3.527 \(\int (a+b \tan (c+d x))^{7/2} \, dx\)

Optimal. Leaf size=167 \[ \frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

[Out]

((-I)*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (I*(a + I*b)^(7/2)*ArcTanh[Sqrt[a +
b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(3*a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (4*a*b*(a + b*Tan[c + d*
x])^(3/2))/(3*d) + (2*b*(a + b*Tan[c + d*x])^(5/2))/(5*d)

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Rubi [A]  time = 0.350049, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3482, 3528, 3539, 3537, 63, 208} \[ \frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[c + d*x])^(7/2),x]

[Out]

((-I)*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d + (I*(a + I*b)^(7/2)*ArcTanh[Sqrt[a +
b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*b*(3*a^2 - b^2)*Sqrt[a + b*Tan[c + d*x]])/d + (4*a*b*(a + b*Tan[c + d*
x])^(3/2))/(3*d) + (2*b*(a + b*Tan[c + d*x])^(5/2))/(5*d)

Rule 3482

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 + b^2, 0] && GtQ[n, 1]

Rule 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 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 (a+b \tan (c+d x))^{7/2} \, dx &=\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int (a+b \tan (c+d x))^{3/2} \left (a^2-b^2+2 a b \tan (c+d x)\right ) \, dx\\ &=\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int \sqrt{a+b \tan (c+d x)} \left (a \left (a^2-3 b^2\right )+b \left (3 a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=\frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\int \frac{a^4-6 a^2 b^2+b^4+4 a b \left (a^2-b^2\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{1}{2} (a-i b)^4 \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx+\frac{1}{2} (a+i b)^4 \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=\frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}+\frac{\left (i (a-i b)^4\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}-\frac{\left (i (a+i b)^4\right ) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}\\ &=\frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}-\frac{(a-i b)^4 \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 )}{b d}-\frac{(a+i b)^4 \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 )}{b d}\\ &=-\frac{i (a-i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}+\frac{i (a+i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}+\frac{2 b \left (3 a^2-b^2\right ) \sqrt{a+b \tan (c+d x)}}{d}+\frac{4 a b (a+b \tan (c+d x))^{3/2}}{3 d}+\frac{2 b (a+b \tan (c+d x))^{5/2}}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.902664, size = 143, normalized size = 0.86 \[ \frac{2 b \sqrt{a+b \tan (c+d x)} \left (58 a^2+16 a b \tan (c+d x)+3 b^2 \tan ^2(c+d x)-15 b^2\right )-15 i (a-i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )+15 i (a+i b)^{7/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{15 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-15*I)*(a - I*b)^(7/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] + (15*I)*(a + I*b)^(7/2)*ArcTanh[Sqrt
[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]*(58*a^2 - 15*b^2 + 16*a*b*Tan[c + d*x] + 3*
b^2*Tan[c + d*x]^2))/(15*d)

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Maple [B]  time = 0.046, size = 1552, normalized size = 9.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*b*(a+b*tan(d*x+c))^(5/2)/d+1/4/d*b^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)-1/4/d*b^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)+4/d*b^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta
n(((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-1/4/d/b*ln((a+b*ta
n(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+3/2/d*b*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^2-4/d*b/(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*b^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^2+b^2)^(1/2)+1/4/d/b*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+b^2
)^(1/2)*a^3-3/4/d*b*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+b^2)^(1/2)*a+3/d*b/(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^2+b^2)^(1/2)*a^2+1/4/d/b*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^4-3/2
/d*b*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+4/d*b/(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*b^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^2+b^2)^(1/2)-4/d*b^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-1/4/
d/b*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+b^2)^(1/2)*a^3+3/4/d*b*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+b^2)^(1/2)*a-3/d*b/(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^2+b^2)^(1/2)*a^2+4/3
*a*b*(a+b*tan(d*x+c))^(3/2)/d+6/d*b*a^2*(a+b*tan(d*x+c))^(1/2)-2/d*b^3*(a+b*tan(d*x+c))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 105.155, size = 21893, normalized size = 131.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60*(60*sqrt(2)*d^5*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^
12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 +
35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8
+ 511*a^4*b^10 - 42*a^2*b^12 + b^14))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^
10 + 7*a^2*b^12 + b^14)/d^4)^(3/4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^
10 - 42*a^2*b^12 + b^14)/d^4)*arctan(-((7*a^26 + 35*a^24*b^2 - 14*a^22*b^4 - 526*a^20*b^6 - 1795*a^18*b^8 - 31
11*a^16*b^10 - 3060*a^14*b^12 - 1428*a^12*b^14 + 273*a^10*b^16 + 805*a^8*b^18 + 482*a^6*b^20 + 130*a^4*b^22 +
11*a^2*b^24 - b^26)*d^4*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*
b^12 + b^14)/d^4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12
+ b^14)/d^4) + (7*a^33 + 56*a^31*b^2 + 112*a^29*b^4 - 456*a^27*b^6 - 3380*a^25*b^8 - 10088*a^23*b^10 - 18304*a
^21*b^12 - 21736*a^19*b^14 - 16302*a^17*b^16 - 5720*a^15*b^18 + 2288*a^13*b^20 + 4264*a^11*b^22 + 2652*a^9*b^2
4 + 904*a^7*b^26 + 160*a^5*b^28 + 8*a^3*b^30 - a*b^32)*d^2*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1
484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + sqrt(2)*((21*a^14*b - 49*a^12*b^3 - 175*a^10*b^5 - 45*
a^8*b^7 + 111*a^6*b^9 + 29*a^4*b^11 - 21*a^2*b^13 + b^15)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b
^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484
*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + 4*(7*a^21*b - 91*a^17*b^5 - 176*a^15*b^7 - 26*a^13*b^9 +
208*a^11*b^11 + 170*a^9*b^13 - 16*a^7*b^15 - 61*a^5*b^17 - 16*a^3*b^19 + a*b^21)*d^5*sqrt((49*a^12*b^2 - 490*a
^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4))*sqrt((a^14 + 7*a^12*b^2 + 21*
a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^
6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)
)/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(
d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*
b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4) + sqrt(2)*((3*a^2 - b^2)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a
^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 -
1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + 4*(a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((49*
a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4))*sqrt((a^14 +
7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*
a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b
^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^1
4))*sqrt(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a^10*b^12 - 1518*
a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^
8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + sqrt(2)*(4*(49*a^15*b^3 - 539*a^13*b
^5 + 2009*a^11*b^7 - 3003*a^9*b^9 + 1995*a^7*b^11 - 553*a^5*b^13 + 43*a^3*b^15 - a*b^17)*d^3*sqrt((a^14 + 7*a^
12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (147*a^2
2*b^3 - 931*a^20*b^5 - 147*a^18*b^7 + 5691*a^16*b^9 + 3486*a^14*b^11 - 5726*a^12*b^13 - 3238*a^10*b^15 + 2454*
a^8*b^17 + 735*a^6*b^19 - 463*a^4*b^21 + 41*a^2*b^23 - b^25)*d*cos(d*x + c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10
*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d
^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(4
9*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10
+ 7*a^2*b^12 + b^14)/d^4)^(1/4) + (49*a^27*b^2 - 147*a^25*b^4 - 882*a^23*b^6 + 574*a^21*b^8 + 6587*a^19*b^10
+ 9415*a^17*b^12 + 1716*a^15*b^14 - 6412*a^13*b^16 - 4585*a^11*b^18 + 427*a^9*b^20 + 1246*a^7*b^22 + 238*a^5*b
^24 - 35*a^3*b^26 + a*b^28)*cos(d*x + c) + (49*a^26*b^3 - 147*a^24*b^5 - 882*a^22*b^7 + 574*a^20*b^9 + 6587*a^
18*b^11 + 9415*a^16*b^13 + 1716*a^14*b^15 - 6412*a^12*b^17 - 4585*a^10*b^19 + 427*a^8*b^21 + 1246*a^6*b^23 + 2
38*a^4*b^25 - 35*a^2*b^27 + b^29)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4
+ 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4))/(49*a^38*b^2 + 147*a^36*b^4 - 1029*a^
34*b^6 - 5943*a^32*b^8 - 5404*a^30*b^10 + 37996*a^28*b^12 + 154428*a^26*b^14 + 280020*a^24*b^16 + 272350*a^22*
b^18 + 92378*a^20*b^20 - 104390*a^18*b^22 - 154050*a^16*b^24 - 76908*a^14*b^26 + 764*a^12*b^28 + 20908*a^10*b^
30 + 10788*a^8*b^32 + 2169*a^6*b^34 + 43*a^4*b^36 - 29*a^2*b^38 + b^40))*cos(d*x + c)^2 + 60*sqrt(2)*d^5*sqrt(
(a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b
^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 +
7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^
12 + b^14))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4
)^(3/4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^
4)*arctan(((7*a^26 + 35*a^24*b^2 - 14*a^22*b^4 - 526*a^20*b^6 - 1795*a^18*b^8 - 3111*a^16*b^10 - 3060*a^14*b^1
2 - 1428*a^12*b^14 + 273*a^10*b^16 + 805*a^8*b^18 + 482*a^6*b^20 + 130*a^4*b^22 + 11*a^2*b^24 - b^26)*d^4*sqrt
((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*sqrt((49*a
^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4) + (7*a^33 + 56*
a^31*b^2 + 112*a^29*b^4 - 456*a^27*b^6 - 3380*a^25*b^8 - 10088*a^23*b^10 - 18304*a^21*b^12 - 21736*a^19*b^14 -
16302*a^17*b^16 - 5720*a^15*b^18 + 2288*a^13*b^20 + 4264*a^11*b^22 + 2652*a^9*b^24 + 904*a^7*b^26 + 160*a^5*b
^28 + 8*a^3*b^30 - a*b^32)*d^2*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 -
42*a^2*b^12 + b^14)/d^4) - sqrt(2)*((21*a^14*b - 49*a^12*b^3 - 175*a^10*b^5 - 45*a^8*b^7 + 111*a^6*b^9 + 29*a
^4*b^11 - 21*a^2*b^13 + b^15)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^1
0 + 7*a^2*b^12 + b^14)/d^4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42
*a^2*b^12 + b^14)/d^4) + 4*(7*a^21*b - 91*a^17*b^5 - 176*a^15*b^7 - 26*a^13*b^9 + 208*a^11*b^11 + 170*a^9*b^13
- 16*a^7*b^15 - 61*a^5*b^17 - 16*a^3*b^19 + a*b^21)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 148
4*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a
^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b
^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^
4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/c
os(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^
4)^(3/4) - sqrt(2)*((3*a^2 - b^2)*d^7*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4
*b^10 + 7*a^2*b^12 + b^14)/d^4)*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10
- 42*a^2*b^12 + b^14)/d^4) + 4*(a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^5*sqrt((49*a^12*b^2 - 490*a^10*b^4 + 15
19*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14)/d^4))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 3
5*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt(
(a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b
^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt(((49*a^20*b^2 - 294
*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 34
9*a^4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*
b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) - sqrt(2)*(4*(49*a^15*b^3 - 539*a^13*b^5 + 2009*a^11*b^7 - 3003*a^
9*b^9 + 1995*a^7*b^11 - 553*a^5*b^13 + 43*a^3*b^15 - a*b^17)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^
8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (147*a^22*b^3 - 931*a^20*b^5 - 147*a
^18*b^7 + 5691*a^16*b^9 + 3486*a^14*b^11 - 5726*a^12*b^13 - 3238*a^10*b^15 + 2454*a^8*b^17 + 735*a^6*b^19 - 46
3*a^4*b^21 + 41*a^2*b^23 - b^25)*d*cos(d*x + c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b
^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 +
21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 +
1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d
*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(
1/4) + (49*a^27*b^2 - 147*a^25*b^4 - 882*a^23*b^6 + 574*a^21*b^8 + 6587*a^19*b^10 + 9415*a^17*b^12 + 1716*a^15
*b^14 - 6412*a^13*b^16 - 4585*a^11*b^18 + 427*a^9*b^20 + 1246*a^7*b^22 + 238*a^5*b^24 - 35*a^3*b^26 + a*b^28)*
cos(d*x + c) + (49*a^26*b^3 - 147*a^24*b^5 - 882*a^22*b^7 + 574*a^20*b^9 + 6587*a^18*b^11 + 9415*a^16*b^13 + 1
716*a^14*b^15 - 6412*a^12*b^17 - 4585*a^10*b^19 + 427*a^8*b^21 + 1246*a^6*b^23 + 238*a^4*b^25 - 35*a^2*b^27 +
b^29)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 +
21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(3/4))/(49*a^38*b^2 + 147*a^36*b^4 - 1029*a^34*b^6 - 5943*a^32*b^8 - 540
4*a^30*b^10 + 37996*a^28*b^12 + 154428*a^26*b^14 + 280020*a^24*b^16 + 272350*a^22*b^18 + 92378*a^20*b^20 - 104
390*a^18*b^22 - 154050*a^16*b^24 - 76908*a^14*b^26 + 764*a^12*b^28 + 20908*a^10*b^30 + 10788*a^8*b^32 + 2169*a
^6*b^34 + 43*a^4*b^36 - 29*a^2*b^38 + b^40))*cos(d*x + c)^2 - 15*sqrt(2)*((a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a
*b^6)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d
^4)*cos(d*x + c)^2 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b
^14)*d*cos(d*x + c)^2)*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b
^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6
+ 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^
8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b
^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4)*log(((49*a^20*b^2 - 294*a^18*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^1
2*b^10 - 1316*a^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 349*a^4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt((a^14 + 7
*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + sqrt(
2)*(4*(49*a^15*b^3 - 539*a^13*b^5 + 2009*a^11*b^7 - 3003*a^9*b^9 + 1995*a^7*b^11 - 553*a^5*b^13 + 43*a^3*b^15
- a*b^17)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^1
4)/d^4)*cos(d*x + c) + (147*a^22*b^3 - 931*a^20*b^5 - 147*a^18*b^7 + 5691*a^16*b^9 + 3486*a^14*b^11 - 5726*a^1
2*b^13 - 3238*a^10*b^15 + 2454*a^8*b^17 + 735*a^6*b^19 - 463*a^4*b^21 + 41*a^2*b^23 - b^25)*d*cos(d*x + c))*sq
rt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^
5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^1
0 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2
*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8
*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4) + (49*a^27*b^2 - 147*a^25*b^4 - 882*a^23*b^6 +
574*a^21*b^8 + 6587*a^19*b^10 + 9415*a^17*b^12 + 1716*a^15*b^14 - 6412*a^13*b^16 - 4585*a^11*b^18 + 427*a^9*b
^20 + 1246*a^7*b^22 + 238*a^5*b^24 - 35*a^3*b^26 + a*b^28)*cos(d*x + c) + (49*a^26*b^3 - 147*a^24*b^5 - 882*a^
22*b^7 + 574*a^20*b^9 + 6587*a^18*b^11 + 9415*a^16*b^13 + 1716*a^14*b^15 - 6412*a^12*b^17 - 4585*a^10*b^19 + 4
27*a^8*b^21 + 1246*a^6*b^23 + 238*a^4*b^25 - 35*a^2*b^27 + b^29)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) + 1
5*sqrt(2)*((a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 +
35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c)^2 - (a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*
b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)*d*cos(d*x + c)^2)*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 +
35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt
((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*
b^2 - 490*a^10*b^4 + 1519*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*((a^14 + 7*a^12*b^2 + 2
1*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4)*log(((49*a^20*b^2 - 294*a^1
8*b^4 - 147*a^16*b^6 + 1848*a^14*b^8 + 1778*a^12*b^10 - 1316*a^10*b^12 - 1518*a^8*b^14 + 312*a^6*b^16 + 349*a^
4*b^18 - 38*a^2*b^20 + b^22)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10
+ 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) - sqrt(2)*(4*(49*a^15*b^3 - 539*a^13*b^5 + 2009*a^11*b^7 - 3003*a^9*b^
9 + 1995*a^7*b^11 - 553*a^5*b^13 + 43*a^3*b^15 - a*b^17)*d^3*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^
6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)*cos(d*x + c) + (147*a^22*b^3 - 931*a^20*b^5 - 147*a^18*
b^7 + 5691*a^16*b^9 + 3486*a^14*b^11 - 5726*a^12*b^13 - 3238*a^10*b^15 + 2454*a^8*b^17 + 735*a^6*b^19 - 463*a^
4*b^21 + 41*a^2*b^23 - b^25)*d*cos(d*x + c))*sqrt((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 +
21*a^4*b^10 + 7*a^2*b^12 + b^14 + (a^7 - 21*a^5*b^2 + 35*a^3*b^4 - 7*a*b^6)*d^2*sqrt((a^14 + 7*a^12*b^2 + 21*
a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4))/(49*a^12*b^2 - 490*a^10*b^4 + 1519
*a^8*b^6 - 1484*a^6*b^8 + 511*a^4*b^10 - 42*a^2*b^12 + b^14))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x +
c))*((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^4*b^10 + 7*a^2*b^12 + b^14)/d^4)^(1/4)
+ (49*a^27*b^2 - 147*a^25*b^4 - 882*a^23*b^6 + 574*a^21*b^8 + 6587*a^19*b^10 + 9415*a^17*b^12 + 1716*a^15*b^1
4 - 6412*a^13*b^16 - 4585*a^11*b^18 + 427*a^9*b^20 + 1246*a^7*b^22 + 238*a^5*b^24 - 35*a^3*b^26 + a*b^28)*cos(
d*x + c) + (49*a^26*b^3 - 147*a^24*b^5 - 882*a^22*b^7 + 574*a^20*b^9 + 6587*a^18*b^11 + 9415*a^16*b^13 + 1716*
a^14*b^15 - 6412*a^12*b^17 - 4585*a^10*b^19 + 427*a^8*b^21 + 1246*a^6*b^23 + 238*a^4*b^25 - 35*a^2*b^27 + b^29
)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) - 8*(3*a^14*b^3 + 21*a^12*b^5 + 63*a^10*b^7 + 105*a^8*b^9 + 105*a^
6*b^11 + 63*a^4*b^13 + 21*a^2*b^15 + 3*b^17 + 2*(29*a^16*b + 194*a^14*b^3 + 546*a^12*b^5 + 826*a^10*b^7 + 700*
a^8*b^9 + 294*a^6*b^11 + 14*a^4*b^13 - 34*a^2*b^15 - 9*b^17)*cos(d*x + c)^2 + 16*(a^15*b^2 + 7*a^13*b^4 + 21*a
^11*b^6 + 35*a^9*b^8 + 35*a^7*b^10 + 21*a^5*b^12 + 7*a^3*b^14 + a*b^16)*cos(d*x + c)*sin(d*x + c))*sqrt((a*cos
(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((a^14 + 7*a^12*b^2 + 21*a^10*b^4 + 35*a^8*b^6 + 35*a^6*b^8 + 21*a^
4*b^10 + 7*a^2*b^12 + b^14)*d*cos(d*x + c)^2)

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Sympy [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((a+b*tan(d*x+c))**(7/2),x)

[Out]

Timed out

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

[Out]

Timed out