3.6.60 \(\int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx\) [560]

Optimal. Leaf size=235 \[ \frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}+\frac {5 a \left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d} \]

[Out]

5/8*(8*a^4+12*a^2*b^2+3*b^4)*arcsinh(tan(d*x+c))*sec(d*x+c)/b^6/d/(sec(d*x+c)^2)^(1/2)+5*a*(a^2+b^2)^(3/2)*arc
tanh((b-a*tan(d*x+c))/(a^2+b^2)^(1/2)/(sec(d*x+c)^2)^(1/2))*sec(d*x+c)/b^6/d/(sec(d*x+c)^2)^(1/2)-5/12*sec(d*x
+c)^3*(4*a-3*b*tan(d*x+c))/b^3/d-sec(d*x+c)^5/b/d/(a+b*tan(d*x+c))-5/8*sec(d*x+c)*(8*a*(a^2+b^2)-b*(4*a^2+3*b^
2)*tan(d*x+c))/b^5/d

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Rubi [A]
time = 0.19, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3593, 747, 829, 858, 221, 739, 212} \begin {gather*} \frac {5 a \left (a^2+b^2\right )^{3/2} \sec (c+d x) \tanh ^{-1}\left (\frac {b-a \tan (c+d x)}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right )}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sec (c+d x) \sinh ^{-1}(\tan (c+d x))}{8 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^7/(a + b*Tan[c + d*x])^2,x]

[Out]

(5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*ArcSinh[Tan[c + d*x]]*Sec[c + d*x])/(8*b^6*d*Sqrt[Sec[c + d*x]^2]) + (5*a*(a^2
+ b^2)^(3/2)*ArcTanh[(b - a*Tan[c + d*x])/(Sqrt[a^2 + b^2]*Sqrt[Sec[c + d*x]^2])]*Sec[c + d*x])/(b^6*d*Sqrt[S
ec[c + d*x]^2]) - (5*Sec[c + d*x]^3*(4*a - 3*b*Tan[c + d*x]))/(12*b^3*d) - Sec[c + d*x]^5/(b*d*(a + b*Tan[c +
d*x])) - (5*Sec[c + d*x]*(8*a*(a^2 + b^2) - b*(4*a^2 + 3*b^2)*Tan[c + d*x]))/(8*b^5*d)

Rule 212

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

Rule 221

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

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 747

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 1))), x] - Dist[2*c*(p/(e*(m + 1))), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 829

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m +
2*p + 2))), x] + Dist[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), Int[(d + e*x)^m*(a + c*x^2)^(p - 1)*Simp[f*a
*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f*d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))
*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !R
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*
m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 3593

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[d^(2*
IntPart[m/2])*((d*Sec[e + f*x])^(2*FracPart[m/2])/(b*f*(Sec[e + f*x]^2)^FracPart[m/2])), Subst[Int[(a + x)^n*(
1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && NeQ[a^2 + b^2, 0] &&
!IntegerQ[m/2]

Rubi steps

\begin {align*} \int \frac {\sec ^7(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=\frac {\sec (c+d x) \text {Subst}\left (\int \frac {\left (1+\frac {x^2}{b^2}\right )^{5/2}}{(a+x)^2} \, dx,x,b \tan (c+d x)\right )}{b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {x \left (1+\frac {x^2}{b^2}\right )^{3/2}}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}+\frac {(5 \sec (c+d x)) \text {Subst}\left (\int \frac {\left (-\frac {a}{b^2}+\frac {\left (4 a^2+3 b^2\right ) x}{b^4}\right ) \sqrt {1+\frac {x^2}{b^2}}}{a+x} \, dx,x,b \tan (c+d x)\right )}{4 b d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {(5 b \sec (c+d x)) \text {Subst}\left (\int \frac {-\frac {a \left (4 a^2+5 b^2\right )}{b^6}+\frac {\left (8 a^4+12 a^2 b^2+3 b^4\right ) x}{b^8}}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 d \sqrt {\sec ^2(c+d x)}}\\ &=-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}-\frac {\left (5 a \left (a^2+b^2\right )^2 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{(a+x) \sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{b^7 d \sqrt {\sec ^2(c+d x)}}+\frac {\left (5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{b^2}}} \, dx,x,b \tan (c+d x)\right )}{8 b^7 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}+\frac {\left (5 a \left (a^2+b^2\right )^2 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a^2}{b^2}-x^2} \, dx,x,\frac {1-\frac {a \tan (c+d x)}{b}}{\sqrt {\sec ^2(c+d x)}}\right )}{b^7 d \sqrt {\sec ^2(c+d x)}}\\ &=\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \sinh ^{-1}(\tan (c+d x)) \sec (c+d x)}{8 b^6 d \sqrt {\sec ^2(c+d x)}}+\frac {5 a \left (a^2+b^2\right )^{3/2} \tanh ^{-1}\left (\frac {b \left (1-\frac {a \tan (c+d x)}{b}\right )}{\sqrt {a^2+b^2} \sqrt {\sec ^2(c+d x)}}\right ) \sec (c+d x)}{b^6 d \sqrt {\sec ^2(c+d x)}}-\frac {5 \sec ^3(c+d x) (4 a-3 b \tan (c+d x))}{12 b^3 d}-\frac {\sec ^5(c+d x)}{b d (a+b \tan (c+d x))}-\frac {5 \sec (c+d x) \left (8 a \left (a^2+b^2\right )-b \left (4 a^2+3 b^2\right ) \tan (c+d x)\right )}{8 b^5 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 6.23, size = 1152, normalized size = 4.90 \begin {gather*} -\frac {(a-i b)^2 (a+i b)^2 \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))}{b^5 d (a+b \tan (c+d x))^2}-\frac {a \left (12 a^2+13 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d (a+b \tan (c+d x))^2}+\frac {10 i a (a+i b) (i a+b) \sqrt {a^2+b^2} \tanh ^{-1}\left (\frac {\sqrt {a^2+b^2} \left (-b \cos \left (\frac {1}{2} (c+d x)\right )+a \sin \left (\frac {1}{2} (c+d x)\right )\right )}{a^2 \cos \left (\frac {1}{2} (c+d x)\right )+b^2 \cos \left (\frac {1}{2} (c+d x)\right )}\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{b^6 d (a+b \tan (c+d x))^2}-\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{8 b^6 d (a+b \tan (c+d x))^2}+\frac {5 \left (8 a^4+12 a^2 b^2+3 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{8 b^6 d (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{16 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a+b \tan (c+d x))^2}+\frac {\left (36 a^2-8 a b+21 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{48 b^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}-\frac {a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+b \tan (c+d x))^2}-\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{16 b^2 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4 (a+b \tan (c+d x))^2}+\frac {a \sec ^2(c+d x) \sin \left (\frac {1}{2} (c+d x)\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3 (a+b \tan (c+d x))^2}+\frac {\left (-36 a^2-8 a b-21 b^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2}{48 b^4 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2 (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) \left (-12 a^3 \sin \left (\frac {1}{2} (c+d x)\right )-13 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2}+\frac {\sec ^2(c+d x) \left (12 a^3 \sin \left (\frac {1}{2} (c+d x)\right )+13 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \cos (c+d x)+b \sin (c+d x))^2}{3 b^5 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a+b \tan (c+d x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^7/(a + b*Tan[c + d*x])^2,x]

[Out]

-(((a - I*b)^2*(a + I*b)^2*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x]))/(b^5*d*(a + b*Tan[c + d*x])^2)) -
(a*(12*a^2 + 13*b^2)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^5*d*(a + b*Tan[c + d*x])^2) + (
(10*I)*a*(a + I*b)*(I*a + b)*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a^2 + b^2]*(-(b*Cos[(c + d*x)/2]) + a*Sin[(c + d*x)
/2]))/(a^2*Cos[(c + d*x)/2] + b^2*Cos[(c + d*x)/2])]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(b^6*
d*(a + b*Tan[c + d*x])^2) - (5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d
*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(8*b^6*d*(a + b*Tan[c + d*x])^2) + (5*(8*a^4 + 12*a^2*b^2 + 3*b^4)*
Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(8*b^6*d*(a + b*T
an[c + d*x])^2) + (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(16*b^2*d*(Cos[(c + d*x)/2] - Sin[(c +
d*x)/2])^4*(a + b*Tan[c + d*x])^2) + ((36*a^2 - 8*a*b + 21*b^2)*Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x
])^2)/(48*b^4*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) - (a*Sec[c + d*x]^2*Sin[(c + d
*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3*(a + b*Tan[c + d*
x])^2) - (Sec[c + d*x]^2*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(16*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^
4*(a + b*Tan[c + d*x])^2) + (a*Sec[c + d*x]^2*Sin[(c + d*x)/2]*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^3*d*(
Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3*(a + b*Tan[c + d*x])^2) + ((-36*a^2 - 8*a*b - 21*b^2)*Sec[c + d*x]^2*(a
*Cos[c + d*x] + b*Sin[c + d*x])^2)/(48*b^4*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2*(a + b*Tan[c + d*x])^2) +
(Sec[c + d*x]^2*(-12*a^3*Sin[(c + d*x)/2] - 13*a*b^2*Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(
3*b^5*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(a + b*Tan[c + d*x])^2) + (Sec[c + d*x]^2*(12*a^3*Sin[(c + d*x)/
2] + 13*a*b^2*Sin[(c + d*x)/2])*(a*Cos[c + d*x] + b*Sin[c + d*x])^2)/(3*b^5*d*(Cos[(c + d*x)/2] + Sin[(c + d*x
)/2])*(a + b*Tan[c + d*x])^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(219)=438\).
time = 0.52, size = 454, normalized size = 1.93 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^7/(a+b*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(2/b^6*(((a^4+2*a^2*b^2+b^4)*b^2/a*tan(1/2*d*x+1/2*c)+b*(a^4+2*a^2*b^2+b^4))/(a*tan(1/2*d*x+1/2*c)^2-2*b*t
an(1/2*d*x+1/2*c)-a)-5*a*(a^4+2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2
)^(1/2)))+1/4/b^2/(tan(1/2*d*x+1/2*c)-1)^4-1/6*(-4*a-3*b)/b^3/(tan(1/2*d*x+1/2*c)-1)^3-1/8*(-12*a^2-8*a*b-11*b
^2)/b^4/(tan(1/2*d*x+1/2*c)-1)^2+1/8/b^6*(-40*a^4-60*a^2*b^2-15*b^4)*ln(tan(1/2*d*x+1/2*c)-1)-1/8*(-32*a^3-12*
a^2*b-40*a*b^2-9*b^3)/b^5/(tan(1/2*d*x+1/2*c)-1)-1/4/b^2/(tan(1/2*d*x+1/2*c)+1)^4-1/6*(4*a-3*b)/b^3/(tan(1/2*d
*x+1/2*c)+1)^3-1/8*(12*a^2-8*a*b+11*b^2)/b^4/(tan(1/2*d*x+1/2*c)+1)^2+1/8/b^6*(40*a^4+60*a^2*b^2+15*b^4)*ln(ta
n(1/2*d*x+1/2*c)+1)-1/8*(32*a^3-12*a^2*b+40*a*b^2-9*b^3)/b^5/(tan(1/2*d*x+1/2*c)+1))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (220) = 440\).
time = 0.50, size = 827, normalized size = 3.52 \begin {gather*} -\frac {\frac {2 \, {\left (120 \, a^{5} + 160 \, a^{3} b^{2} + 24 \, a b^{4} + \frac {{\left (180 \, a^{4} b + 245 \, a^{2} b^{3} + 24 \, b^{5}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {10 \, {\left (48 \, a^{5} + 68 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (300 \, a^{4} b + 385 \, a^{2} b^{3} + 48 \, b^{5}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, {\left (72 \, a^{5} + 100 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {48 \, {\left (15 \, a^{4} b + 20 \, a^{2} b^{3} + 3 \, b^{5}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {30 \, {\left (16 \, a^{5} + 20 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {6 \, {\left (60 \, a^{4} b + 85 \, a^{2} b^{3} + 16 \, b^{5}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {30 \, {\left (4 \, a^{5} + 4 \, a^{3} b^{2} - a b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {3 \, {\left (20 \, a^{4} b + 25 \, a^{2} b^{3} + 8 \, b^{5}\right )} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}}{a^{2} b^{5} + \frac {2 \, a b^{6} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {5 \, a^{2} b^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {8 \, a b^{6} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {10 \, a^{2} b^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {12 \, a b^{6} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {10 \, a^{2} b^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {8 \, a b^{6} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {5 \, a^{2} b^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {2 \, a b^{6} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {a^{2} b^{5} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {120 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} a \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} - \frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{6}} + \frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{6}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/24*(2*(120*a^5 + 160*a^3*b^2 + 24*a*b^4 + (180*a^4*b + 245*a^2*b^3 + 24*b^5)*sin(d*x + c)/(cos(d*x + c) + 1
) - 10*(48*a^5 + 68*a^3*b^2 + 15*a*b^4)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(300*a^4*b + 385*a^2*b^3 + 48*
b^5)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*(72*a^5 + 100*a^3*b^2 + 15*a*b^4)*sin(d*x + c)^4/(cos(d*x + c) +
1)^4 + 48*(15*a^4*b + 20*a^2*b^3 + 3*b^5)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 30*(16*a^5 + 20*a^3*b^2 + 3*a
*b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 6*(60*a^4*b + 85*a^2*b^3 + 16*b^5)*sin(d*x + c)^7/(cos(d*x + c) +
1)^7 + 30*(4*a^5 + 4*a^3*b^2 - a*b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 3*(20*a^4*b + 25*a^2*b^3 + 8*b^5)*
sin(d*x + c)^9/(cos(d*x + c) + 1)^9)/(a^2*b^5 + 2*a*b^6*sin(d*x + c)/(cos(d*x + c) + 1) - 5*a^2*b^5*sin(d*x +
c)^2/(cos(d*x + c) + 1)^2 - 8*a*b^6*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 10*a^2*b^5*sin(d*x + c)^4/(cos(d*x +
c) + 1)^4 + 12*a*b^6*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 10*a^2*b^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 8
*a*b^6*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 5*a^2*b^5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 2*a*b^6*sin(d*x +
c)^9/(cos(d*x + c) + 1)^9 - a^2*b^5*sin(d*x + c)^10/(cos(d*x + c) + 1)^10) - 120*(a^4 + 2*a^2*b^2 + b^4)*a*lo
g((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - sqrt(a^2
+ b^2)))/(sqrt(a^2 + b^2)*b^6) - 15*(8*a^4 + 12*a^2*b^2 + 3*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/b^6
+ 15*(8*a^4 + 12*a^2*b^2 + 3*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^6)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (220) = 440\).
time = 0.59, size = 472, normalized size = 2.01 \begin {gather*} \frac {12 \, b^{5} - 30 \, {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (4 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + 120 \, {\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{5} + {\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) + 15 \, {\left ({\left (8 \, a^{5} + 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left ({\left (8 \, a^{5} + 12 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} + {\left (8 \, a^{4} b + 12 \, a^{2} b^{3} + 3 \, b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + 3 \, {\left (4 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (a b^{6} d \cos \left (d x + c\right )^{5} + b^{7} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/48*(12*b^5 - 30*(8*a^4*b + 12*a^2*b^3 + 3*b^5)*cos(d*x + c)^4 + 10*(4*a^2*b^3 + 3*b^5)*cos(d*x + c)^2 + 120*
((a^4 + a^2*b^2)*cos(d*x + c)^5 + (a^3*b + a*b^3)*cos(d*x + c)^4*sin(d*x + c))*sqrt(a^2 + b^2)*log((2*a*b*cos(
d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 - 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d
*x + c)))/(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) + 15*((8*a^5 + 12*a^3*b^2 + 3*
a*b^4)*cos(d*x + c)^5 + (8*a^4*b + 12*a^2*b^3 + 3*b^5)*cos(d*x + c)^4*sin(d*x + c))*log(sin(d*x + c) + 1) - 15
*((8*a^5 + 12*a^3*b^2 + 3*a*b^4)*cos(d*x + c)^5 + (8*a^4*b + 12*a^2*b^3 + 3*b^5)*cos(d*x + c)^4*sin(d*x + c))*
log(-sin(d*x + c) + 1) - 10*(2*a*b^4*cos(d*x + c) + 3*(4*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(a*b
^6*d*cos(d*x + c)^5 + b^7*d*cos(d*x + c)^4*sin(d*x + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{7}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**7/(a+b*tan(d*x+c))**2,x)

[Out]

Integral(sec(c + d*x)**7/(a + b*tan(c + d*x))**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (220) = 440\).
time = 0.65, size = 530, normalized size = 2.26 \begin {gather*} \frac {\frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (8 \, a^{4} + 12 \, a^{2} b^{2} + 3 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {120 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} + \frac {48 \, {\left (a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a\right )} a b^{5}} + \frac {2 \, {\left (36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 96 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 144 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 336 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 288 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 304 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 27 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, a^{3} - 112 \, a b^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4} b^{5}}}{24 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^7/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/24*(15*(8*a^4 + 12*a^2*b^2 + 3*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(8*a^4 + 12*a^2*b^2 + 3*b^4)
*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^6 + 120*(a^5 + 2*a^3*b^2 + a*b^4)*log(abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b
- 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6) + 48*(a^4
*b*tan(1/2*d*x + 1/2*c) + 2*a^2*b^3*tan(1/2*d*x + 1/2*c) + b^5*tan(1/2*d*x + 1/2*c) + a^5 + 2*a^3*b^2 + a*b^4)
/((a*tan(1/2*d*x + 1/2*c)^2 - 2*b*tan(1/2*d*x + 1/2*c) - a)*a*b^5) + 2*(36*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 27*b
^3*tan(1/2*d*x + 1/2*c)^7 + 96*a^3*tan(1/2*d*x + 1/2*c)^6 + 144*a*b^2*tan(1/2*d*x + 1/2*c)^6 - 36*a^2*b*tan(1/
2*d*x + 1/2*c)^5 - 3*b^3*tan(1/2*d*x + 1/2*c)^5 - 288*a^3*tan(1/2*d*x + 1/2*c)^4 - 336*a*b^2*tan(1/2*d*x + 1/2
*c)^4 - 36*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 3*b^3*tan(1/2*d*x + 1/2*c)^3 + 288*a^3*tan(1/2*d*x + 1/2*c)^2 + 304*
a*b^2*tan(1/2*d*x + 1/2*c)^2 + 36*a^2*b*tan(1/2*d*x + 1/2*c) + 27*b^3*tan(1/2*d*x + 1/2*c) - 96*a^3 - 112*a*b^
2)/((tan(1/2*d*x + 1/2*c)^2 - 1)^4*b^5))/d

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Mupad [B]
time = 6.54, size = 2500, normalized size = 10.64 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^7*(a + b*tan(c + d*x))^2),x)

[Out]

-((9*a*b^5)/64 + (15*a^5*b)/8 + (b^6*sin(c + d*x))/8 + (115*a^3*b^3)/48 + (3*b^6*sin(3*c + 3*d*x))/16 + (b^6*s
in(5*c + 5*d*x))/16 + (a^6*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*25i)/4 + (5*a*b^5*cos
(2*c + 2*d*x))/8 + (5*a^5*b*cos(2*c + 2*d*x))/2 + (5*a*b^5*cos(3*c + 3*d*x))/16 + (25*a^5*b*cos(3*c + 3*d*x))/
16 + (15*a*b^5*cos(4*c + 4*d*x))/64 + (5*a^5*b*cos(4*c + 4*d*x))/8 + (a*b^5*cos(5*c + 5*d*x))/16 + (5*a^5*b*co
s(5*c + 5*d*x))/16 + (25*a^3*b^3*cos(c + d*x))/6 + (5*a^2*b^4*sin(c + d*x))/6 + (5*a^4*b^2*sin(c + d*x))/8 + (
a^6*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*25i)/8 + (a^6*atan((sin(c/2 + (d*x)/2)*1
i)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x)*5i)/8 + (10*a^3*b^3*cos(2*c + 2*d*x))/3 + (25*a^3*b^3*cos(3*c + 3*d*x)
)/12 + (15*a^3*b^3*cos(4*c + 4*d*x))/16 + (5*a^3*b^3*cos(5*c + 5*d*x))/12 + (95*a^2*b^4*sin(2*c + 2*d*x))/96 +
(5*a^4*b^2*sin(2*c + 2*d*x))/8 + (5*a^2*b^4*sin(3*c + 3*d*x))/4 + (15*a^4*b^2*sin(3*c + 3*d*x))/16 + (25*a^2*
b^4*sin(4*c + 4*d*x))/64 + (5*a^4*b^2*sin(4*c + 4*d*x))/16 + (5*a^2*b^4*sin(5*c + 5*d*x))/12 + (5*a^4*b^2*sin(
5*c + 5*d*x))/16 + (5*a*b^5*cos(c + d*x))/8 + (25*a^5*b*cos(c + d*x))/8 + (a*b^5*atan((sin(c/2 + (d*x)/2)*1i)/
cos(c/2 + (d*x)/2))*sin(3*c + 3*d*x)*45i)/64 + (a^5*b*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*sin(3*c
+ 3*d*x)*15i)/8 + (a*b^5*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*sin(5*c + 5*d*x)*15i)/64 + (a^5*b*a
tan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*sin(5*c + 5*d*x)*5i)/8 + (a^3*b^3*sin(c + d*x)*atan((sin(c/2 +
(d*x)/2)*1i)/cos(c/2 + (d*x)/2))*15i)/8 + (a^2*b^4*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c +
3*d*x)*75i)/64 + (a^4*b^2*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(3*c + 3*d*x)*75i)/16 + (a^2*b^
4*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x)*15i)/64 + (a^4*b^2*atan((sin(c/2 + (d*x)/2
)*1i)/cos(c/2 + (d*x)/2))*cos(5*c + 5*d*x)*15i)/16 + (a^3*b^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))
*sin(3*c + 3*d*x)*45i)/16 + (a^3*b^3*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*sin(5*c + 5*d*x)*15i)/16
+ (25*a^3*cos(c + d*x)*atanh((a^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + 2*b^2*sin(c/
2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b*cos(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4
*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)/2) + a*b^4*cos(c/2 + (d*x)/2) + 2*a^4*b*sin(c/2 +
(d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*sin(c/2 + (d*x)/2)))*((a^2 + b^2)^3)^(1/2))/4 + (a*b^5*si
n(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*15i)/32 + (a^5*b*sin(c + d*x)*atan((sin(c/2 + (d*x
)/2)*1i)/cos(c/2 + (d*x)/2))*5i)/4 + (25*a^3*atanh((a^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)
^(1/2) + 2*b^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b*cos(c/2 + (d*x)/2)*(a^6 + b^
6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)/2) + a*b^4*cos(c/2 + (d*x)/2
) + 2*a^4*b*sin(c/2 + (d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*sin(c/2 + (d*x)/2)))*cos(3*c + 3*d*x
)*((a^2 + b^2)^3)^(1/2))/8 + (5*a^3*atanh((a^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) +
2*b^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b*cos(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2
*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)/2) + a*b^4*cos(c/2 + (d*x)/2) + 2*a^4
*b*sin(c/2 + (d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*sin(c/2 + (d*x)/2)))*cos(5*c + 5*d*x)*((a^2 +
b^2)^3)^(1/2))/8 + (a^2*b^4*cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*75i)/32 + (a^4*b^2*
cos(c + d*x)*atan((sin(c/2 + (d*x)/2)*1i)/cos(c/2 + (d*x)/2))*75i)/8 + (15*a^2*b*atanh((a^2*sin(c/2 + (d*x)/2)
*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + 2*b^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2
) + a*b*cos(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2
+ (d*x)/2) + a*b^4*cos(c/2 + (d*x)/2) + 2*a^4*b*sin(c/2 + (d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*
sin(c/2 + (d*x)/2)))*sin(3*c + 3*d*x)*((a^2 + b^2)^3)^(1/2))/8 + (5*a^2*b*atanh((a^2*sin(c/2 + (d*x)/2)*(a^6 +
b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + 2*b^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b
*cos(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)
/2) + a*b^4*cos(c/2 + (d*x)/2) + 2*a^4*b*sin(c/2 + (d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*sin(c/2
+ (d*x)/2)))*sin(5*c + 5*d*x)*((a^2 + b^2)^3)^(1/2))/8 + (5*a^2*b*atanh((a^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 +
3*a^2*b^4 + 3*a^4*b^2)^(1/2) + 2*b^2*sin(c/2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2) + a*b*cos(c/
2 + (d*x)/2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^(1/2))/(a^5*cos(c/2 + (d*x)/2) + 2*b^5*sin(c/2 + (d*x)/2) + a
*b^4*cos(c/2 + (d*x)/2) + 2*a^4*b*sin(c/2 + (d*x)/2) + 2*a^3*b^2*cos(c/2 + (d*x)/2) + 4*a^2*b^3*sin(c/2 + (d*x
)/2)))*sin(c + d*x)*((a^2 + b^2)^3)^(1/2))/4)/(...

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