\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx\) [253]

Optimal result

Integrand size = 31, antiderivative size = 170 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\frac {d^3 \text {arctanh}(\sin (e+f x))}{2 b f}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^3 f}+\frac {2 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} f}+\frac {d^2 (3 b c-a d) \tan (e+f x)}{b^2 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 b f} \]

[Out]

1/2*d^3*arctanh(sin(f*x+e))/b/f+d*(a^2*d^2-3*a*b*c*d+3*b^2*c^2)*arctanh(sin(f*x+e))/b^3/f+2*(-a*d+b*c)^3*arcta
nh((a-b)^(1/2)*tan(1/2*f*x+1/2*e)/(a+b)^(1/2))/b^3/f/(a-b)^(1/2)/(a+b)^(1/2)+d^2*(-a*d+3*b*c)*tan(f*x+e)/b^2/f
+1/2*d^3*sec(f*x+e)*tan(f*x+e)/b/f

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4073, 3031, 2738, 214, 3855, 3852, 8, 3853} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\frac {d \left (a^2 d^2-3 a b c d+3 b^2 c^2\right ) \text {arctanh}(\sin (e+f x))}{b^3 f}+\frac {2 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{b^3 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^2 (3 b c-a d) \tan (e+f x)}{b^2 f}+\frac {d^3 \text {arctanh}(\sin (e+f x))}{2 b f}+\frac {d^3 \tan (e+f x) \sec (e+f x)}{2 b f} \]

[In]

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

[Out]

(d^3*ArcTanh[Sin[e + f*x]])/(2*b*f) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*ArcTanh[Sin[e + f*x]])/(b^3*f) + (2
*(b*c - a*d)^3*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*f) + (d^2*(3*
b*c - a*d)*Tan[e + f*x])/(b^2*f) + (d^3*Sec[e + f*x]*Tan[e + f*x])/(2*b*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3031

Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p
] || IntegersQ[n, p]) && NeQ[p, 2]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4073

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+c \cos (e+f x))^3 \sec ^3(e+f x)}{b+a \cos (e+f x)} \, dx \\ & = \int \left (\frac {(b c-a d)^3}{b^3 (b+a \cos (e+f x))}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \sec (e+f x)}{b^3}+\frac {d^2 (3 b c-a d) \sec ^2(e+f x)}{b^2}+\frac {d^3 \sec ^3(e+f x)}{b}\right ) \, dx \\ & = \frac {d^3 \int \sec ^3(e+f x) \, dx}{b}+\frac {(b c-a d)^3 \int \frac {1}{b+a \cos (e+f x)} \, dx}{b^3}+\frac {\left (d^2 (3 b c-a d)\right ) \int \sec ^2(e+f x) \, dx}{b^2}+\frac {\left (d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right )\right ) \int \sec (e+f x) \, dx}{b^3} \\ & = \frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^3 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 b f}+\frac {d^3 \int \sec (e+f x) \, dx}{2 b}+\frac {\left (2 (b c-a d)^3\right ) \text {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{b^3 f}-\frac {\left (d^2 (3 b c-a d)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b^2 f} \\ & = \frac {d^3 \text {arctanh}(\sin (e+f x))}{2 b f}+\frac {d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^3 f}+\frac {2 (b c-a d)^3 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^3 \sqrt {a+b} f}+\frac {d^2 (3 b c-a d) \tan (e+f x)}{b^2 f}+\frac {d^3 \sec (e+f x) \tan (e+f x)}{2 b f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(389\) vs. \(2(170)=340\).

Time = 3.11 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.29 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\frac {\cos ^2(e+f x) (b+a \cos (e+f x)) (c+d \sec (e+f x))^3 \left (\frac {8 (-b c+a d)^3 \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-2 d \left (-6 a b c d+2 a^2 d^2+b^2 \left (6 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 d \left (-6 a b c d+2 a^2 d^2+b^2 \left (6 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {b^2 d^3}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {4 b d^2 (3 b c-a d) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}-\frac {b^2 d^3}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {4 b d^2 (3 b c-a d) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}\right )}{4 b^3 f (d+c \cos (e+f x))^3 (a+b \sec (e+f x))} \]

[In]

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

[Out]

(Cos[e + f*x]^2*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^3*((8*(-(b*c) + a*d)^3*ArcTanh[((-a + b)*Tan[(e + f*
x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 2*d*(-6*a*b*c*d + 2*a^2*d^2 + b^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/
2] - Sin[(e + f*x)/2]] + 2*d*(-6*a*b*c*d + 2*a^2*d^2 + b^2*(6*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)
/2]] + (b^2*d^3)/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (4*b*d^2*(3*b*c - a*d)*Sin[(e + f*x)/2])/(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2]) - (b^2*d^3)/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*b*d^2*(3*b*c - a*d)*Sin[(
e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(4*b^3*f*(d + c*Cos[e + f*x])^3*(a + b*Sec[e + f*x]))

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.70

method	result	size
derivativedivides	\(\frac {-\frac {d^{3}}{2 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {d \left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{3}}+\frac {d^{2} \left (2 a d -6 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{3}}{2 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d \left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{3}}+\frac {d^{2} \left (2 a d -6 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{f}\)	\(289\)
default	\(\frac {-\frac {d^{3}}{2 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}+\frac {d \left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{3}}+\frac {d^{2} \left (2 a d -6 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{3}}{2 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {d \left (2 a^{2} d^{2}-6 a b c d +6 b^{2} c^{2}+b^{2} d^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{3}}+\frac {d^{2} \left (2 a d -6 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {2 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{3} \sqrt {\left (a -b \right ) \left (a +b \right )}}}{f}\)	\(289\)
risch	\(-\frac {i d^{2} \left (b d \,{\mathrm e}^{3 i \left (f x +e \right )}+2 a d \,{\mathrm e}^{2 i \left (f x +e \right )}-6 b c \,{\mathrm e}^{2 i \left (f x +e \right )}-b d \,{\mathrm e}^{i \left (f x +e \right )}+2 a d -6 b c \right )}{f \,b^{2} \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{3} d^{3}}{\sqrt {a^{2}-b^{2}}\, f \,b^{3}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} c \,d^{2}}{\sqrt {a^{2}-b^{2}}\, f \,b^{2}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a \,c^{2} d}{\sqrt {a^{2}-b^{2}}\, f b}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}-\frac {i a^{2}-i b^{2}-b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{3}}{\sqrt {a^{2}-b^{2}}\, f}-\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{3} d^{3}}{\sqrt {a^{2}-b^{2}}\, f \,b^{3}}+\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a^{2} c \,d^{2}}{\sqrt {a^{2}-b^{2}}\, f \,b^{2}}-\frac {3 \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) a \,c^{2} d}{\sqrt {a^{2}-b^{2}}\, f b}+\frac {\ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i a^{2}-i b^{2}+b \sqrt {a^{2}-b^{2}}}{\sqrt {a^{2}-b^{2}}\, a}\right ) c^{3}}{\sqrt {a^{2}-b^{2}}\, f}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a^{2}}{f \,b^{3}}+\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) a c}{f \,b^{2}}-\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right ) c^{2}}{f b}-\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{2 f b}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a^{2}}{f \,b^{3}}-\frac {3 d^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a c}{f \,b^{2}}+\frac {3 d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) c^{2}}{f b}+\frac {d^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{2 f b}\)	\(900\)

[In]

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

[Out]

1/f*(-1/2*d^3/b/(tan(1/2*f*x+1/2*e)+1)^2+1/2*d*(2*a^2*d^2-6*a*b*c*d+6*b^2*c^2+b^2*d^2)/b^3*ln(tan(1/2*f*x+1/2*
e)+1)+1/2*d^2*(2*a*d-6*b*c+b*d)/b^2/(tan(1/2*f*x+1/2*e)+1)+1/2*d^3/b/(tan(1/2*f*x+1/2*e)-1)^2-1/2*d*(2*a^2*d^2
-6*a*b*c*d+6*b^2*c^2+b^2*d^2)/b^3*ln(tan(1/2*f*x+1/2*e)-1)+1/2*d^2*(2*a*d-6*b*c+b*d)/b^2/(tan(1/2*f*x+1/2*e)-1
)-2*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^3/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*f*x+1/2*e)/((a
-b)*(a+b))^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (157) = 314\).

Time = 22.06 (sec) , antiderivative size = 779, normalized size of antiderivative = 4.58 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\left [-\frac {2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a^{2} - b^{2}} \cos \left (f x + e\right )^{2} \log \left (\frac {2 \, a b \cos \left (f x + e\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (f x + e\right )^{2} + 2 \, a b \cos \left (f x + e\right ) + b^{2}}\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) + {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left ({\left (a^{2} b^{2} - b^{4}\right )} d^{3} + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left (a^{2} b^{3} - b^{5}\right )} f \cos \left (f x + e\right )^{2}}, \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (f x + e\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{2} + {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (\sin \left (f x + e\right ) + 1\right ) - {\left (6 \, {\left (a^{2} b^{2} - b^{4}\right )} c^{2} d - 6 \, {\left (a^{3} b - a b^{3}\right )} c d^{2} + {\left (2 \, a^{4} - a^{2} b^{2} - b^{4}\right )} d^{3}\right )} \cos \left (f x + e\right )^{2} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left ({\left (a^{2} b^{2} - b^{4}\right )} d^{3} + 2 \, {\left (3 \, {\left (a^{2} b^{2} - b^{4}\right )} c d^{2} - {\left (a^{3} b - a b^{3}\right )} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left (a^{2} b^{3} - b^{5}\right )} f \cos \left (f x + e\right )^{2}}\right ] \]

[In]

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

[Out]

[-1/4*(2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(a^2 - b^2)*cos(f*x + e)^2*log((2*a*b*cos(f*x
+ e) - (a^2 - 2*b^2)*cos(f*x + e)^2 - 2*sqrt(a^2 - b^2)*(b*cos(f*x + e) + a)*sin(f*x + e) + 2*a^2 - b^2)/(a^2
*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + b^2)) - (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2*a^4 - a
^2*b^2 - b^4)*d^3)*cos(f*x + e)^2*log(sin(f*x + e) + 1) + (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 +
(2*a^4 - a^2*b^2 - b^4)*d^3)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) - 2*((a^2*b^2 - b^4)*d^3 + 2*(3*(a^2*b^2 -
b^4)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^2*b^3 - b^5)*f*cos(f*x + e)^2), 1/4*(4*(b^3
*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cos(f*x + e) + a)
/((a^2 - b^2)*sin(f*x + e)))*cos(f*x + e)^2 + (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 + (2*a^4 - a^
2*b^2 - b^4)*d^3)*cos(f*x + e)^2*log(sin(f*x + e) + 1) - (6*(a^2*b^2 - b^4)*c^2*d - 6*(a^3*b - a*b^3)*c*d^2 +
(2*a^4 - a^2*b^2 - b^4)*d^3)*cos(f*x + e)^2*log(-sin(f*x + e) + 1) + 2*((a^2*b^2 - b^4)*d^3 + 2*(3*(a^2*b^2 -
b^4)*c*d^2 - (a^3*b - a*b^3)*d^3)*cos(f*x + e))*sin(f*x + e))/((a^2*b^3 - b^5)*f*cos(f*x + e)^2)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (157) = 314\).

Time = 0.40 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.99 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\frac {\frac {{\left (6 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 2 \, a^{2} d^{3} + b^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{3}} - \frac {{\left (6 \, b^{2} c^{2} d - 6 \, a b c d^{2} + 2 \, a^{2} d^{3} + b^{2} d^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{3}} - \frac {4 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{\sqrt {-a^{2} + b^{2}} b^{3}} - \frac {2 \, {\left (6 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, b c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} b^{2}}}{2 \, f} \]

[In]

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

[Out]

1/2*((6*b^2*c^2*d - 6*a*b*c*d^2 + 2*a^2*d^3 + b^2*d^3)*log(abs(tan(1/2*f*x + 1/2*e) + 1))/b^3 - (6*b^2*c^2*d -
6*a*b*c*d^2 + 2*a^2*d^3 + b^2*d^3)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/b^3 - 4*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^
2*b*c*d^2 - a^3*d^3)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(2*a - 2*b) + arctan((a*tan(1/2*f*x + 1/2*e) - b*tan
(1/2*f*x + 1/2*e))/sqrt(-a^2 + b^2)))/(sqrt(-a^2 + b^2)*b^3) - 2*(6*b*c*d^2*tan(1/2*f*x + 1/2*e)^3 - 2*a*d^3*t
an(1/2*f*x + 1/2*e)^3 - b*d^3*tan(1/2*f*x + 1/2*e)^3 - 6*b*c*d^2*tan(1/2*f*x + 1/2*e) + 2*a*d^3*tan(1/2*f*x +
1/2*e) - b*d^3*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^2*b^2))/f

Mupad [B] (verification not implemented)

Time = 21.04 (sec) , antiderivative size = 6730, normalized size of antiderivative = 39.59 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^3}{a+b \sec (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((tan(e/2 + (f*x)/2)*(b*d^3 - 2*a*d^3 + 6*b*c*d^2))/b^2 + (tan(e/2 + (f*x)/2)^3*(2*a*d^3 + b*d^3 - 6*b*c*d^2))
/b^2)/(f*(tan(e/2 + (f*x)/2)^4 - 2*tan(e/2 + (f*x)/2)^2 + 1)) - (atan(((((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*
b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6
+ 16*a^5*b^2*d^6 - 12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2
- 36*a^2*b^5*c*d^5 - 24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^
2*d^4 - 216*a^2*b^5*c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4
*d^2 - 240*a^4*b^3*c^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*
a^6*b*c*d^5))/b^4 + (((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3
+ 2*a^2*b^8*d^3 - 6*a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*
a*b^9*c*d^2 - 24*a*b^9*c^2*d))/b^6 - (8*tan(e/2 + (f*x)/2)*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6)*(b^2*(3*c^2*d +
d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^7)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^3)*(b^2*(3*c^2*d + d^
3/2) + a^2*d^3 - 3*a*b*c*d^2)*1i)/b^3 + (((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6
+ 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 - 12*b^7*c^2*
d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 - 24*a^2*b^5
*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*c^3*d^3 - 1
68*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c^2*d^4 - 15
2*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 - (((8*(4*b^1
0*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6*a^3*b^7*d^3
+ 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*b^9*c^2*d))/
b^6 + (8*tan(e/2 + (f*x)/2)*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2)
)/b^7)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^3)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2)*1
i)/b^3)/((16*(4*a^8*d^9 - 6*a^7*b*d^9 - 12*b^8*c^8*d - a^3*b^5*d^9 + 2*a^4*b^4*d^9 - 5*a^5*b^3*d^9 + 6*a^6*b^2
*d^9 + b^8*c^3*d^6 + 12*b^8*c^5*d^4 - 2*b^8*c^6*d^3 + 36*b^8*c^7*d^2 - 3*a*b^7*c^2*d^7 - 2*a*b^7*c^3*d^6 - 48*
a*b^7*c^4*d^5 - 12*a*b^7*c^5*d^4 - 178*a*b^7*c^6*d^3 + 12*a*b^7*c^7*d^2 + 3*a^2*b^6*c*d^8 - 6*a^3*b^5*c*d^8 +
27*a^4*b^4*c*d^8 - 36*a^5*b^3*c*d^8 + 48*a^6*b^2*c*d^8 + 6*a^2*b^6*c^2*d^7 + 77*a^2*b^6*c^3*d^6 + 66*a^2*b^6*c
^4*d^5 + 384*a^2*b^6*c^5*d^4 + 104*a^2*b^6*c^6*d^3 - 48*a^2*b^6*c^7*d^2 - 63*a^3*b^5*c^2*d^7 - 112*a^3*b^5*c^3
*d^6 - 474*a^3*b^5*c^4*d^5 - 324*a^3*b^5*c^5*d^4 + 76*a^3*b^5*c^6*d^3 + 90*a^4*b^4*c^2*d^7 + 364*a^4*b^4*c^3*d
^6 + 432*a^4*b^4*c^4*d^5 - 60*a^4*b^4*c^5*d^4 - 174*a^5*b^3*c^2*d^7 - 324*a^5*b^3*c^3*d^6 + 24*a^5*b^3*c^4*d^5
+ 144*a^6*b^2*c^2*d^7 - 4*a^6*b^2*c^3*d^6 + 12*a*b^7*c^8*d - 36*a^7*b*c*d^8))/b^6 - (((8*tan(e/2 + (f*x)/2)*(
8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 -
16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 - 12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*
a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 - 24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 -
96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 +
96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b
^6*c^5*d - 48*a^6*b*c*d^5))/b^4 + (((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d +
4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6*a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b
^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*b^9*c^2*d))/b^6 - (8*tan(e/2 + (f*x)/2)*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6)*(b
^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^7)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^3)*(b^2
*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^3 + (((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 +
4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 -
12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 -
24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*
c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c
^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 -
(((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6*
a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*b
^9*c^2*d))/b^6 + (8*tan(e/2 + (f*x)/2)*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3
*a*b*c*d^2))/b^7)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2))/b^3)*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a
*b*c*d^2))/b^3))*(b^2*(3*c^2*d + d^3/2) + a^2*d^3 - 3*a*b*c*d^2)*2i)/(b^3*f) - (atan(((((a + b)*(a - b))^(1/2)
*(a*d - b*c)^3*((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*
d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 - 12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*
a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 - 24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5
- 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a
^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^
5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 + (((a + b)*(a - b))^(1/2)*(a*d - b*c)^
3*((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6
*a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*
b^9*c^2*d))/b^6 - (8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^
6))/(b^4*(b^5 - a^2*b^3))))/(b^5 - a^2*b^3))*1i)/(b^5 - a^2*b^3) + (((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*
tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6
+ 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 - 12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*
a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 - 24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5
+ 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 29
6*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*
a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 - (((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(4*b^10*c^3 +
2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6*a^3*b^7*d^3 + 4*a^
4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*b^9*c^2*d))/b^6 + (
8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6))/(b^4*(b^5 - a^2
*b^3))))/(b^5 - a^2*b^3))*1i)/(b^5 - a^2*b^3))/((16*(4*a^8*d^9 - 6*a^7*b*d^9 - 12*b^8*c^8*d - a^3*b^5*d^9 + 2*
a^4*b^4*d^9 - 5*a^5*b^3*d^9 + 6*a^6*b^2*d^9 + b^8*c^3*d^6 + 12*b^8*c^5*d^4 - 2*b^8*c^6*d^3 + 36*b^8*c^7*d^2 -
3*a*b^7*c^2*d^7 - 2*a*b^7*c^3*d^6 - 48*a*b^7*c^4*d^5 - 12*a*b^7*c^5*d^4 - 178*a*b^7*c^6*d^3 + 12*a*b^7*c^7*d^2
+ 3*a^2*b^6*c*d^8 - 6*a^3*b^5*c*d^8 + 27*a^4*b^4*c*d^8 - 36*a^5*b^3*c*d^8 + 48*a^6*b^2*c*d^8 + 6*a^2*b^6*c^2*
d^7 + 77*a^2*b^6*c^3*d^6 + 66*a^2*b^6*c^4*d^5 + 384*a^2*b^6*c^5*d^4 + 104*a^2*b^6*c^6*d^3 - 48*a^2*b^6*c^7*d^2
- 63*a^3*b^5*c^2*d^7 - 112*a^3*b^5*c^3*d^6 - 474*a^3*b^5*c^4*d^5 - 324*a^3*b^5*c^5*d^4 + 76*a^3*b^5*c^6*d^3 +
90*a^4*b^4*c^2*d^7 + 364*a^4*b^4*c^3*d^6 + 432*a^4*b^4*c^4*d^5 - 60*a^4*b^4*c^5*d^4 - 174*a^5*b^3*c^2*d^7 - 3
24*a^5*b^3*c^3*d^6 + 24*a^5*b^3*c^4*d^5 + 144*a^6*b^2*c^2*d^7 - 4*a^6*b^2*c^3*d^6 + 12*a*b^7*c^8*d - 36*a^7*b*
c*d^8))/b^6 - (((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 +
4*a*b^6*c^6 + 3*a*b^6*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 -
12*b^7*c^2*d^4 - 36*b^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5
- 24*a^2*b^5*c^5*d + 60*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5
*c^3*d^3 - 168*a^2*b^5*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*
c^2*d^4 - 152*a^4*b^3*c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 +
(((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2
*d + 4*a^2*b^8*c^3 + 2*a^2*b^8*d^3 - 6*a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*
a^3*b^7*c*d^2 - 12*a*b^9*c*d^2 - 24*a*b^9*c^2*d))/b^6 - (8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b
*c)^3*(8*a*b^8 - 16*a^2*b^7 + 8*a^3*b^6))/(b^4*(b^5 - a^2*b^3))))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3) + (((a + b
)*(a - b))^(1/2)*(a*d - b*c)^3*((8*tan(e/2 + (f*x)/2)*(8*a^7*d^6 - 4*b^7*c^6 - b^7*d^6 + 4*a*b^6*c^6 + 3*a*b^6
*d^6 - 16*a^6*b*d^6 - 7*a^2*b^5*d^6 + 13*a^3*b^4*d^6 - 16*a^4*b^3*d^6 + 16*a^5*b^2*d^6 - 12*b^7*c^2*d^4 - 36*b
^7*c^4*d^2 + 36*a*b^6*c^2*d^4 + 72*a*b^6*c^3*d^3 + 108*a*b^6*c^4*d^2 - 36*a^2*b^5*c*d^5 - 24*a^2*b^5*c^5*d + 6
0*a^3*b^4*c*d^5 - 84*a^4*b^3*c*d^5 + 96*a^5*b^2*c*d^5 - 96*a^2*b^5*c^2*d^4 - 216*a^2*b^5*c^3*d^3 - 168*a^2*b^5
*c^4*d^2 + 192*a^3*b^4*c^2*d^4 + 296*a^3*b^4*c^3*d^3 + 96*a^3*b^4*c^4*d^2 - 240*a^4*b^3*c^2*d^4 - 152*a^4*b^3*
c^3*d^3 + 120*a^5*b^2*c^2*d^4 + 12*a*b^6*c*d^5 + 24*a*b^6*c^5*d - 48*a^6*b*c*d^5))/b^4 - (((a + b)*(a - b))^(1
/2)*(a*d - b*c)^3*((8*(4*b^10*c^3 + 2*b^10*d^3 - 8*a*b^9*c^3 - 2*a*b^9*d^3 + 12*b^10*c^2*d + 4*a^2*b^8*c^3 + 2
*a^2*b^8*d^3 - 6*a^3*b^7*d^3 + 4*a^4*b^6*d^3 + 24*a^2*b^8*c*d^2 + 12*a^2*b^8*c^2*d - 12*a^3*b^7*c*d^2 - 12*a*b
^9*c*d^2 - 24*a*b^9*c^2*d))/b^6 + (8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^3*(8*a*b^8 - 16*a^
2*b^7 + 8*a^3*b^6))/(b^4*(b^5 - a^2*b^3))))/(b^5 - a^2*b^3)))/(b^5 - a^2*b^3)))*((a + b)*(a - b))^(1/2)*(a*d -
b*c)^3*2i)/(f*(b^5 - a^2*b^3))