3.548 \(\int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=172 \[ -\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}} \]

[Out]

ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a
+ I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^2*(a^
2 + 4*b^2))/(3*b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])

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Rubi [A]  time = 0.367891, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3565, 3628, 3539, 3537, 63, 208} \[ -\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}-\frac{2 a^2 \tan (c+d x)}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]/((a - I*b)^(5/2)*d) + ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a
+ I*b]]/((a + I*b)^(5/2)*d) - (2*a^2*Tan[c + d*x])/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - (4*a^2*(a^
2 + 4*b^2))/(3*b^2*(a^2 + b^2)^2*d*Sqrt[a + b*Tan[c + d*x]])

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3628

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[((A*b^2 - a*b*B + a^2*C)*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)*(a^2 + b^2
)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +
f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2
+ b^2, 0]

Rule 3539

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

Rule 3537

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\tan ^3(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac{2 \int \frac{a^2-\frac{3}{2} a b \tan (c+d x)+\frac{1}{2} \left (2 a^2+3 b^2\right ) \tan ^2(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 b \left (a^2+b^2\right )}\\ &=-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{2 \int \frac{-3 a b^2-\frac{3}{2} b \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 b \left (a^2+b^2\right )^2}\\ &=-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{i \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac{i \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}\\ &=-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}-\frac{\operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}\\ &=-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{i \operatorname{Subst}\left (\int \frac{1}{-1-\frac{i a}{b}+\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{(a-i b)^2 b d}+\frac{i \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i a}{b}-\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{(a+i b)^2 b d}\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{(a-i b)^{5/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{5/2} d}-\frac{2 a^2 \tan (c+d x)}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{4 a^2 \left (a^2+4 b^2\right )}{3 b^2 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}

Mathematica [C]  time = 1.32415, size = 220, normalized size = 1.28 \[ \frac{-\frac{a \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right )}{b+i a}+\frac{a \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right )}{-b+i a}+\frac{3 (a+b \tan (c+d x)) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a-i b}\right )}{b+i a}+\frac{3 i (a+b \tan (c+d x)) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{a+b \tan (c+d x)}{a+i b}\right )}{a+i b}-\frac{4 a}{b}-6 \tan (c+d x)}{3 b d (a+b \tan (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^3/(a + b*Tan[c + d*x])^(5/2),x]

[Out]

((-4*a)/b - (a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)])/(I*a + b) + (a*Hypergeometric
2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a + I*b)])/(I*a - b) - 6*Tan[c + d*x] + (3*Hypergeometric2F1[-1/2, 1,
1/2, (a + b*Tan[c + d*x])/(a - I*b)]*(a + b*Tan[c + d*x]))/(I*a + b) + ((3*I)*Hypergeometric2F1[-1/2, 1, 1/2,
(a + b*Tan[c + d*x])/(a + I*b)]*(a + b*Tan[c + d*x]))/(a + I*b))/(3*b*d*(a + b*Tan[c + d*x])^(3/2))

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Maple [B]  time = 0.045, size = 2165, normalized size = 12.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x)

[Out]

5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))
^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a-6/d*a^2/(a^2+b^2)^2/(a+b*tan(d*x+c))^(1/2)-1/d/(a^2+b^2)^3/(2*(a^2+b^
2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1
/2))*a^4+1/2/d/(a^2+b^2)^3*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1
/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+1/d/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2
)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^4-2/d/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/
2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a
^3+1/d/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^
(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^5-3/4/d/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2
*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+3/4/d/(a^2+b^2)^(7/2)*ln(b*tan(d*x
+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+
2/d/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/
2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3-1/d/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*
x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^5+1/d*b^4/(a^2+b^2)^3/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2
))-1/d*b^4/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^
(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))-2/d/b^2*a^4/(a^2+b^2)^2/(a+b*tan(d*x+c))^(1/2)-1/4/d*b^4/(a^2+b^2)^(7/2)
*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*
a)^(1/2)+1/4/d*b^4/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2
+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-1/2/d/(a^2+b^2)^3*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+
b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3+2/3/d/b^2*a^3/(a^2+b^2)/(a+b*tan(d*x+
c))^(3/2)-1/2/d*b^2/(a^2+b^2)^3*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^
2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/2/d*b^2/(a^2+b^2)^3*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2
*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d*b^2/(a^2+b^2)^(5/2)/(2*(a^2+b^2)
^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2
))*a-1/2/d*b^2/(a^2+b^2)^(7/2)*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2
)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+1/2/d*b^2/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*
(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^2+2/d*b^2/(a^2+b^2)^(5/2)/(2*(a
^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*
a)^(1/2))*a-6/d*b^2/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2
)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a^3-5/d*b^4/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*a
rctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*a+6/d*b^2/(a^2+b
^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^
2+b^2)^(1/2)-2*a)^(1/2))*a^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 6.91104, size = 22680, normalized size = 131.86 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-1/12*(12*sqrt(2)*((a^18*b^2 + a^16*b^4 - 20*a^14*b^6 - 84*a^12*b^8 - 154*a^10*b^10 - 154*a^8*b^12 - 84*a^6*b^
14 - 20*a^4*b^16 + a^2*b^18 + b^20)*d^5*cos(d*x + c)^4 + 2*(3*a^16*b^4 + 20*a^14*b^6 + 56*a^12*b^8 + 84*a^10*b
^10 + 70*a^8*b^12 + 28*a^6*b^14 - 4*a^2*b^18 - b^20)*d^5*cos(d*x + c)^2 + (a^14*b^6 + 7*a^12*b^8 + 21*a^10*b^1
0 + 35*a^8*b^12 + 35*a^6*b^14 + 21*a^4*b^16 + 7*a^2*b^18 + b^20)*d^5 + 4*((a^17*b^3 + 6*a^15*b^5 + 14*a^13*b^7
+ 14*a^11*b^9 - 14*a^7*b^13 - 14*a^5*b^15 - 6*a^3*b^17 - a*b^19)*d^5*cos(d*x + c)^3 + (a^15*b^5 + 7*a^13*b^7
+ 21*a^11*b^9 + 35*a^9*b^11 + 35*a^7*b^13 + 21*a^5*b^15 + 7*a^3*b^17 + a*b^19)*d^5*cos(d*x + c))*sin(d*x + c))
*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a
^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b
^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 -
100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b
^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a^10 + 5*a^8*b
^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4)*arctan(((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a
^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 - 350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*sq
rt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*
b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqr
t(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4
- 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10 - 5*a^3*b^12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^
6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*
a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) - sqrt(2)*((a^23 + 7*a^21*b^2 + 15*a^19*b^4
- 15*a^17*b^6 - 150*a^15*b^8 - 378*a^13*b^10 - 546*a^11*b^12 - 510*a^9*b^14 - 315*a^7*b^16 - 125*a^5*b^18 - 29
*a^3*b^20 - 3*a*b^22)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b
^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 1
0*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (a^18
+ 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16
- b^18)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*
b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 +
b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^1
1*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^
4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((
(25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt(
1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + sqrt(2)*((25*a^16 - 50
*a^14*b^2 - 90*a^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d^3*sq
rt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (25*a^11 - 175*a^9*
b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^
3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*
b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/
((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9 - 100*a^7*b^2 + 110*a^5
*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x + c) + (25*a^8*b - 100*a^6*b^3 + 110*a^4*b^5 - 20*a^2*b^7 + b^9)*sin(d*x +
c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4) + sqrt(2)*((
5*a^27 + 25*a^25*b^2 + 6*a^23*b^4 - 218*a^21*b^6 - 585*a^19*b^8 - 405*a^17*b^10 + 900*a^15*b^12 + 2532*a^13*b^
14 + 2979*a^11*b^16 + 2015*a^9*b^18 + 790*a^7*b^20 + 150*a^5*b^22 + a^3*b^24 - 3*a*b^26)*d^7*sqrt((25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*
b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*
a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^22 + 25*a^20*b^2 + 31*a^18*b^4 - 53*a^16*b^
6 - 190*a^14*b^8 - 182*a^12*b^10 + 14*a^10*b^12 + 166*a^8*b^14 + 137*a^6*b^16 + 45*a^4*b^18 + 3*a^2*b^20 - b^2
2)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 +
120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*
d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4
- 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10
*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(
d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d
^4))^(3/4))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)) + 12*sqrt(2)*((a^18*b^2 + a^16*b^4 -
20*a^14*b^6 - 84*a^12*b^8 - 154*a^10*b^10 - 154*a^8*b^12 - 84*a^6*b^14 - 20*a^4*b^16 + a^2*b^18 + b^20)*d^5*c
os(d*x + c)^4 + 2*(3*a^16*b^4 + 20*a^14*b^6 + 56*a^12*b^8 + 84*a^10*b^10 + 70*a^8*b^12 + 28*a^6*b^14 - 4*a^2*b
^18 - b^20)*d^5*cos(d*x + c)^2 + (a^14*b^6 + 7*a^12*b^8 + 21*a^10*b^10 + 35*a^8*b^12 + 35*a^6*b^14 + 21*a^4*b^
16 + 7*a^2*b^18 + b^20)*d^5 + 4*((a^17*b^3 + 6*a^15*b^5 + 14*a^13*b^7 + 14*a^11*b^9 - 14*a^7*b^13 - 14*a^5*b^1
5 - 6*a^3*b^17 - a*b^19)*d^5*cos(d*x + c)^3 + (a^15*b^5 + 7*a^13*b^7 + 21*a^11*b^9 + 35*a^9*b^11 + 35*a^7*b^13
+ 21*a^5*b^15 + 7*a^3*b^17 + a*b^19)*d^5*cos(d*x + c))*sin(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10
*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b
^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2
- 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 +
b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a
^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8
+ b^10)*d^4))^(3/4)*arctan(-((5*a^20 + 30*a^18*b^2 + 61*a^16*b^4 + 8*a^14*b^6 - 182*a^12*b^8 - 364*a^10*b^10 -
350*a^8*b^12 - 184*a^6*b^14 - 47*a^4*b^16 - 2*a^2*b^18 + b^20)*d^4*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b
^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210
*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10
*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (5*a^15 + 15*a^13*b^2 + a^11*b^4 - 45*a^9*b^6 - 65*a^7*b^8 - 35*a^5*b^10
- 5*a^3*b^12 + a*b^14)*d^2*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*
b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 +
10*a^2*b^18 + b^20)*d^4)) + sqrt(2)*((a^23 + 7*a^21*b^2 + 15*a^19*b^4 - 15*a^17*b^6 - 150*a^15*b^8 - 378*a^13*
b^10 - 546*a^11*b^12 - 510*a^9*b^14 - 315*a^7*b^16 - 125*a^5*b^18 - 29*a^3*b^20 - 3*a*b^22)*d^7*sqrt((25*a^8*b
^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^
12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 +
5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)) + (a^18 + 7*a^16*b^2 + 20*a^14*b^4 + 28*a^12*b^
6 + 14*a^10*b^8 - 14*a^8*b^10 - 28*a^6*b^12 - 20*a^4*b^14 - 7*a^2*b^16 - b^18)*d^5*sqrt((25*a^8*b^2 - 100*a^6*
b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*
a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10
*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^
10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))
)/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt(((25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 +
35*a^8*b^6 + 171*a^6*b^8 + 53*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a
^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) - sqrt(2)*((25*a^16 - 50*a^14*b^2 - 90*a^12*b^4 + 150*a^10*b^6 +
136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d^3*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 1
0*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (25*a^11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3
*b^8 - 3*a*b^10)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 -
5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5
*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2
*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4
*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9 - 100*a^7*b^2 + 110*a^5*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x + c)
+ (25*a^8*b - 100*a^6*b^3 + 110*a^4*b^5 - 20*a^2*b^7 + b^9)*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2
+ 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4) - sqrt(2)*((5*a^27 + 25*a^25*b^2 + 6*a^23*b^4 - 218*
a^21*b^6 - 585*a^19*b^8 - 405*a^17*b^10 + 900*a^15*b^12 + 2532*a^13*b^14 + 2979*a^11*b^16 + 2015*a^9*b^18 + 79
0*a^7*b^20 + 150*a^5*b^22 + a^3*b^24 - 3*a*b^26)*d^7*sqrt((25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8
+ b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120
*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^
2*b^8 + b^10)*d^4)) + (5*a^22 + 25*a^20*b^2 + 31*a^18*b^4 - 53*a^16*b^6 - 190*a^14*b^8 - 182*a^12*b^10 + 14*a^
10*b^12 + 166*a^8*b^14 + 137*a^6*b^16 + 45*a^4*b^18 + 3*a^2*b^20 - b^22)*d^5*sqrt((25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b
^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b
^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 1
5*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*
a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))
*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(3/4))/(25*a^8*b^2 - 100*a^6*b^4 +
110*a^4*b^6 - 20*a^2*b^8 + b^10)) - 3*sqrt(2)*((a^8*b^2 - 4*a^6*b^4 - 10*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x
+ c)^4 + 2*(3*a^6*b^4 + 5*a^4*b^6 + a^2*b^8 - b^10)*d*cos(d*x + c)^2 + (a^4*b^6 + 2*a^2*b^8 + b^10)*d + 4*((a
^7*b^3 + a^5*b^5 - a^3*b^7 - a*b^9)*d*cos(d*x + c)^3 + (a^5*b^5 + 2*a^3*b^7 + a*b^9)*d*cos(d*x + c))*sin(d*x +
c) + ((a^13*b^2 - 14*a^11*b^4 + 35*a^9*b^6 + 76*a^7*b^8 - 9*a^5*b^10 - 30*a^3*b^12 + 5*a*b^14)*d^3*cos(d*x +
c)^4 + 2*(3*a^11*b^4 - 25*a^9*b^6 - 34*a^7*b^8 + 14*a^5*b^10 + 15*a^3*b^12 - 5*a*b^14)*d^3*cos(d*x + c)^2 + (a
^9*b^6 - 8*a^7*b^8 - 14*a^5*b^10 + 5*a*b^14)*d^3 + 4*((a^12*b^3 - 9*a^10*b^5 - 6*a^8*b^7 + 14*a^6*b^9 + 5*a^4*
b^11 - 5*a^2*b^13)*d^3*cos(d*x + c)^3 + (a^10*b^5 - 8*a^8*b^7 - 14*a^6*b^9 + 5*a^2*b^13)*d^3*cos(d*x + c))*sin
(d*x + c))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))*sqrt((a^10 + 5*a^8*b
^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 +
a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10
)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10
*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4)*log(((25*a^14 - 25*a^12*b^2 - 115*a^10*b^4 + 35*a^8*b^6 + 171*a^6*b^8
+ 53*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^1
0)*d^4))*cos(d*x + c) + sqrt(2)*((25*a^16 - 50*a^14*b^2 - 90*a^12*b^4 + 150*a^10*b^6 + 136*a^8*b^8 - 118*a^6*b
^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d^3*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 +
b^10)*d^4))*cos(d*x + c) + (25*a^11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 + 61*a^3*b^8 - 3*a*b^10)*d*cos(d
*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b^
4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*sqrt((a*co
s(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)
*d^4))^(1/4) + (25*a^9 - 100*a^7*b^2 + 110*a^5*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x + c) + (25*a^8*b - 100*a^6*b^
3 + 110*a^4*b^5 - 20*a^2*b^7 + b^9)*sin(d*x + c))/cos(d*x + c)) + 3*sqrt(2)*((a^8*b^2 - 4*a^6*b^4 - 10*a^4*b^6
- 4*a^2*b^8 + b^10)*d*cos(d*x + c)^4 + 2*(3*a^6*b^4 + 5*a^4*b^6 + a^2*b^8 - b^10)*d*cos(d*x + c)^2 + (a^4*b^6
+ 2*a^2*b^8 + b^10)*d + 4*((a^7*b^3 + a^5*b^5 - a^3*b^7 - a*b^9)*d*cos(d*x + c)^3 + (a^5*b^5 + 2*a^3*b^7 + a*
b^9)*d*cos(d*x + c))*sin(d*x + c) + ((a^13*b^2 - 14*a^11*b^4 + 35*a^9*b^6 + 76*a^7*b^8 - 9*a^5*b^10 - 30*a^3*b
^12 + 5*a*b^14)*d^3*cos(d*x + c)^4 + 2*(3*a^11*b^4 - 25*a^9*b^6 - 34*a^7*b^8 + 14*a^5*b^10 + 15*a^3*b^12 - 5*a
*b^14)*d^3*cos(d*x + c)^2 + (a^9*b^6 - 8*a^7*b^8 - 14*a^5*b^10 + 5*a*b^14)*d^3 + 4*((a^12*b^3 - 9*a^10*b^5 - 6
*a^8*b^7 + 14*a^6*b^9 + 5*a^4*b^11 - 5*a^2*b^13)*d^3*cos(d*x + c)^3 + (a^10*b^5 - 8*a^8*b^7 - 14*a^6*b^9 + 5*a
^2*b^13)*d^3*cos(d*x + c))*sin(d*x + c))*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^1
0)*d^4)))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (a^15 - 5*a^13*b^2 - 35*a^11*b
^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10))*(1/((a^10
+ 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4)*log(((25*a^14 - 25*a^12*b^2 - 115*a^10*
b^4 + 35*a^8*b^6 + 171*a^6*b^8 + 53*a^4*b^10 - 17*a^2*b^12 + b^14)*d^2*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4
+ 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) - sqrt(2)*((25*a^16 - 50*a^14*b^2 - 90*a^12*b^4 + 150*a^10
*b^6 + 136*a^8*b^8 - 118*a^6*b^10 - 70*a^4*b^12 + 18*a^2*b^14 - b^16)*d^3*sqrt(1/((a^10 + 5*a^8*b^2 + 10*a^6*b
^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))*cos(d*x + c) + (25*a^11 - 175*a^9*b^2 + 410*a^7*b^4 - 350*a^5*b^6 +
61*a^3*b^8 - 3*a*b^10)*d*cos(d*x + c))*sqrt((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10 - (
a^15 - 5*a^13*b^2 - 35*a^11*b^4 - 65*a^9*b^6 - 45*a^7*b^8 + a^5*b^10 + 15*a^3*b^12 + 5*a*b^14)*d^2*sqrt(1/((a^
10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4)))/(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 -
20*a^2*b^8 + b^10))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(1/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 +
10*a^4*b^6 + 5*a^2*b^8 + b^10)*d^4))^(1/4) + (25*a^9 - 100*a^7*b^2 + 110*a^5*b^4 - 20*a^3*b^6 + a*b^8)*cos(d*x
+ c) + (25*a^8*b - 100*a^6*b^3 + 110*a^4*b^5 - 20*a^2*b^7 + b^9)*sin(d*x + c))/cos(d*x + c)) + 8*(2*(a^7 - 13
*a^3*b^4)*cos(d*x + c)^4 + 2*(4*a^5*b^2 + 13*a^3*b^4)*cos(d*x + c)^2 + ((7*a^6*b + 22*a^4*b^3 - 9*a^2*b^5)*cos
(d*x + c)^3 + 3*(a^4*b^3 + 3*a^2*b^5)*cos(d*x + c))*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d
*x + c)))/((a^8*b^2 - 4*a^6*b^4 - 10*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x + c)^4 + 2*(3*a^6*b^4 + 5*a^4*b^6 +
a^2*b^8 - b^10)*d*cos(d*x + c)^2 + (a^4*b^6 + 2*a^2*b^8 + b^10)*d + 4*((a^7*b^3 + a^5*b^5 - a^3*b^7 - a*b^9)*
d*cos(d*x + c)^3 + (a^5*b^5 + 2*a^3*b^7 + a*b^9)*d*cos(d*x + c))*sin(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{3}{\left (c + d x \right )}}{\left (a + b \tan{\left (c + d x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**3/(a+b*tan(d*x+c))**(5/2),x)

[Out]

Integral(tan(c + d*x)**3/(a + b*tan(c + d*x))**(5/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (d x + c\right )^{3}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^3/(a+b*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(tan(d*x + c)^3/(b*tan(d*x + c) + a)^(5/2), x)