3.3.24 \(\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^3} \, dx\) [224]

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

[Out]

d^2*(12*c^2+16*c*d+7*d^2)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c+d)^(1/2))/a^2/(c-d)^(9/2)/(c+d)^(5/2)/f+1/
6*d*(2*c^2-16*c*d-21*d^2)*tan(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sec(f*x+e))^2+1/3*(c-8*d)*tan(f*x+e)/a^2/(c-d)^2
/f/(1+sec(f*x+e))/(c+d*sec(f*x+e))^2+1/3*tan(f*x+e)/(c-d)/f/(a+a*sec(f*x+e))^2/(c+d*sec(f*x+e))^2+1/6*d*(2*c^3
-16*c^2*d-59*c*d^2-32*d^3)*tan(f*x+e)/a^2/(c-d)^4/(c+d)^2/f/(c+d*sec(f*x+e))

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Rubi [A]
time = 0.38, 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 = {4072, 105, 156, 157, 12, 95, 211} \begin {gather*} \frac {\left (2 c^3-16 c^2 d-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) \text {ArcTan}\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} \end {gather*}

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 12

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

Rule 95

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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] && ILtQ[m, -1]

Rule 157

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 211

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

Rule 4072

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])

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 ) \text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 7.39, size = 2220, normalized size = 7.82 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.48, size = 280, normalized size = 0.99

method	result	size
derivativedivides	\(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+7 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right ) \left (c -d \right )}-\frac {8 d^{2} \left (\frac {-\frac {d \left (8 c^{2}-3 c d -5 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c +3 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 c +4 d}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (12 c^{2}+16 c d +7 d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{2 f \,a^{2}}\)	\(280\)
default	\(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+7 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right ) \left (c -d \right )}-\frac {8 d^{2} \left (\frac {-\frac {d \left (8 c^{2}-3 c d -5 d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 \left (c^{2}+2 c d +d^{2}\right )}+\frac {d \left (8 c +3 d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 c +4 d}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (12 c^{2}+16 c d +7 d^{2}\right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{4 \left (c^{2}+2 c d +d^{2}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{4}}}{2 f \,a^{2}}\)	\(280\)
risch	\(\text {Expression too large to display}\)	\(1368\)

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,method=_RETURNVERBOSE)

[Out]

1/2/f/a^2*(-1/(c^3-3*c^2*d+3*c*d^2-d^3)/(c-d)*(1/3*c*tan(1/2*f*x+1/2*e)^3-1/3*d*tan(1/2*f*x+1/2*e)^3-c*tan(1/2
*f*x+1/2*e)+7*d*tan(1/2*f*x+1/2*e))-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))/(c*tan(1/2*f*x+1/2*e)^2-d*tan(1/2*f*x+1/2*e)^2-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((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*c^2-4*d^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1000 vs. \(2 (278) = 556\).
time = 2.37, size = 2062, normalized size = 7.26 \begin {gather*} \text {Too large to display} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec {\left (e + f x \right )}}{c^{3} \sec ^{2}{\left (e + f x \right )} + 2 c^{3} \sec {\left (e + f x \right )} + c^{3} + 3 c^{2} d \sec ^{3}{\left (e + f x \right )} + 6 c^{2} d \sec ^{2}{\left (e + f x \right )} + 3 c^{2} d \sec {\left (e + f x \right )} + 3 c d^{2} \sec ^{4}{\left (e + f x \right )} + 6 c d^{2} \sec ^{3}{\left (e + f x \right )} + 3 c d^{2} \sec ^{2}{\left (e + f x \right )} + d^{3} \sec ^{5}{\left (e + f x \right )} + 2 d^{3} \sec ^{4}{\left (e + f x \right )} + d^{3} \sec ^{3}{\left (e + f x \right )}}\, dx}{a^{2}} \end {gather*}

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]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 751 vs. \(2 (267) = 534\).
time = 0.56, size = 751, normalized size = 2.64 \begin {gather*} \frac {\frac {6 \, {\left (12 \, c^{2} d^{2} + 16 \, c d^{3} + 7 \, d^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (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}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a^{4} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{4} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 20 \, a^{4} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, a^{4} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, a^{4} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 36 \, a^{4} c^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 135 \, a^{4} c^{4} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 240 \, a^{4} c^{3} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 225 \, a^{4} c^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 108 \, a^{4} c d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 21 \, a^{4} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{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}} + \frac {6 \, {\left (8 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 11 \, c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (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}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{2}}}{6 \, f} \end {gather*}

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

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Mupad [B]
time = 2.28, size = 505, normalized size = 1.78 \begin {gather*} \frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-8\,c^2\,d^3+3\,c\,d^4+5\,d^5\right )}{{\left (c+d\right )}^2}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,d^4+8\,c\,d^3\right )}{c+d}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^2\,c^6-8\,a^2\,c^5\,d+10\,a^2\,c^4\,d^2-10\,a^2\,c^2\,d^4+8\,a^2\,c\,d^5-2\,a^2\,d^6\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (a^2\,c^6-6\,a^2\,c^5\,d+15\,a^2\,c^4\,d^2-20\,a^2\,c^3\,d^3+15\,a^2\,c^2\,d^4-6\,a^2\,c\,d^5+a^2\,d^6\right )-a^2\,c^6-a^2\,d^6+2\,a^2\,c\,d^5+2\,a^2\,c^5\,d+a^2\,c^2\,d^4-4\,a^2\,c^3\,d^3+a^2\,c^4\,d^2\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {2}{a^2\,{\left (c-d\right )}^3}-\frac {3\,\left (c+d\right )}{2\,a^2\,{\left (c-d\right )}^4}\right )}{f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{6\,a^2\,f\,{\left (c-d\right )}^3}-\frac {d^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^5-5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4\,d+10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d^2-10{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^3+5{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^4-1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^5}{\sqrt {c+d}\,{\left (c-d\right )}^{9/2}}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )\,1{}\mathrm {i}}{a^2\,f\,{\left (c+d\right )}^{5/2}\,{\left (c-d\right )}^{9/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(e + f*x)*(a + a/cos(e + f*x))^2*(c + d/cos(e + f*x))^3),x)

[Out]

((tan(e/2 + (f*x)/2)^3*(3*c*d^4 + 5*d^5 - 8*c^2*d^3))/(c + d)^2 + (tan(e/2 + (f*x)/2)*(8*c*d^3 + 3*d^4))/(c +
d))/(f*(tan(e/2 + (f*x)/2)^2*(2*a^2*c^6 - 2*a^2*d^6 + 8*a^2*c*d^5 - 8*a^2*c^5*d - 10*a^2*c^2*d^4 + 10*a^2*c^4*
d^2) - tan(e/2 + (f*x)/2)^4*(a^2*c^6 + a^2*d^6 - 6*a^2*c*d^5 - 6*a^2*c^5*d + 15*a^2*c^2*d^4 - 20*a^2*c^3*d^3 +
15*a^2*c^4*d^2) - a^2*c^6 - a^2*d^6 + 2*a^2*c*d^5 + 2*a^2*c^5*d + a^2*c^2*d^4 - 4*a^2*c^3*d^3 + a^2*c^4*d^2))
+ (tan(e/2 + (f*x)/2)*(2/(a^2*(c - d)^3) - (3*(c + d))/(2*a^2*(c - d)^4)))/f - tan(e/2 + (f*x)/2)^3/(6*a^2*f*
(c - d)^3) - (d^2*atan((c^5*tan(e/2 + (f*x)/2)*1i - d^5*tan(e/2 + (f*x)/2)*1i + c*d^4*tan(e/2 + (f*x)/2)*5i -
c^4*d*tan(e/2 + (f*x)/2)*5i - c^2*d^3*tan(e/2 + (f*x)/2)*10i + c^3*d^2*tan(e/2 + (f*x)/2)*10i)/((c + d)^(1/2)*
(c - d)^(9/2)))*(16*c*d + 12*c^2 + 7*d^2)*1i)/(a^2*f*(c + d)^(5/2)*(c - d)^(9/2))

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