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

Optimal. Leaf size=284 \[ \frac{d^2 \left (12 c^2+16 c d+7 d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c-d} \tan \left (\frac{1}{2} (e+f x)\right )}{\sqrt{c+d}}\right )}{a^2 f (c-d)^{9/2} (c+d)^{5/2}}+\frac{d \left (-16 c^2 d+2 c^3-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 a^2 f (c-d)^4 (c+d)^2 (c+d \sec (e+f x))}+\frac{d \left (2 c^2-16 c d-21 d^2\right ) \tan (e+f x)}{6 a^2 f (c-d)^3 (c+d) (c+d \sec (e+f x))^2}+\frac{(c-8 d) \tan (e+f x)}{3 a^2 f (c-d)^2 (\sec (e+f x)+1) (c+d \sec (e+f x))^2}+\frac{\tan (e+f x)}{3 f (c-d) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2} \]

[Out]

(d^2*(12*c^2 + 16*c*d + 7*d^2)*ArcTanh[(Sqrt[c - d]*Tan[(e + f*x)/2])/Sqrt[c + d]])/(a^2*(c - d)^(9/2)*(c + d)
^(5/2)*f) + (d*(2*c^2 - 16*c*d - 21*d^2)*Tan[e + f*x])/(6*a^2*(c - d)^3*(c + d)*f*(c + d*Sec[e + f*x])^2) + ((
c - 8*d)*Tan[e + f*x])/(3*a^2*(c - d)^2*f*(1 + Sec[e + f*x])*(c + d*Sec[e + f*x])^2) + Tan[e + f*x]/(3*(c - d)
*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) + (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*Tan[e + f*x])/(6
*a^2*(c - d)^4*(c + d)^2*f*(c + d*Sec[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.55765, antiderivative size = 346, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {3987, 103, 151, 152, 12, 93, 205} \[ \frac{\left (-16 c^2 d+2 c^3-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 f (c-d)^4 (c+d)^2 \left (a^2 \sec (e+f x)+a^2\right )}-\frac{d^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x) \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a \sec (e+f x)+a}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right )}{a f (c-d)^{9/2} (c+d)^{5/2} \sqrt{a-a \sec (e+f x)} \sqrt{a \sec (e+f x)+a}}-\frac{d (5 c+2 d) \tan (e+f x)}{2 f \left (c^2-d^2\right )^2 (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}-\frac{d \tan (e+f x)}{2 f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}+\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 f (c-d)^3 (c+d)^2 (a \sec (e+f x)+a)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*c^2 + 22*c*d + 11*d^2)*Tan[e + f*x])/(6*(c - d)^3*(c + d)^2*f*(a + a*Sec[e + f*x])^2) - (d^2*(12*c^2 + 16*
c*d + 7*d^2)*ArcTan[(Sqrt[c + d]*Sqrt[a + a*Sec[e + f*x]])/(Sqrt[c - d]*Sqrt[a - a*Sec[e + f*x]])]*Tan[e + f*x
])/(a*(c - d)^(9/2)*(c + d)^(5/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((2*c^3 - 16*c^2*d -
59*c*d^2 - 32*d^3)*Tan[e + f*x])/(6*(c - d)^4*(c + d)^2*f*(a^2 + a^2*Sec[e + f*x])) - (d*Tan[e + f*x])/(2*(c^2
- d^2)*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^2) - (d*(5*c + 2*d)*Tan[e + f*x])/(2*(c^2 - d^2)^2*f*(a
+ a*Sec[e + f*x])^2*(c + d*Sec[e + f*x]))

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 103

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))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

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

Rule 152

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3} \, dx &=-\frac{\left (a^2 \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} (a+a x)^{5/2} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{2 a^2 (c+d)-3 a^2 d x}{\sqrt{a-a x} (a+a x)^{5/2} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{2 \left (c^2-d^2\right ) f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{a^4 \left (2 c^2+12 c d+7 d^2\right )-2 a^4 d (5 c+2 d) x}{\sqrt{a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 a^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^2}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}+\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{-a^6 (c+d) \left (2 c^2-16 c d-21 d^2\right )-a^6 d \left (2 c^2+22 c d+11 d^2\right ) x}{\sqrt{a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{6 a^5 (c-d) \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 (c-d)^4 (c+d)^2 f \left (a^2+a^2 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac{\tan (e+f x) \operatorname{Subst}\left (\int \frac{3 a^8 d^2 \left (12 c^2+16 c d+7 d^2\right )}{\sqrt{a-a x} \sqrt{a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{6 a^8 (c-d)^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 (c-d)^4 (c+d)^2 f \left (a^2+a^2 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac{\left (d^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a-a x} \sqrt{a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 (c-d)^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^2}+\frac{\left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 (c-d)^4 (c+d)^2 f \left (a^2+a^2 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac{\left (d^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{1}{a c-a d-(-a c-a d) x^2} \, dx,x,\frac{\sqrt{a+a \sec (e+f x)}}{\sqrt{a-a \sec (e+f x)}}\right )}{(c-d)^2 \left (c^2-d^2\right )^2 f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}\\ &=\frac{\left (2 c^2+22 c d+11 d^2\right ) \tan (e+f x)}{6 (c-d)^3 (c+d)^2 f (a+a \sec (e+f x))^2}-\frac{d^2 \left (12 c^2+16 c d+7 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+a \sec (e+f x)}}{\sqrt{c-d} \sqrt{a-a \sec (e+f x)}}\right ) \tan (e+f x)}{a (c-d)^{9/2} (c+d)^{5/2} f \sqrt{a-a \sec (e+f x)} \sqrt{a+a \sec (e+f x)}}+\frac{\left (2 c^3-16 c^2 d-59 c d^2-32 d^3\right ) \tan (e+f x)}{6 (c-d)^4 (c+d)^2 f \left (a^2+a^2 \sec (e+f x)\right )}-\frac{d \tan (e+f x)}{2 \left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2}-\frac{d (5 c+2 d) \tan (e+f x)}{2 \left (c^2-d^2\right )^2 f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}\\ \end{align*}

Mathematica [C]  time = 7.29495, size = 2220, normalized size = 7.82 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((12*c^2 + 16*c*d + 7*d^2)*Cos[e/2 + (f*x)/2]^4*(d + c*Cos[e + f*x])^3*Sec[e + f*x]^5*(((-4*I)*d^2*ArcTan[Sec[
(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I
*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Cos[e])/(Sqrt[c^2 - d^2]*f*Sqrt[Cos[2*e] - I*Sin[2*
e]]) - (4*d^2*ArcTan[Sec[(f*x)/2]*(Cos[e]/(Sqrt[c^2 - d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]) - (I*Sin[e])/(Sqrt[c^2
- d^2]*Sqrt[Cos[2*e] - I*Sin[2*e]]))*((-I)*d*Sin[(f*x)/2] + I*c*Sin[e + (f*x)/2])]*Sin[e])/(Sqrt[c^2 - d^2]*f
*Sqrt[Cos[2*e] - I*Sin[2*e]])))/((-c + d)^4*(c + d)^2*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3) + (Cos[e/
2 + (f*x)/2]*(d + c*Cos[e + f*x])*Sec[e/2]*Sec[e]*Sec[e + f*x]^5*(-16*c^7*Sin[(f*x)/2] + 14*c^6*d*Sin[(f*x)/2]
+ 220*c^5*d^2*Sin[(f*x)/2] + 334*c^4*d^3*Sin[(f*x)/2] + 54*c^3*d^4*Sin[(f*x)/2] - 156*c^2*d^5*Sin[(f*x)/2] -
48*c*d^6*Sin[(f*x)/2] + 18*d^7*Sin[(f*x)/2] + 14*c^7*Sin[(3*f*x)/2] - 16*c^6*d*Sin[(3*f*x)/2] - 226*c^5*d^2*Si
n[(3*f*x)/2] - 532*c^4*d^3*Sin[(3*f*x)/2] - 583*c^3*d^4*Sin[(3*f*x)/2] - 232*c^2*d^5*Sin[(3*f*x)/2] - 6*c*d^6*
Sin[(3*f*x)/2] + 6*d^7*Sin[(3*f*x)/2] - 12*c^7*Sin[e - (f*x)/2] + 20*c^6*d*Sin[e - (f*x)/2] + 236*c^5*d^2*Sin[
e - (f*x)/2] + 628*c^4*d^3*Sin[e - (f*x)/2] + 778*c^3*d^4*Sin[e - (f*x)/2] + 420*c^2*d^5*Sin[e - (f*x)/2] + 48
*c*d^6*Sin[e - (f*x)/2] - 18*d^7*Sin[e - (f*x)/2] + 12*c^7*Sin[e + (f*x)/2] - 20*c^6*d*Sin[e + (f*x)/2] - 236*
c^5*d^2*Sin[e + (f*x)/2] - 460*c^4*d^3*Sin[e + (f*x)/2] - 310*c^3*d^4*Sin[e + (f*x)/2] + 39*c^2*d^5*Sin[e + (f
*x)/2] + 48*c*d^6*Sin[e + (f*x)/2] - 18*d^7*Sin[e + (f*x)/2] - 16*c^7*Sin[2*e + (f*x)/2] + 14*c^6*d*Sin[2*e +
(f*x)/2] + 220*c^5*d^2*Sin[2*e + (f*x)/2] + 502*c^4*d^3*Sin[2*e + (f*x)/2] + 522*c^3*d^4*Sin[2*e + (f*x)/2] +
303*c^2*d^5*Sin[2*e + (f*x)/2] + 48*c*d^6*Sin[2*e + (f*x)/2] - 18*d^7*Sin[2*e + (f*x)/2] - 6*c^7*Sin[e + (3*f*
x)/2] + 6*c^6*d*Sin[e + (3*f*x)/2] + 126*c^5*d^2*Sin[e + (3*f*x)/2] + 114*c^4*d^3*Sin[e + (3*f*x)/2] - 159*c^3
*d^4*Sin[e + (3*f*x)/2] - 144*c^2*d^5*Sin[e + (3*f*x)/2] - 6*c*d^6*Sin[e + (3*f*x)/2] + 6*d^7*Sin[e + (3*f*x)/
2] + 14*c^7*Sin[2*e + (3*f*x)/2] - 16*c^6*d*Sin[2*e + (3*f*x)/2] - 226*c^5*d^2*Sin[2*e + (3*f*x)/2] - 412*c^4*
d^3*Sin[2*e + (3*f*x)/2] - 235*c^3*d^4*Sin[2*e + (3*f*x)/2] - 7*c^2*d^5*Sin[2*e + (3*f*x)/2] + 6*c*d^6*Sin[2*e
+ (3*f*x)/2] - 6*d^7*Sin[2*e + (3*f*x)/2] - 6*c^7*Sin[3*e + (3*f*x)/2] + 6*c^6*d*Sin[3*e + (3*f*x)/2] + 126*c
^5*d^2*Sin[3*e + (3*f*x)/2] + 234*c^4*d^3*Sin[3*e + (3*f*x)/2] + 189*c^3*d^4*Sin[3*e + (3*f*x)/2] + 81*c^2*d^5
*Sin[3*e + (3*f*x)/2] + 6*c*d^6*Sin[3*e + (3*f*x)/2] - 6*d^7*Sin[3*e + (3*f*x)/2] + 6*c^7*Sin[e + (5*f*x)/2] -
14*c^6*d*Sin[e + (5*f*x)/2] - 134*c^5*d^2*Sin[e + (5*f*x)/2] - 274*c^4*d^3*Sin[e + (5*f*x)/2] - 193*c^3*d^4*S
in[e + (5*f*x)/2] - 27*c^2*d^5*Sin[e + (5*f*x)/2] + 6*c*d^6*Sin[e + (5*f*x)/2] - 6*c^7*Sin[2*e + (5*f*x)/2] +
12*c^6*d*Sin[2*e + (5*f*x)/2] + 42*c^5*d^2*Sin[2*e + (5*f*x)/2] - 48*c^4*d^3*Sin[2*e + (5*f*x)/2] - 105*c^3*d^
4*Sin[2*e + (5*f*x)/2] - 27*c^2*d^5*Sin[2*e + (5*f*x)/2] + 6*c*d^6*Sin[2*e + (5*f*x)/2] + 6*c^7*Sin[3*e + (5*f
*x)/2] - 14*c^6*d*Sin[3*e + (5*f*x)/2] - 134*c^5*d^2*Sin[3*e + (5*f*x)/2] - 202*c^4*d^3*Sin[3*e + (5*f*x)/2] -
61*c^3*d^4*Sin[3*e + (5*f*x)/2] + 12*c^2*d^5*Sin[3*e + (5*f*x)/2] - 6*c*d^6*Sin[3*e + (5*f*x)/2] - 6*c^7*Sin[
4*e + (5*f*x)/2] + 12*c^6*d*Sin[4*e + (5*f*x)/2] + 42*c^5*d^2*Sin[4*e + (5*f*x)/2] + 24*c^4*d^3*Sin[4*e + (5*f
*x)/2] + 27*c^3*d^4*Sin[4*e + (5*f*x)/2] + 12*c^2*d^5*Sin[4*e + (5*f*x)/2] - 6*c*d^6*Sin[4*e + (5*f*x)/2] + 4*
c^7*Sin[2*e + (7*f*x)/2] - 14*c^6*d*Sin[2*e + (7*f*x)/2] - 40*c^5*d^2*Sin[2*e + (7*f*x)/2] - 46*c^4*d^3*Sin[2*
e + (7*f*x)/2] - 12*c^3*d^4*Sin[2*e + (7*f*x)/2] + 3*c^2*d^5*Sin[2*e + (7*f*x)/2] - 24*c^4*d^3*Sin[3*e + (7*f*
x)/2] - 12*c^3*d^4*Sin[3*e + (7*f*x)/2] + 3*c^2*d^5*Sin[3*e + (7*f*x)/2] + 4*c^7*Sin[4*e + (7*f*x)/2] - 14*c^6
*d*Sin[4*e + (7*f*x)/2] - 40*c^5*d^2*Sin[4*e + (7*f*x)/2] - 22*c^4*d^3*Sin[4*e + (7*f*x)/2]))/(48*c^2*(-c + d)
^4*(c + d)^2*f*(a + a*Sec[e + f*x])^2*(c + d*Sec[e + f*x])^3)

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Maple [A]  time = 0.105, size = 280, normalized size = 1. \begin{align*}{\frac{1}{2\,f{a}^{2}} \left ( -{\frac{1}{ \left ({c}^{3}-3\,{c}^{2}d+3\,{d}^{2}c-{d}^{3} \right ) \left ( c-d \right ) } \left ({\frac{c}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-{\frac{d}{3} \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{3}}-c\tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) +7\,\tan \left ( 1/2\,fx+e/2 \right ) d \right ) }-8\,{\frac{{d}^{2}}{ \left ( c-d \right ) ^{4}} \left ({\frac{1}{ \left ( \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}c- \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{2}d-c-d \right ) ^{2}} \left ( -1/4\,{\frac{d \left ( 8\,{c}^{2}-3\,cd-5\,{d}^{2} \right ) \left ( \tan \left ( 1/2\,fx+e/2 \right ) \right ) ^{3}}{{c}^{2}+2\,cd+{d}^{2}}}+1/4\,{\frac{d \left ( 8\,c+3\,d \right ) \tan \left ( 1/2\,fx+e/2 \right ) }{c+d}} \right ) }-1/4\,{\frac{12\,{c}^{2}+16\,cd+7\,{d}^{2}}{ \left ({c}^{2}+2\,cd+{d}^{2} \right ) \sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}{\it Artanh} \left ({\frac{\tan \left ( 1/2\,fx+e/2 \right ) \left ( c-d \right ) }{\sqrt{ \left ( c+d \right ) \left ( c-d \right ) }}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/f/a^2*(-1/(c^3-3*c^2*d+3*c*d^2-d^3)/(c-d)*(1/3*tan(1/2*f*x+1/2*e)^3*c-1/3*tan(1/2*f*x+1/2*e)^3*d-c*tan(1/2
*f*x+1/2*e)+7*tan(1/2*f*x+1/2*e)*d)-8*d^2/(c-d)^4*((-1/4*d*(8*c^2-3*c*d-5*d^2)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2
*e)^3+1/4*d*(8*c+3*d)/(c+d)*tan(1/2*f*x+1/2*e))/(tan(1/2*f*x+1/2*e)^2*c-tan(1/2*f*x+1/2*e)^2*d-c-d)^2-1/4*(12*
c^2+16*c*d+7*d^2)/(c^2+2*c*d+d^2)/((c+d)*(c-d))^(1/2)*arctanh(tan(1/2*f*x+1/2*e)*(c-d)/((c+d)*(c-d))^(1/2))))

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 0.78321, size = 4329, normalized size = 15.24 \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))^2/(c+d*sec(f*x+e))^3,x, algorithm="fricas")

[Out]

[1/12*(3*(12*c^2*d^4 + 16*c*d^5 + 7*d^6 + (12*c^4*d^2 + 16*c^3*d^3 + 7*c^2*d^4)*cos(f*x + e)^4 + 2*(12*c^4*d^2
+ 28*c^3*d^3 + 23*c^2*d^4 + 7*c*d^5)*cos(f*x + e)^3 + (12*c^4*d^2 + 64*c^3*d^3 + 83*c^2*d^4 + 44*c*d^5 + 7*d^
6)*cos(f*x + e)^2 + 2*(12*c^3*d^3 + 28*c^2*d^4 + 23*c*d^5 + 7*d^6)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*co
s(f*x + e) - (c^2 - 2*d^2)*cos(f*x + e)^2 + 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)
/(c^2*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*(2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2*d^5 + 59*c
*d^6 + 32*d^7 + (4*c^7 - 14*c^6*d - 44*c^5*d^2 - 32*c^4*d^3 + 28*c^3*d^4 + 49*c^2*d^5 + 12*c*d^6 - 3*d^7)*cos(
f*x + e)^3 + (2*c^7 - 8*c^6*d - 68*c^5*d^2 - 140*c^4*d^3 - 23*c^3*d^4 + 142*c^2*d^5 + 89*c*d^6 + 6*d^7)*cos(f*
x + e)^2 + (4*c^6*d - 28*c^5*d^2 - 118*c^4*d^3 - 106*c^3*d^4 + 71*c^2*d^5 + 134*c*d^6 + 43*d^7)*cos(f*x + e))*
sin(f*x + e))/((a^2*c^10 - 2*a^2*c^9*d - 2*a^2*c^8*d^2 + 6*a^2*c^7*d^3 - 6*a^2*c^5*d^5 + 2*a^2*c^4*d^6 + 2*a^2
*c^3*d^7 - a^2*c^2*d^8)*f*cos(f*x + e)^4 + 2*(a^2*c^10 - a^2*c^9*d - 4*a^2*c^8*d^2 + 4*a^2*c^7*d^3 + 6*a^2*c^6
*d^4 - 6*a^2*c^5*d^5 - 4*a^2*c^4*d^6 + 4*a^2*c^3*d^7 + a^2*c^2*d^8 - a^2*c*d^9)*f*cos(f*x + e)^3 + (a^2*c^10 +
2*a^2*c^9*d - 9*a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 22*a^2*c^6*d^4 - 22*a^2*c^4*d^6 + 4*a^2*c^3*d^7 + 9*a^2*c^2*d^8
- 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^6*d^4 + 6*a
^2*c^5*d^5 - 6*a^2*c^4*d^6 - 4*a^2*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e) + (a^2*c^8*d
^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 - a^2*d^10)*f
), 1/6*(3*(12*c^2*d^4 + 16*c*d^5 + 7*d^6 + (12*c^4*d^2 + 16*c^3*d^3 + 7*c^2*d^4)*cos(f*x + e)^4 + 2*(12*c^4*d^
2 + 28*c^3*d^3 + 23*c^2*d^4 + 7*c*d^5)*cos(f*x + e)^3 + (12*c^4*d^2 + 64*c^3*d^3 + 83*c^2*d^4 + 44*c*d^5 + 7*d
^6)*cos(f*x + e)^2 + 2*(12*c^3*d^3 + 28*c^2*d^4 + 23*c*d^5 + 7*d^6)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqr
t(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (2*c^5*d^2 - 16*c^4*d^3 - 61*c^3*d^4 - 16*c^2
*d^5 + 59*c*d^6 + 32*d^7 + (4*c^7 - 14*c^6*d - 44*c^5*d^2 - 32*c^4*d^3 + 28*c^3*d^4 + 49*c^2*d^5 + 12*c*d^6 -
3*d^7)*cos(f*x + e)^3 + (2*c^7 - 8*c^6*d - 68*c^5*d^2 - 140*c^4*d^3 - 23*c^3*d^4 + 142*c^2*d^5 + 89*c*d^6 + 6*
d^7)*cos(f*x + e)^2 + (4*c^6*d - 28*c^5*d^2 - 118*c^4*d^3 - 106*c^3*d^4 + 71*c^2*d^5 + 134*c*d^6 + 43*d^7)*cos
(f*x + e))*sin(f*x + e))/((a^2*c^10 - 2*a^2*c^9*d - 2*a^2*c^8*d^2 + 6*a^2*c^7*d^3 - 6*a^2*c^5*d^5 + 2*a^2*c^4*
d^6 + 2*a^2*c^3*d^7 - a^2*c^2*d^8)*f*cos(f*x + e)^4 + 2*(a^2*c^10 - a^2*c^9*d - 4*a^2*c^8*d^2 + 4*a^2*c^7*d^3
+ 6*a^2*c^6*d^4 - 6*a^2*c^5*d^5 - 4*a^2*c^4*d^6 + 4*a^2*c^3*d^7 + a^2*c^2*d^8 - a^2*c*d^9)*f*cos(f*x + e)^3 +
(a^2*c^10 + 2*a^2*c^9*d - 9*a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 22*a^2*c^6*d^4 - 22*a^2*c^4*d^6 + 4*a^2*c^3*d^7 + 9*
a^2*c^2*d^8 - 2*a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e)^2 + 2*(a^2*c^9*d - a^2*c^8*d^2 - 4*a^2*c^7*d^3 + 4*a^2*c^
6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^4*d^6 - 4*a^2*c^3*d^7 + 4*a^2*c^2*d^8 + a^2*c*d^9 - a^2*d^10)*f*cos(f*x + e) +
(a^2*c^8*d^2 - 2*a^2*c^7*d^3 - 2*a^2*c^6*d^4 + 6*a^2*c^5*d^5 - 6*a^2*c^3*d^7 + 2*a^2*c^2*d^8 + 2*a^2*c*d^9 -
a^2*d^10)*f)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.31008, size = 1048, normalized size = 3.69 \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))^2/(c+d*sec(f*x+e))^3,x, algorithm="giac")

[Out]

1/6*(6*(12*c^2*d^2 + 16*c*d^3 + 7*d^4)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*
f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((a^2*c^6 - 2*a^2*c^5*d - a^2*c^4*d^2 + 4*a^2*c^3*d^
3 - a^2*c^2*d^4 - 2*a^2*c*d^5 + a^2*d^6)*sqrt(-c^2 + d^2)) - (a^4*c^6*tan(1/2*f*x + 1/2*e)^3 - 6*a^4*c^5*d*tan
(1/2*f*x + 1/2*e)^3 + 15*a^4*c^4*d^2*tan(1/2*f*x + 1/2*e)^3 - 20*a^4*c^3*d^3*tan(1/2*f*x + 1/2*e)^3 + 15*a^4*c
^2*d^4*tan(1/2*f*x + 1/2*e)^3 - 6*a^4*c*d^5*tan(1/2*f*x + 1/2*e)^3 + a^4*d^6*tan(1/2*f*x + 1/2*e)^3 - 3*a^4*c^
6*tan(1/2*f*x + 1/2*e) + 36*a^4*c^5*d*tan(1/2*f*x + 1/2*e) - 135*a^4*c^4*d^2*tan(1/2*f*x + 1/2*e) + 240*a^4*c^
3*d^3*tan(1/2*f*x + 1/2*e) - 225*a^4*c^2*d^4*tan(1/2*f*x + 1/2*e) + 108*a^4*c*d^5*tan(1/2*f*x + 1/2*e) - 21*a^
4*d^6*tan(1/2*f*x + 1/2*e))/(a^6*c^9 - 9*a^6*c^8*d + 36*a^6*c^7*d^2 - 84*a^6*c^6*d^3 + 126*a^6*c^5*d^4 - 126*a
^6*c^4*d^5 + 84*a^6*c^3*d^6 - 36*a^6*c^2*d^7 + 9*a^6*c*d^8 - a^6*d^9) + 6*(8*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 -
3*c*d^4*tan(1/2*f*x + 1/2*e)^3 - 5*d^5*tan(1/2*f*x + 1/2*e)^3 - 8*c^2*d^3*tan(1/2*f*x + 1/2*e) - 11*c*d^4*tan(
1/2*f*x + 1/2*e) - 3*d^5*tan(1/2*f*x + 1/2*e))/((a^2*c^6 - 2*a^2*c^5*d - a^2*c^4*d^2 + 4*a^2*c^3*d^3 - a^2*c^2
*d^4 - 2*a^2*c*d^5 + a^2*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^2))/f