(1/2)-I*b^2*(d*x+c)^3/a^2/(a^2-b^2)/f

Rubi [A] (verified)

Time = 3.54 (sec) , antiderivative size = 1523, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3405, 3402, 2296, 2221, 2611, 6744, 2320, 6724, 4618} \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\frac {(c+d x)^4}{4 a^2 d}+\frac {2 i b \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt {b^2-a^2} f}-\frac {i b^3 \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}-\frac {2 i b \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt {b^2-a^2} f}+\frac {i b^3 \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}+\frac {b^2 \sin (e+f x) (c+d x)^3}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {3 b^2 d \log \left (\frac {e^{i (e+f x)} a}{b-i \sqrt {a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d \log \left (\frac {e^{i (e+f x)} a}{b+i \sqrt {a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac {6 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt {b^2-a^2} f^2}-\frac {3 b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac {6 b d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt {b^2-a^2} f^2}+\frac {3 b^3 d \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac {6 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}+\frac {12 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt {b^2-a^2} f^3}-\frac {6 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}-\frac {12 i b d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt {b^2-a^2} f^3}+\frac {6 i b^3 d^2 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}+\frac {6 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {12 b d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} f^4}+\frac {6 b^3 d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4}+\frac {12 b d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} f^4}-\frac {6 b^3 d^3 \operatorname {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4} \]

[In]

Int[(c + d*x)^3/(a + b*Sec[e + f*x])^2,x]

[Out]

((-I)*b^2*(c + d*x)^3)/(a^2*(a^2 - b^2)*f) + (c + d*x)^4/(4*a^2*d) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e +
f*x)))/(b - I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) + (3*b^2*d*(c + d*x)^2*Log[1 + (a*E^(I*(e + f*x)))/(b
+ I*Sqrt[a^2 - b^2])])/(a^2*(a^2 - b^2)*f^2) - (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 +
b^2])])/(a^2*(-a^2 + b^2)^(3/2)*f) + ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2])]
)/(a^2*Sqrt[-a^2 + b^2]*f) + (I*b^3*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*(-a^
2 + b^2)^(3/2)*f) - ((2*I)*b*(c + d*x)^3*Log[1 + (a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2])])/(a^2*Sqrt[-a^2 +
b^2]*f) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^
2)*f^3) - ((6*I)*b^2*d^2*(c + d*x)*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2
)*f^3) - (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/
2)*f^2) + (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*
f^2) + (3*b^3*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)
*f^2) - (6*b*d*(c + d*x)^2*PolyLog[2, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^
2) + (6*b^2*d^3*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) + (6*b^2*d^3
*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + I*Sqrt[a^2 - b^2]))])/(a^2*(a^2 - b^2)*f^4) - ((6*I)*b^3*d^2*(c + d*x)*
PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) + ((12*I)*b*d^2*(c + d
*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + ((6*I)*b^3*d^2*(c
+ d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^3) - ((12*I)*b*d^2
*(c + d*x)*PolyLog[3, -((a*E^(I*(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^3) + (6*b^3*d^3*
PolyLog[4, -((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) - (12*b*d^3*PolyLog[4,
-((a*E^(I*(e + f*x)))/(b - Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) - (6*b^3*d^3*PolyLog[4, -((a*E^(I*
(e + f*x)))/(b + Sqrt[-a^2 + b^2]))])/(a^2*(-a^2 + b^2)^(3/2)*f^4) + (12*b*d^3*PolyLog[4, -((a*E^(I*(e + f*x))
)/(b + Sqrt[-a^2 + b^2]))])/(a^2*Sqrt[-a^2 + b^2]*f^4) + (b^2*(c + d*x)^3*Sin[e + f*x])/(a*(a^2 - b^2)*f*(b +
a*Cos[e + f*x]))

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 2320

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 2611

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 3402

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 3405

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

Rule 4276

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Sin[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] &
& IGtQ[m, 0]

Rule 4618

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)])/(Cos[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :>
Simp[I*((e + f*x)^(m + 1)/(b*f*(m + 1))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a - Rt[-a^2 + b^2, 2] + I*
b*E^(I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(I*a + Rt[-a^2 + b^2, 2] + I*b*E^(I*(c + d*x)))), x
]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NegQ[a^2 - b^2]

Rule 6724

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]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(c+d x)^3}{a^2}+\frac {b^2 (c+d x)^3}{a^2 (b+a \cos (e+f x))^2}-\frac {2 b (c+d x)^3}{a^2 (b+a \cos (e+f x))}\right ) \, dx \\ & = \frac {(c+d x)^4}{4 a^2 d}-\frac {(2 b) \int \frac {(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a^2}+\frac {b^2 \int \frac {(c+d x)^3}{(b+a \cos (e+f x))^2} \, dx}{a^2} \\ & = \frac {(c+d x)^4}{4 a^2 d}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^3}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2}-\frac {b^3 \int \frac {(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {\left (3 b^2 d\right ) \int \frac {(c+d x)^2 \sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f} \\ & = -\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^3}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^3}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}+\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)^3}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}-\frac {\left (3 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{i b-\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}-\frac {\left (3 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{i b+\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f} \\ & = -\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^3}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)^3}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {i a e^{i (e+f x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {i a e^{i (e+f x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {(6 i b d) \int (c+d x)^2 \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f}+\frac {(6 i b d) \int (c+d x)^2 \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f} \\ & = -\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}+\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}-\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (6 i b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {i a e^{i (e+f x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}+\frac {\left (6 i b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-\frac {i a e^{i (e+f x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}-\frac {\left (12 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {\left (12 b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {\left (3 i b^3 d\right ) \int (c+d x)^2 \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {\left (3 i b^3 d\right ) \int (c+d x)^2 \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f} \\ & = -\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x)^3 \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {3 b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {6 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {12 i b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}-\frac {12 i b d^2 (c+d x) \operatorname {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,\frac {i a x}{-i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,-\frac {i a x}{i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {\left (12 i b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {\left (12 i b d^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {\left (6 b^3 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {\left (6 b^3 d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f^2} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(20116\) vs. \(2(1523)=3046\).

Time = 22.31 (sec) , antiderivative size = 20116, normalized size of antiderivative = 13.21 \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\text {Result too large to show} \]

[In]

Integrate[(c + d*x)^3/(a + b*Sec[e + f*x])^2,x]

[Out]

Result too large to show

Maple [F]

\[\int \frac {\left (d x +c \right )^{3}}{\left (a +b \sec \left (f x +e \right )\right )^{2}}d x\]

[In]

int((d*x+c)^3/(a+b*sec(f*x+e))^2,x)

[Out]

int((d*x+c)^3/(a+b*sec(f*x+e))^2,x)

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. 7008 vs. \(2 (1361) = 2722\).

Time = 0.71 (sec) , antiderivative size = 7008, normalized size of antiderivative = 4.60 \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^3/(a+b*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\int \frac {\left (c + d x\right )^{3}}{\left (a + b \sec {\left (e + f x \right )}\right )^{2}}\, dx \]

[In]

integrate((d*x+c)**3/(a+b*sec(f*x+e))**2,x)

[Out]

Integral((c + d*x)**3/(a + b*sec(e + f*x))**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((d*x+c)^3/(a+b*sec(f*x+e))^2,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*a^2-4*b^2>0)', see `assume?`
for more de

Giac [F]

\[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (b \sec \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate((d*x+c)^3/(a+b*sec(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^3/(b*sec(f*x + e) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx=\text {Hanged} \]

[In]

int((c + d*x)^3/(a + b/cos(e + f*x))^2,x)

[Out]

\text{Hanged}