3.3.23 \(\int \frac {\sec (e+f x)}{(a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2} \, dx\) [223]

Optimal. Leaf size=211 \[ \frac {2 d^2 (3 c+2 d) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{a^2 (c-d)^{7/2} (c+d)^{3/2} f}+\frac {d \left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 a^2 (c-d)^3 (c+d) f (c+d \sec (e+f x))}+\frac {(c-6 d) \tan (e+f x)}{3 a^2 (c-d)^2 f (1+\sec (e+f x)) (c+d \sec (e+f x))}+\frac {\tan (e+f x)}{3 (c-d) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))} \]

[Out]

2*d^2*(3*c+2*d)*arctanh((c-d)^(1/2)*tan(1/2*f*x+1/2*e)/(c+d)^(1/2))/a^2/(c-d)^(7/2)/(c+d)^(3/2)/f+1/3*d*(c^2-6
*c*d-10*d^2)*tan(f*x+e)/a^2/(c-d)^3/(c+d)/f/(c+d*sec(f*x+e))+1/3*(c-6*d)*tan(f*x+e)/a^2/(c-d)^2/f/(1+sec(f*x+e
))/(c+d*sec(f*x+e))+1/3*tan(f*x+e)/(c-d)/f/(a+a*sec(f*x+e))^2/(c+d*sec(f*x+e))

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Rubi [A]
time = 0.25, antiderivative size = 260, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {4072, 105, 157, 12, 95, 211} \begin {gather*} \frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 f (c-d)^3 (c+d) \left (a^2 \sec (e+f x)+a^2\right )}-\frac {2 d^2 (3 c+2 d) \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)^{7/2} (c+d)^{3/2} \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}-\frac {d \tan (e+f x)}{f \left (c^2-d^2\right ) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}+\frac {(c+4 d) \tan (e+f x)}{3 f (c-d)^2 (c+d) (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])^2),x]

[Out]

((c + 4*d)*Tan[e + f*x])/(3*(c - d)^2*(c + d)*f*(a + a*Sec[e + f*x])^2) - (2*d^2*(3*c + 2*d)*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)^(7/2)*(c + d)^(3
/2)*f*Sqrt[a - a*Sec[e + f*x]]*Sqrt[a + a*Sec[e + f*x]]) + ((c^2 - 6*c*d - 10*d^2)*Tan[e + f*x])/(3*(c - d)^3*
(c + d)*f*(a^2 + a^2*Sec[e + f*x])) - (d*Tan[e + f*x])/((c^2 - d^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 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))^2} \, 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)^2} \, 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)}{\left (c^2-d^2\right ) 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^2 (c+2 d)-2 a^2 d x}{\sqrt {a-a x} (a+a x)^{5/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{\left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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 (c-6 d) (c+d)-a^4 d (c+4 d) x}{\sqrt {a-a x} (a+a x)^{3/2} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^3 (c-d) \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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^6 d^2 (3 c+2 d)}{\sqrt {a-a x} \sqrt {a+a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{3 a^6 (c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (d^2 (3 c+2 d) \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 )}{(c-d)^2 \left (c^2-d^2\right ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) f (a+a \sec (e+f x))^2 (c+d \sec (e+f x))}-\frac {\left (2 d^2 (3 c+2 d) \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 ) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {(c+4 d) \tan (e+f x)}{3 (c-d)^2 (c+d) f (a+a \sec (e+f x))^2}-\frac {2 d^2 (3 c+2 d) \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)^{7/2} (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (c^2-6 c d-10 d^2\right ) \tan (e+f x)}{3 (c-d)^3 (c+d) f \left (a^2+a^2 \sec (e+f x)\right )}-\frac {d \tan (e+f x)}{\left (c^2-d^2\right ) 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 = 3.93, size = 376, normalized size = 1.78 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec ^4(e+f x) \left (\frac {12 d^2 (3 c+2 d) \text {ArcTan}\left (\frac {(i \cos (e)+\sin (e)) \left (c \sin (e)+(-d+c \cos (e)) \tan \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) \cos ^3\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) (i \cos (e)+\sin (e))}{(c+d) \sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+(c-d) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )-4 (c-4 d) \cos ^2\left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \sec \left (\frac {e}{2}\right ) \sin \left (\frac {f x}{2}\right )+\frac {6 d^3 \cos ^3\left (\frac {1}{2} (e+f x)\right ) (-d \sin (e)+c \sin (f x))}{c (c+d) \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right )}+(c-d) \cos \left (\frac {1}{2} (e+f x)\right ) (d+c \cos (e+f x)) \tan \left (\frac {e}{2}\right )\right )}{3 a^2 (-c+d)^3 f (1+\sec (e+f x))^2 (c+d \sec (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Sec[e + f*x]^4*((12*d^2*(3*c + 2*d)*ArcTan[((I*Cos[e] + Sin[e])*(c*Si
n[e] + (-d + c*Cos[e])*Tan[(f*x)/2]))/(Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2])]*Cos[(e + f*x)/2]^3*(d + c
*Cos[e + f*x])*(I*Cos[e] + Sin[e]))/((c + d)*Sqrt[c^2 - d^2]*Sqrt[(Cos[e] - I*Sin[e])^2]) + (c - d)*(d + c*Cos
[e + f*x])*Sec[e/2]*Sin[(f*x)/2] - 4*(c - 4*d)*Cos[(e + f*x)/2]^2*(d + c*Cos[e + f*x])*Sec[e/2]*Sin[(f*x)/2] +
(6*d^3*Cos[(e + f*x)/2]^3*(-(d*Sin[e]) + c*Sin[f*x]))/(c*(c + d)*(Cos[e/2] - Sin[e/2])*(Cos[e/2] + Sin[e/2]))
+ (c - d)*Cos[(e + f*x)/2]*(d + c*Cos[e + f*x])*Tan[e/2]))/(3*a^2*(-c + d)^3*f*(1 + Sec[e + f*x])^2*(c + d*Se
c[e + f*x])^2)

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Maple [A]
time = 0.36, size = 203, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (3 c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\)	\(203\)
default	\(\frac {-\frac {\frac {c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-\frac {d \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3}-c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+5 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c^{2}-2 c d +d^{2}\right ) \left (c -d \right )}-\frac {4 d^{2} \left (-\frac {d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (c +d \right ) \left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )}-\frac {\left (3 c +2 d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c +d \right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c -d \right )^{3}}}{2 f \,a^{2}}\)	\(203\)
risch	\(\frac {2 i \left (-3 c^{4} {\mathrm e}^{4 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{4 i \left (f x +e \right )}+9 c^{2} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{4 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{3 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{3 i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+21 c \,d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-5 c^{4} {\mathrm e}^{2 i \left (f x +e \right )}+6 c^{3} d \,{\mathrm e}^{2 i \left (f x +e \right )}+41 c^{2} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+39 c \,d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 d^{4} {\mathrm e}^{2 i \left (f x +e \right )}-3 c^{4} {\mathrm e}^{i \left (f x +e \right )}+8 c^{3} d \,{\mathrm e}^{i \left (f x +e \right )}+27 c^{2} d^{2} {\mathrm e}^{i \left (f x +e \right )}+25 c \,d^{3} {\mathrm e}^{i \left (f x +e \right )}+3 d^{4} {\mathrm e}^{i \left (f x +e \right )}-2 c^{4}+6 c^{3} d +8 c^{2} d^{2}+3 c \,d^{3}\right )}{3 \left ({\mathrm e}^{2 i \left (f x +e \right )} c +2 d \,{\mathrm e}^{i \left (f x +e \right )}+c \right ) \left (-c^{2}+d^{2}\right ) c f \,a^{2} \left (-c +d \right )^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{3}}+\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c^{2}-i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{\sqrt {c^{2}-d^{2}}\, c}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right ) c}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {-i c^{2}+i d^{2}+\sqrt {c^{2}-d^{2}}\, d}{c \sqrt {c^{2}-d^{2}}}\right )}{\sqrt {c^{2}-d^{2}}\, \left (c +d \right ) \left (c -d \right )^{3} f \,a^{2}}\)	\(729\)

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))^2,x,method=_RETURNVERBOSE)

[Out]

1/2/f/a^2*(-1/(c^2-2*c*d+d^2)/(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
)+5*d*tan(1/2*f*x+1/2*e))-4*d^2/(c-d)^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)-(3*c+2*d)/(c+d)/((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))^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*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. 603 vs. \(2 (205) = 410\).
time = 2.84, size = 1268, normalized size = 6.01 \begin {gather*} \left [-\frac {3 \, {\left (3 \, c d^{3} + 2 \, d^{4} + {\left (3 \, c^{2} d^{2} + 2 \, c d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{2} d^{2} + 7 \, c d^{3} + 2 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) - 2 \, {\left (c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5} + {\left (2 \, c^{5} - 6 \, c^{4} d - 10 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 8 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{6 \, {\left ({\left (a^{2} c^{7} - 2 \, a^{2} c^{6} d - a^{2} c^{5} d^{2} + 4 \, a^{2} c^{4} d^{3} - a^{2} c^{3} d^{4} - 2 \, a^{2} c^{2} d^{5} + a^{2} c d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (2 \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - 4 \, a^{2} c^{5} d^{2} + 7 \, a^{2} c^{4} d^{3} + 2 \, a^{2} c^{3} d^{4} - 5 \, a^{2} c^{2} d^{5} + a^{2} d^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right )} f\right )}}, \frac {3 \, {\left (3 \, c d^{3} + 2 \, d^{4} + {\left (3 \, c^{2} d^{2} + 2 \, c d^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (6 \, c^{2} d^{2} + 7 \, c d^{3} + 2 \, d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, c^{2} d^{2} + 8 \, c d^{3} + 4 \, d^{4}\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left (c^{4} d - 6 \, c^{3} d^{2} - 11 \, c^{2} d^{3} + 6 \, c d^{4} + 10 \, d^{5} + {\left (2 \, c^{5} - 6 \, c^{4} d - 10 \, c^{3} d^{2} + 3 \, c^{2} d^{3} + 8 \, c d^{4} + 3 \, d^{5}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{5} - 4 \, c^{4} d - 14 \, c^{3} d^{2} - 10 \, c^{2} d^{3} + 13 \, c d^{4} + 14 \, d^{5}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left ({\left (a^{2} c^{7} - 2 \, a^{2} c^{6} d - a^{2} c^{5} d^{2} + 4 \, a^{2} c^{4} d^{3} - a^{2} c^{3} d^{4} - 2 \, a^{2} c^{2} d^{5} + a^{2} c d^{6}\right )} f \cos \left (f x + e\right )^{3} + {\left (2 \, a^{2} c^{7} - 3 \, a^{2} c^{6} d - 4 \, a^{2} c^{5} d^{2} + 7 \, a^{2} c^{4} d^{3} + 2 \, a^{2} c^{3} d^{4} - 5 \, a^{2} c^{2} d^{5} + a^{2} d^{7}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{2} c^{7} - 5 \, a^{2} c^{5} d^{2} + 2 \, a^{2} c^{4} d^{3} + 7 \, a^{2} c^{3} d^{4} - 4 \, a^{2} c^{2} d^{5} - 3 \, a^{2} c d^{6} + 2 \, a^{2} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (a^{2} c^{6} d - 2 \, a^{2} c^{5} d^{2} - a^{2} c^{4} d^{3} + 4 \, a^{2} c^{3} d^{4} - a^{2} c^{2} d^{5} - 2 \, a^{2} c d^{6} + a^{2} d^{7}\right )} f\right )}}\right ] \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))^2,x, algorithm="fricas")

[Out]

[-1/6*(3*(3*c*d^3 + 2*d^4 + (3*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^3 + (6*c^2*d^2 + 7*c*d^3 + 2*d^4)*cos(f*x + e)^
2 + (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(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*(c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*d^5 + (2*c^5 - 6*c^4*d - 10*c^3*d^2 + 3*c^
2*d^3 + 8*c*d^4 + 3*d^5)*cos(f*x + e)^2 + (c^5 - 4*c^4*d - 14*c^3*d^2 - 10*c^2*d^3 + 13*c*d^4 + 14*d^5)*cos(f*
x + e))*sin(f*x + e))/((a^2*c^7 - 2*a^2*c^6*d - a^2*c^5*d^2 + 4*a^2*c^4*d^3 - a^2*c^3*d^4 - 2*a^2*c^2*d^5 + a^
2*c*d^6)*f*cos(f*x + e)^3 + (2*a^2*c^7 - 3*a^2*c^6*d - 4*a^2*c^5*d^2 + 7*a^2*c^4*d^3 + 2*a^2*c^3*d^4 - 5*a^2*c
^2*d^5 + a^2*d^7)*f*cos(f*x + e)^2 + (a^2*c^7 - 5*a^2*c^5*d^2 + 2*a^2*c^4*d^3 + 7*a^2*c^3*d^4 - 4*a^2*c^2*d^5
- 3*a^2*c*d^6 + 2*a^2*d^7)*f*cos(f*x + e) + (a^2*c^6*d - 2*a^2*c^5*d^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2
*d^5 - 2*a^2*c*d^6 + a^2*d^7)*f), 1/3*(3*(3*c*d^3 + 2*d^4 + (3*c^2*d^2 + 2*c*d^3)*cos(f*x + e)^3 + (6*c^2*d^2
+ 7*c*d^3 + 2*d^4)*cos(f*x + e)^2 + (3*c^2*d^2 + 8*c*d^3 + 4*d^4)*cos(f*x + e))*sqrt(-c^2 + d^2)*arctan(-sqrt(
-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (c^4*d - 6*c^3*d^2 - 11*c^2*d^3 + 6*c*d^4 + 10*
d^5 + (2*c^5 - 6*c^4*d - 10*c^3*d^2 + 3*c^2*d^3 + 8*c*d^4 + 3*d^5)*cos(f*x + e)^2 + (c^5 - 4*c^4*d - 14*c^3*d^
2 - 10*c^2*d^3 + 13*c*d^4 + 14*d^5)*cos(f*x + e))*sin(f*x + e))/((a^2*c^7 - 2*a^2*c^6*d - a^2*c^5*d^2 + 4*a^2*
c^4*d^3 - a^2*c^3*d^4 - 2*a^2*c^2*d^5 + a^2*c*d^6)*f*cos(f*x + e)^3 + (2*a^2*c^7 - 3*a^2*c^6*d - 4*a^2*c^5*d^2
+ 7*a^2*c^4*d^3 + 2*a^2*c^3*d^4 - 5*a^2*c^2*d^5 + a^2*d^7)*f*cos(f*x + e)^2 + (a^2*c^7 - 5*a^2*c^5*d^2 + 2*a^
2*c^4*d^3 + 7*a^2*c^3*d^4 - 4*a^2*c^2*d^5 - 3*a^2*c*d^6 + 2*a^2*d^7)*f*cos(f*x + e) + (a^2*c^6*d - 2*a^2*c^5*d
^2 - a^2*c^4*d^3 + 4*a^2*c^3*d^4 - a^2*c^2*d^5 - 2*a^2*c*d^6 + a^2*d^7)*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^{2} \sec ^{2}{\left (e + f x \right )} + 2 c^{2} \sec {\left (e + f x \right )} + c^{2} + 2 c d \sec ^{3}{\left (e + f x \right )} + 4 c d \sec ^{2}{\left (e + f x \right )} + 2 c d \sec {\left (e + f x \right )} + d^{2} \sec ^{4}{\left (e + f x \right )} + 2 d^{2} \sec ^{3}{\left (e + f x \right )} + d^{2} \sec ^{2}{\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))**2,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (196) = 392\).
time = 0.53, size = 474, normalized size = 2.25 \begin {gather*} \frac {\frac {12 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (a^{2} c^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\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 )}} + \frac {12 \, {\left (3 \, c d^{2} + 2 \, d^{3}\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^{4} - 2 \, a^{2} c^{3} d + 2 \, a^{2} c d^{3} - a^{2} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} - \frac {a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{4} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{4} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 54 \, a^{4} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 48 \, a^{4} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 15 \, a^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{a^{6} c^{6} - 6 \, a^{6} c^{5} d + 15 \, a^{6} c^{4} d^{2} - 20 \, a^{6} c^{3} d^{3} + 15 \, a^{6} c^{2} d^{4} - 6 \, a^{6} c d^{5} + a^{6} d^{6}}}{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))^2,x, algorithm="giac")

[Out]

1/6*(12*d^3*tan(1/2*f*x + 1/2*e)/((a^2*c^4 - 2*a^2*c^3*d + 2*a^2*c*d^3 - a^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 -
d*tan(1/2*f*x + 1/2*e)^2 - c - d)) + 12*(3*c*d^2 + 2*d^3)*(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^4 - 2*a^2*c^3*d + 2*a^2*c
*d^3 - a^2*d^4)*sqrt(-c^2 + d^2)) - (a^4*c^4*tan(1/2*f*x + 1/2*e)^3 - 4*a^4*c^3*d*tan(1/2*f*x + 1/2*e)^3 + 6*a
^4*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 - 4*a^4*c*d^3*tan(1/2*f*x + 1/2*e)^3 + a^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 3*a^
4*c^4*tan(1/2*f*x + 1/2*e) + 24*a^4*c^3*d*tan(1/2*f*x + 1/2*e) - 54*a^4*c^2*d^2*tan(1/2*f*x + 1/2*e) + 48*a^4*
c*d^3*tan(1/2*f*x + 1/2*e) - 15*a^4*d^4*tan(1/2*f*x + 1/2*e))/(a^6*c^6 - 6*a^6*c^5*d + 15*a^6*c^4*d^2 - 20*a^6
*c^3*d^3 + 15*a^6*c^2*d^4 - 6*a^6*c*d^5 + a^6*d^6))/f

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Mupad [B]
time = 2.18, size = 314, normalized size = 1.49 \begin {gather*} \frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3}{2\,a^2\,{\left (c-d\right )}^2}-\frac {c^2-d^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 )}^2}+\frac {2\,d^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (c+d\right )\,\left (a^2\,d^4-a^2\,c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (a^2\,c^4-4\,a^2\,c^3\,d+6\,a^2\,c^2\,d^2-4\,a^2\,c\,d^3+a^2\,d^4\right )-2\,a^2\,c\,d^3+2\,a^2\,c^3\,d\right )}-\frac {d^2\,\mathrm {atan}\left (\frac {1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^4-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^3\,d+6{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c^2\,d^2-4{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,c\,d^3+1{}\mathrm {i}\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,d^4}{\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )\,\left (3\,c+2\,d\right )\,2{}\mathrm {i}}{a^2\,f\,{\left (c+d\right )}^{3/2}\,{\left (c-d\right )}^{7/2}} \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))^2),x)

[Out]

(tan(e/2 + (f*x)/2)*(3/(2*a^2*(c - d)^2) - (c^2 - d^2)/(a^2*(c - d)^4)))/f - tan(e/2 + (f*x)/2)^3/(6*a^2*f*(c
- d)^2) + (2*d^3*tan(e/2 + (f*x)/2))/(f*(c + d)*(a^2*d^4 - a^2*c^4 + tan(e/2 + (f*x)/2)^2*(a^2*c^4 + a^2*d^4 -
4*a^2*c*d^3 - 4*a^2*c^3*d + 6*a^2*c^2*d^2) - 2*a^2*c*d^3 + 2*a^2*c^3*d)) - (d^2*atan((c^4*tan(e/2 + (f*x)/2)*
1i + d^4*tan(e/2 + (f*x)/2)*1i - c*d^3*tan(e/2 + (f*x)/2)*4i - c^3*d*tan(e/2 + (f*x)/2)*4i + c^2*d^2*tan(e/2 +
(f*x)/2)*6i)/((c + d)^(1/2)*(c - d)^(7/2)))*(3*c + 2*d)*2i)/(a^2*f*(c + d)^(3/2)*(c - d)^(7/2))

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