3.202 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx\)

Optimal. Leaf size=288 \[ \frac{a^3 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a^3 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac{\left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{48 f}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2 \left (d \left (6 c^2+62 c d+31 d^2\right ) \sec (e+f x)+2 \left (74 c^2 d+4 c^3+66 c d^2+21 d^3\right )\right )}{120 f}+\frac{a \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^3}{6 f}+\frac{a (3 c+8 d) \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}{30 f} \]

[Out]

(a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*ArcTanh[Sin[e + f*x]])/(16*f) + (a^3*(40*c^3 + 90*c^2*d + 78*c*d^
2 + 23*d^3)*Tan[e + f*x])/(16*f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f
*x])/(48*f) + (a*(3*c + 8*d)*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(30*f) + (a*(a + a*Se
c[e + f*x])^2*(c + d*Sec[e + f*x])^3*Tan[e + f*x])/(6*f) + (a*(a + a*Sec[e + f*x])^2*(2*(4*c^3 + 74*c^2*d + 66
*c*d^2 + 21*d^3) + d*(6*c^2 + 62*c*d + 31*d^2)*Sec[e + f*x])*Tan[e + f*x])/(120*f)

________________________________________________________________________________________

Rubi [A]  time = 0.427078, antiderivative size = 333, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3987, 100, 147, 50, 63, 217, 203} \[ \frac{a^3 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a^4 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a (\sec (e+f x)+1)}}\right )}{8 f \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}+\frac{\left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{48 f}+\frac{a \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) (a \sec (e+f x)+a)^2}{120 f}+\frac{d \tan (e+f x) (a \sec (e+f x)+a)^3 \left (70 c^2+4 d (8 c+3 d) \sec (e+f x)+54 c d+19 d^2\right )}{120 f}+\frac{d \tan (e+f x) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}{6 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Tan[e + f*x])/(16*f) + (a^4*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^
3)*ArcTan[Sqrt[a - a*Sec[e + f*x]]/Sqrt[a*(1 + Sec[e + f*x])]]*Tan[e + f*x])/(8*f*Sqrt[a - a*Sec[e + f*x]]*Sqr
t[a + a*Sec[e + f*x]]) + (a*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(120*
f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*Sec[e + f*x])*Tan[e + f*x])/(48*f) + (d*(a + a*Sec[e
+ f*x])^3*(c + d*Sec[e + f*x])^2*Tan[e + f*x])/(6*f) + (d*(a + a*Sec[e + f*x])^3*(70*c^2 + 54*c*d + 19*d^2 + 4
*d*(8*c + 3*d)*Sec[e + f*x])*Tan[e + f*x])/(120*f)

Rule 3987

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[(a^2*g*Cot[e + f*x])/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x
]]), Subst[Int[((g*x)^(p - 1)*(a + b*x)^(m - 1/2)*(c + d*x)^n)/Sqrt[a - b*x], x], x, Csc[e + f*x]], x] /; Free
Q[{a, b, c, d, e, f, g, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && (EqQ[p,
1] || IntegerQ[m - 1/2])

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \sec (e+f x) (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^3 \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{5/2} (c+d x)^3}{\sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{(a+a x)^{5/2} (c+d x) \left (-a^2 \left (6 c^2+3 c d+2 d^2\right )-a^2 d (8 c+3 d) x\right )}{\sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{6 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac{\left (a^2 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{\sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{40 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac{\left (a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{\sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{24 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac{\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{\sqrt{a-a x}} \, dx,x,\sec (e+f x)\right )}{16 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac{\left (a^5 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x}} \, dx,x,\sec (e+f x)\right )}{16 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac{\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2 a-x^2}} \, dx,x,\sqrt{a-a \sec (e+f x)}\right )}{8 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac{\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right )}{8 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac{a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan ^{-1}\left (\frac{\sqrt{a-a \sec (e+f x)}}{\sqrt{a+a \sec (e+f x)}}\right ) \tan (e+f x)}{8 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac{\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac{d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac{d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}\\ \end{align*}

Mathematica [A]  time = 2.77836, size = 380, normalized size = 1.32 \[ -\frac{a^3 (\cos (e+f x)+1)^3 \sec ^6\left (\frac{1}{2} (e+f x)\right ) \sec ^6(e+f x) \left (240 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \cos ^6(e+f x) \left (\log \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )\right )-2 \sin (e+f x) \left (16 \left (945 c^2 d+305 c^3+984 c d^2+344 d^3\right ) \cos (e+f x)+20 \left (306 c^2 d+72 c^3+342 c d^2+115 d^3\right ) \cos (2 (e+f x))+6840 c^2 d \cos (3 (e+f x))+1350 c^2 d \cos (4 (e+f x))+1080 c^2 d \cos (5 (e+f x))+4770 c^2 d+2360 c^3 \cos (3 (e+f x))+360 c^3 \cos (4 (e+f x))+440 c^3 \cos (5 (e+f x))+1080 c^3+6384 c d^2 \cos (3 (e+f x))+1170 c d^2 \cos (4 (e+f x))+912 c d^2 \cos (5 (e+f x))+5670 c d^2+1904 d^3 \cos (3 (e+f x))+345 d^3 \cos (4 (e+f x))+272 d^3 \cos (5 (e+f x))+2275 d^3\right )\right )}{30720 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*(1 + Cos[e + f*x])^3*Sec[(e + f*x)/2]^6*Sec[e + f*x]^6*(240*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*Cos[
e + f*x]^6*(Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]) - 2*(1080*c^3
+ 4770*c^2*d + 5670*c*d^2 + 2275*d^3 + 16*(305*c^3 + 945*c^2*d + 984*c*d^2 + 344*d^3)*Cos[e + f*x] + 20*(72*c
^3 + 306*c^2*d + 342*c*d^2 + 115*d^3)*Cos[2*(e + f*x)] + 2360*c^3*Cos[3*(e + f*x)] + 6840*c^2*d*Cos[3*(e + f*x
)] + 6384*c*d^2*Cos[3*(e + f*x)] + 1904*d^3*Cos[3*(e + f*x)] + 360*c^3*Cos[4*(e + f*x)] + 1350*c^2*d*Cos[4*(e
+ f*x)] + 1170*c*d^2*Cos[4*(e + f*x)] + 345*d^3*Cos[4*(e + f*x)] + 440*c^3*Cos[5*(e + f*x)] + 1080*c^2*d*Cos[5
*(e + f*x)] + 912*c*d^2*Cos[5*(e + f*x)] + 272*d^3*Cos[5*(e + f*x)])*Sin[e + f*x]))/(30720*f)

________________________________________________________________________________________

Maple [A]  time = 0.075, size = 523, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/15/f*a^3*d^3*tan(f*x+e)+17/15/f*a^3*d^3*tan(f*x+e)*sec(f*x+e)^2+3/5/f*a^3*d^3*tan(f*x+e)*sec(f*x+e)^4+1/3/f
*a^3*c^3*tan(f*x+e)*sec(f*x+e)^2+3/2*a^3*c^3*sec(f*x+e)*tan(f*x+e)/f+1/6/f*a^3*d^3*tan(f*x+e)*sec(f*x+e)^5+39/
8/f*a^3*d^2*c*sec(f*x+e)*tan(f*x+e)+45/8/f*a^3*c^2*d*sec(f*x+e)*tan(f*x+e)+9/4/f*a^3*d^2*c*tan(f*x+e)*sec(f*x+
e)^3+9/f*a^3*c^2*d*tan(f*x+e)+39/8/f*a^3*d^2*c*ln(sec(f*x+e)+tan(f*x+e))+45/8/f*a^3*c^2*d*ln(sec(f*x+e)+tan(f*
x+e))+23/24/f*a^3*d^3*tan(f*x+e)*sec(f*x+e)^3+23/16/f*a^3*d^3*sec(f*x+e)*tan(f*x+e)+5/2/f*a^3*c^3*ln(sec(f*x+e
)+tan(f*x+e))+11/3/f*a^3*c^3*tan(f*x+e)+23/16/f*a^3*d^3*ln(sec(f*x+e)+tan(f*x+e))+3/4/f*a^3*c^2*d*tan(f*x+e)*s
ec(f*x+e)^3+38/5/f*a^3*d^2*c*tan(f*x+e)+19/5/f*a^3*d^2*c*tan(f*x+e)*sec(f*x+e)^2+3/f*a^3*c^2*d*tan(f*x+e)*sec(
f*x+e)^2+3/5/f*a^3*d^2*c*tan(f*x+e)*sec(f*x+e)^4

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Maxima [B]  time = 1.01484, size = 946, normalized size = 3.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(a+a*sec(f*x+e))^3*(c+d*sec(f*x+e))^3,x, algorithm="maxima")

[Out]

1/480*(160*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^3 + 1440*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3*c^2*d + 96*(
3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*c*d^2 + 1440*(tan(f*x + e)^3 + 3*tan(f*x + e))*a^3
*c*d^2 + 96*(3*tan(f*x + e)^5 + 10*tan(f*x + e)^3 + 15*tan(f*x + e))*a^3*d^3 + 160*(tan(f*x + e)^3 + 3*tan(f*x
+ e))*a^3*d^3 - 5*a^3*d^3*(2*(15*sin(f*x + e)^5 - 40*sin(f*x + e)^3 + 33*sin(f*x + e))/(sin(f*x + e)^6 - 3*si
n(f*x + e)^4 + 3*sin(f*x + e)^2 - 1) - 15*log(sin(f*x + e) + 1) + 15*log(sin(f*x + e) - 1)) - 90*a^3*c^2*d*(2*
(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(
sin(f*x + e) - 1)) - 270*a^3*c*d^2*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(sin(f*x + e)^4 - 2*sin(f*x + e)^2 +
1) - 3*log(sin(f*x + e) + 1) + 3*log(sin(f*x + e) - 1)) - 90*a^3*d^3*(2*(3*sin(f*x + e)^3 - 5*sin(f*x + e))/(
sin(f*x + e)^4 - 2*sin(f*x + e)^2 + 1) - 3*log(sin(f*x + e) + 1) + 3*log(sin(f*x + e) - 1)) - 360*a^3*c^3*(2*s
in(f*x + e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 1080*a^3*c^2*d*(2*sin(f*x
+ e)/(sin(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) - 360*a^3*c*d^2*(2*sin(f*x + e)/(si
n(f*x + e)^2 - 1) - log(sin(f*x + e) + 1) + log(sin(f*x + e) - 1)) + 480*a^3*c^3*log(sec(f*x + e) + tan(f*x +
e)) + 1440*a^3*c^3*tan(f*x + e) + 1440*a^3*c^2*d*tan(f*x + e))/f

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Fricas [A]  time = 0.554277, size = 791, normalized size = 2.75 \begin{align*} \frac{15 \,{\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \,{\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \,{\left (40 \, a^{3} d^{3} + 16 \,{\left (55 \, a^{3} c^{3} + 135 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 34 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 15 \,{\left (24 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} + 16 \,{\left (5 \, a^{3} c^{3} + 45 \, a^{3} c^{2} d + 57 \, a^{3} c d^{2} + 17 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 10 \,{\left (18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2} + 144 \,{\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(15*(40*a^3*c^3 + 90*a^3*c^2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*cos(f*x + e)^6*log(sin(f*x + e) + 1) - 15*(4
0*a^3*c^3 + 90*a^3*c^2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*cos(f*x + e)^6*log(-sin(f*x + e) + 1) + 2*(40*a^3*d^3 +
16*(55*a^3*c^3 + 135*a^3*c^2*d + 114*a^3*c*d^2 + 34*a^3*d^3)*cos(f*x + e)^5 + 15*(24*a^3*c^3 + 90*a^3*c^2*d +
78*a^3*c*d^2 + 23*a^3*d^3)*cos(f*x + e)^4 + 16*(5*a^3*c^3 + 45*a^3*c^2*d + 57*a^3*c*d^2 + 17*a^3*d^3)*cos(f*x
+ e)^3 + 10*(18*a^3*c^2*d + 54*a^3*c*d^2 + 23*a^3*d^3)*cos(f*x + e)^2 + 144*(a^3*c*d^2 + a^3*d^3)*cos(f*x + e)
)*sin(f*x + e))/(f*cos(f*x + e)^6)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int c^{3} \sec{\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{3}{\left (e + f x \right )}\, dx + \int c^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{6}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{7}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{6}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{5}{\left (e + f x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*(Integral(c**3*sec(e + f*x), x) + Integral(3*c**3*sec(e + f*x)**2, x) + Integral(3*c**3*sec(e + f*x)**3,
x) + Integral(c**3*sec(e + f*x)**4, x) + Integral(d**3*sec(e + f*x)**4, x) + Integral(3*d**3*sec(e + f*x)**5,
x) + Integral(3*d**3*sec(e + f*x)**6, x) + Integral(d**3*sec(e + f*x)**7, x) + Integral(3*c*d**2*sec(e + f*x)*
*3, x) + Integral(9*c*d**2*sec(e + f*x)**4, x) + Integral(9*c*d**2*sec(e + f*x)**5, x) + Integral(3*c*d**2*sec
(e + f*x)**6, x) + Integral(3*c**2*d*sec(e + f*x)**2, x) + Integral(9*c**2*d*sec(e + f*x)**3, x) + Integral(9*
c**2*d*sec(e + f*x)**4, x) + Integral(3*c**2*d*sec(e + f*x)**5, x))

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Giac [B]  time = 1.33745, size = 825, normalized size = 2.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(15*(40*a^3*c^3 + 90*a^3*c^2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*log(abs(tan(1/2*f*x + 1/2*e) + 1)) - 15*(40*
a^3*c^3 + 90*a^3*c^2*d + 78*a^3*c*d^2 + 23*a^3*d^3)*log(abs(tan(1/2*f*x + 1/2*e) - 1)) - 2*(600*a^3*c^3*tan(1/
2*f*x + 1/2*e)^11 + 1350*a^3*c^2*d*tan(1/2*f*x + 1/2*e)^11 + 1170*a^3*c*d^2*tan(1/2*f*x + 1/2*e)^11 + 345*a^3*
d^3*tan(1/2*f*x + 1/2*e)^11 - 3400*a^3*c^3*tan(1/2*f*x + 1/2*e)^9 - 7650*a^3*c^2*d*tan(1/2*f*x + 1/2*e)^9 - 66
30*a^3*c*d^2*tan(1/2*f*x + 1/2*e)^9 - 1955*a^3*d^3*tan(1/2*f*x + 1/2*e)^9 + 7920*a^3*c^3*tan(1/2*f*x + 1/2*e)^
7 + 17820*a^3*c^2*d*tan(1/2*f*x + 1/2*e)^7 + 15444*a^3*c*d^2*tan(1/2*f*x + 1/2*e)^7 + 4554*a^3*d^3*tan(1/2*f*x
+ 1/2*e)^7 - 9360*a^3*c^3*tan(1/2*f*x + 1/2*e)^5 - 22500*a^3*c^2*d*tan(1/2*f*x + 1/2*e)^5 - 17964*a^3*c*d^2*t
an(1/2*f*x + 1/2*e)^5 - 5814*a^3*d^3*tan(1/2*f*x + 1/2*e)^5 + 5560*a^3*c^3*tan(1/2*f*x + 1/2*e)^3 + 15390*a^3*
c^2*d*tan(1/2*f*x + 1/2*e)^3 + 12570*a^3*c*d^2*tan(1/2*f*x + 1/2*e)^3 + 3165*a^3*d^3*tan(1/2*f*x + 1/2*e)^3 -
1320*a^3*c^3*tan(1/2*f*x + 1/2*e) - 4410*a^3*c^2*d*tan(1/2*f*x + 1/2*e) - 4590*a^3*c*d^2*tan(1/2*f*x + 1/2*e)
- 1575*a^3*d^3*tan(1/2*f*x + 1/2*e))/(tan(1/2*f*x + 1/2*e)^2 - 1)^6)/f