3.5.17 \(\int \frac {\sec ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [417]

Optimal. Leaf size=161 \[ \frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}+\frac {b \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}+\frac {(a-3 b) \tan (c+d x)}{(a-b)^2 d}+\frac {\tan ^3(c+d x)}{3 (a-b) d} \]

[Out]

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

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Rubi [A]
time = 0.22, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3303, 1184, 1180, 211} \begin {gather*} \frac {b \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {b \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\tan ^3(c+d x)}{3 d (a-b)}+\frac {(a-3 b) \tan (c+d x)}{d (a-b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]

[Out]

(b*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) + (b*ArcTan
[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) + ((a - 3*b)*Tan[c +
d*x])/((a - b)^2*d) + Tan[c + d*x]^3/(3*(a - b)*d)

Rule 211

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

Rule 1180

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1184

Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]

Rule 3303

Int[cos[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(m/2 + 2
*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

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

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Mathematica [A]
time = 0.71, size = 205, normalized size = 1.27 \begin {gather*} \frac {\frac {3 \left (\sqrt {a}-\sqrt {b}\right )^2 b \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}}-\frac {3 \left (\sqrt {a}+\sqrt {b}\right )^2 b \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+4 (a-4 b) \tan (c+d x)+2 (a-b) \sec ^2(c+d x) \tan (c+d x)}{6 (a-b)^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4/(a - b*Sin[c + d*x]^4),x]

[Out]

((3*(Sqrt[a] - Sqrt[b])^2*b*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqr
t[a + Sqrt[a]*Sqrt[b]]) - (3*(Sqrt[a] + Sqrt[b])^2*b*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt
[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 4*(a - 4*b)*Tan[c + d*x] + 2*(a - b)*Sec[c + d*x]^2*Tan[
c + d*x])/(6*(a - b)^2*d)

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Maple [A]
time = 1.38, size = 230, normalized size = 1.43

method	result	size
derivativedivides	\(\frac {\frac {\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right ) a -3 b \tan \left (d x +c \right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b +3 a b +b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b -3 a b -b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a -b}}{d}\)	\(230\)
default	\(\frac {\frac {\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{3}-\frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3}+\tan \left (d x +c \right ) a -3 b \tan \left (d x +c \right )}{\left (a -b \right )^{2}}+\frac {b \left (\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b +3 a b +b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (a \sqrt {a b}+3 \sqrt {a b}\, b -3 a b -b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{a -b}}{d}\)	\(230\)
risch	\(-\frac {4 i \left (3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-3 a \,{\mathrm e}^{2 i \left (d x +c \right )}+9 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a +4 b \right )}{3 d \left (a -b \right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+16 \left (\munderset {\textit {\_R} =\RootOf \left (\left (16777216 a^{8} d^{4}-83886080 a^{7} b \,d^{4}[phone] a^{6} b^{2} d^{4}-167772160 a^{5} b^{3} d^{4}[phone] a^{4} b^{4} d^{4}-16777216 a^{3} b^{5} d^{4}\right ) \textit {\_Z}^{4}+\left (8192 a^{4} b^{2} d^{2}+81920 a^{3} b^{3} d^{2}+40960 a^{2} b^{4} d^{2}\right ) \textit {\_Z}^{2}+b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {524288 i d^{3} a^{9}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {1048576 i a^{8} b \,d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {2621440 i a^{7} b^{2} d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {10485760 i a^{6} b^{3} d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {13107200 i a^{5} b^{4} d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {7340032 i a^{4} b^{5} d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {1572864 i a^{3} b^{6} d^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}\right ) \textit {\_R}^{3}+\left (-\frac {8192 d^{2} a^{7} b}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {40960 a^{6} b^{2} d^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {81920 a^{5} b^{3} d^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {81920 a^{4} b^{4} d^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {40960 a^{3} b^{5} d^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {8192 a^{2} b^{6} d^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}\right ) \textit {\_R}^{2}+\left (\frac {128 i d \,a^{5} b^{2}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {3584 i a^{4} b^{3} d}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {8960 i a^{3} b^{4} d}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {3584 i a^{2} b^{5} d}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}+\frac {128 i d \,b^{6} a}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}\right ) \textit {\_R} -\frac {2 a^{3} b^{3}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {25 a^{2} b^{4}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {20 a \,b^{5}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}-\frac {b^{6}}{5 a^{2} b^{4}+10 a \,b^{5}+b^{6}}\right )\right )\)	\(869\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/(a-b)^2*(1/3*a*tan(d*x+c)^3-1/3*b*tan(d*x+c)^3+tan(d*x+c)*a-3*b*tan(d*x+c))+b/(a-b)*(1/2*(a*(a*b)^(1/2)
+3*(a*b)^(1/2)*b+3*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1
/2)+a)*(a-b))^(1/2))+1/2*(a*(a*b)^(1/2)+3*(a*b)^(1/2)*b-3*a*b-b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(
1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

-1/3*(36*(a - 2*b)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) - 12*(b*sin(4*d*x + 4*c) - (a - 3*b)*sin(2*d*x + 2*c))*co
s(6*d*x + 6*c) - 3*((a^2 - 2*a*b + b^2)*d*cos(6*d*x + 6*c)^2 + 9*(a^2 - 2*a*b + b^2)*d*cos(4*d*x + 4*c)^2 + 9*
(a^2 - 2*a*b + b^2)*d*cos(2*d*x + 2*c)^2 + (a^2 - 2*a*b + b^2)*d*sin(6*d*x + 6*c)^2 + 9*(a^2 - 2*a*b + b^2)*d*
sin(4*d*x + 4*c)^2 + 18*(a^2 - 2*a*b + b^2)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*(a^2 - 2*a*b + b^2)*d*sin(
2*d*x + 2*c)^2 + 6*(a^2 - 2*a*b + b^2)*d*cos(2*d*x + 2*c) + (a^2 - 2*a*b + b^2)*d + 2*(3*(a^2 - 2*a*b + b^2)*d
*cos(4*d*x + 4*c) + 3*(a^2 - 2*a*b + b^2)*d*cos(2*d*x + 2*c) + (a^2 - 2*a*b + b^2)*d)*cos(6*d*x + 6*c) + 6*(3*
(a^2 - 2*a*b + b^2)*d*cos(2*d*x + 2*c) + (a^2 - 2*a*b + b^2)*d)*cos(4*d*x + 4*c) + 6*((a^2 - 2*a*b + b^2)*d*si
n(4*d*x + 4*c) + (a^2 - 2*a*b + b^2)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-8*(4*b^3*cos(6*d*x + 6*c
)^2 + 4*b^3*cos(2*d*x + 2*c)^2 + 4*b^3*sin(6*d*x + 6*c)^2 + 4*b^3*sin(2*d*x + 2*c)^2 - b^3*cos(2*d*x + 2*c) -
4*(8*a^2*b + 13*a*b^2 - 6*b^3)*cos(4*d*x + 4*c)^2 - 4*(8*a^2*b + 13*a*b^2 - 6*b^3)*sin(4*d*x + 4*c)^2 + 2*(4*a
*b^2 - 11*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^3*cos(6*d*x + 6*c) + b^3*cos(2*d*x + 2*c) - 2*(a*b^2 + 2
*b^3)*cos(4*d*x + 4*c))*cos(8*d*x + 8*c) + (8*b^3*cos(2*d*x + 2*c) - b^3 + 2*(4*a*b^2 - 11*b^3)*cos(4*d*x + 4*
c))*cos(6*d*x + 6*c) + 2*(a*b^2 + 2*b^3 + (4*a*b^2 - 11*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (b^3*sin(6*d
*x + 6*c) + b^3*sin(2*d*x + 2*c) - 2*(a*b^2 + 2*b^3)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 2*(4*b^3*sin(2*d*x +
2*c) + (4*a*b^2 - 11*b^3)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c))/(a^2*b^2 - 2*a*b^3 + b^4 + (a^2*b^2 - 2*a*b^3 +
b^4)*cos(8*d*x + 8*c)^2 + 16*(a^2*b^2 - 2*a*b^3 + b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a^4 - 176*a^3*b + 169*a^2*b
^2 - 66*a*b^3 + 9*b^4)*cos(4*d*x + 4*c)^2 + 16*(a^2*b^2 - 2*a*b^3 + b^4)*cos(2*d*x + 2*c)^2 + (a^2*b^2 - 2*a*b
^3 + b^4)*sin(8*d*x + 8*c)^2 + 16*(a^2*b^2 - 2*a*b^3 + b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^4 - 176*a^3*b + 169*a
^2*b^2 - 66*a*b^3 + 9*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a^3*b - 19*a^2*b^2 + 14*a*b^3 - 3*b^4)*sin(4*d*x + 4*c)*
sin(2*d*x + 2*c) + 16*(a^2*b^2 - 2*a*b^3 + b^4)*sin(2*d*x + 2*c)^2 + 2*(a^2*b^2 - 2*a*b^3 + b^4 - 4*(a^2*b^2 -
2*a*b^3 + b^4)*cos(6*d*x + 6*c) - 2*(8*a^3*b - 19*a^2*b^2 + 14*a*b^3 - 3*b^4)*cos(4*d*x + 4*c) - 4*(a^2*b^2 -
2*a*b^3 + b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^2*b^2 - 2*a*b^3 + b^4 - 2*(8*a^3*b - 19*a^2*b^2 + 14
*a*b^3 - 3*b^4)*cos(4*d*x + 4*c) - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^3*b
- 19*a^2*b^2 + 14*a*b^3 - 3*b^4 - 4*(8*a^3*b - 19*a^2*b^2 + 14*a*b^3 - 3*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4
*c) - 8*(a^2*b^2 - 2*a*b^3 + b^4)*cos(2*d*x + 2*c) - 4*(2*(a^2*b^2 - 2*a*b^3 + b^4)*sin(6*d*x + 6*c) + (8*a^3*
b - 19*a^2*b^2 + 14*a*b^3 - 3*b^4)*sin(4*d*x + 4*c) + 2*(a^2*b^2 - 2*a*b^3 + b^4)*sin(2*d*x + 2*c))*sin(8*d*x
+ 8*c) + 16*((8*a^3*b - 19*a^2*b^2 + 14*a*b^3 - 3*b^4)*sin(4*d*x + 4*c) + 2*(a^2*b^2 - 2*a*b^3 + b^4)*sin(2*d*
x + 2*c))*sin(6*d*x + 6*c)), x) + 4*(3*b*cos(4*d*x + 4*c) - 3*(a - 3*b)*cos(2*d*x + 2*c) - a + 4*b)*sin(6*d*x
+ 6*c) - 12*(3*(a - 2*b)*cos(2*d*x + 2*c) + a - 3*b)*sin(4*d*x + 4*c) + 12*b*sin(2*d*x + 2*c))/((a^2 - 2*a*b +
b^2)*d*cos(6*d*x + 6*c)^2 + 9*(a^2 - 2*a*b + b^2)*d*cos(4*d*x + 4*c)^2 + 9*(a^2 - 2*a*b + b^2)*d*cos(2*d*x +
2*c)^2 + (a^2 - 2*a*b + b^2)*d*sin(6*d*x + 6*c)^2 + 9*(a^2 - 2*a*b + b^2)*d*sin(4*d*x + 4*c)^2 + 18*(a^2 - 2*a
*b + b^2)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 9*(a^2 - 2*a*b + b^2)*d*sin(2*d*x + 2*c)^2 + 6*(a^2 - 2*a*b +
b^2)*d*cos(2*d*x + 2*c) + (a^2 - 2*a*b + b^2)*d + 2*(3*(a^2 - 2*a*b + b^2)*d*cos(4*d*x + 4*c) + 3*(a^2 - 2*a*b
+ b^2)*d*cos(2*d*x + 2*c) + (a^2 - 2*a*b + b^2)*d)*cos(6*d*x + 6*c) + 6*(3*(a^2 - 2*a*b + b^2)*d*cos(2*d*x +
2*c) + (a^2 - 2*a*b + b^2)*d)*cos(4*d*x + 4*c) + 6*((a^2 - 2*a*b + b^2)*d*sin(4*d*x + 4*c) + (a^2 - 2*a*b + b^
2)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4113 vs. \(2 (123) = 246\).
time = 0.92, size = 4113, normalized size = 25.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/24*(3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5
*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a
^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3
*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(5/4*a^2*
b^4 + 5/2*a*b^5 + 1/4*b^6 - 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 + 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2
+ 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^
8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b
^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b
^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*
a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^
12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^
4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b
- 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*
b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((
a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5
*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) - 3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 - (a^6 - 5*a^
5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8
+ b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^
7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*
d^2))*cos(d*x + c)^3*log(5/4*a^2*b^4 + 5/2*a*b^5 + 1/4*b^6 - 1/4*(5*a^2*b^4 + 10*a*b^5 + b^6)*cos(d*x + c)^2 -
1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 10
0*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*
a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (1
5*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 -
(a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7
+ 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 -
120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^
4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)
^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2)*sqrt((25*a^4*b^5 + 100*a^3*b^6 +
110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 2
10*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4))) + 3*(a^2 - 2*a*b + b^2)*d*sqrt(-(a^2*b^2
+ 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a^4*b^5 + 100
*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9*b^4 - 252*a
^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b + 10*a^4*b^2 -
10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2))*cos(d*x + c)^3*log(-5/4*a^2*b^4 - 5/2*a*b^5 - 1/4*b^6 + 1/4*(5*a^2*b^4
+ 10*a*b^5 + b^6)*cos(d*x + c)^2 + 1/2*((a^9 - 2*a^8*b - 5*a^7*b^2 + 20*a^6*b^3 - 25*a^5*b^4 + 14*a^4*b^5 - 3*
a^3*b^6)*d^3*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 -
120*a^10*b^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^
4))*cos(d*x + c)*sin(d*x + c) - (15*a^4*b^3 + 35*a^3*b^4 + 13*a^2*b^5 + a*b^6)*d*cos(d*x + c)*sin(d*x + c))*sq
rt(-(a^2*b^2 + 10*a*b^3 + 5*b^4 + (a^6 - 5*a^5*b + 10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2*sqrt((25*a
^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b^3 + 210*a^9
*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - 120*a^6*b^7 + 45*a^5*b^8 - 10*a^4*b^9 + a^3*b^10)*d^4)))/((a^6 - 5*a^5*b +
10*a^4*b^2 - 10*a^3*b^3 + 5*a^2*b^4 - a*b^5)*d^2)) - 1/4*(2*(a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a
^3*b^5 - a^2*b^6)*d^2*cos(d*x + c)^2 - (a^7*b - 5*a^6*b^2 + 10*a^5*b^3 - 10*a^4*b^4 + 5*a^3*b^5 - a^2*b^6)*d^2
)*sqrt((25*a^4*b^5 + 100*a^3*b^6 + 110*a^2*b^7 + 20*a*b^8 + b^9)/((a^13 - 10*a^12*b + 45*a^11*b^2 - 120*a^10*b
^3 + 210*a^9*b^4 - 252*a^8*b^5 + 210*a^7*b^6 - ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{4}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(sec(c + d*x)**4/(a - b*sin(c + d*x)**4), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2183 vs. \(2 (123) = 246\).
time = 0.90, size = 2183, normalized size = 13.56 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

1/6*(2*(a^2*tan(d*x + c)^3 - 2*a*b*tan(d*x + c)^3 + b^2*tan(d*x + c)^3 + 3*a^2*tan(d*x + c) - 12*a*b*tan(d*x +
c) + 9*b^2*tan(d*x + c))/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - 3*((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b
)*a^3*b + 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*a*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)^2*abs(-a +
b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b^2 + 23*sqrt(
a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^3 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^4 - 23*sqrt(a^2 - a*b + s
qrt(a*b)*(a - b))*a^3*b^5 + 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^2*b^6 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*a*b^7 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^8)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*abs(-a + b) - (9*sqrt
(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b - 69*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 2
16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 352*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a
^6*b^4 + 306*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 114*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*s
qrt(a*b)*a^4*b^6 - 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + 24*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*a^2*b^8 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^9 - sqrt(a^2 - a*b + sqrt(a*b)*
(a - b))*sqrt(a*b)*b^10)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 3*a^3*b
+ 3*a^2*b^2 - a*b^3 + sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)^2 - (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^4 -
4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4))))/((3*a^11 - 30*a^10*b + 1
31*a^9*b^2 - 328*a^8*b^3 + 518*a^7*b^4 - 532*a^6*b^5 + 350*a^5*b^6 - 136*a^4*b^7 + 23*a^3*b^8 + 2*a^2*b^9 - a*
b^10)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)) + 3*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b + 3*sqrt
(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^2 - 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^3 - 3
*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^4)*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)^2*abs(-a + b) + (3*sqrt(a^
2 - a*b - sqrt(a*b)*(a - b))*a^7*b - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^2 + 23*sqrt(a^2 - a*b - sqrt
(a*b)*(a - b))*a^5*b^3 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^4*b^4 - 23*sqrt(a^2 - a*b - sqrt(a*b)*(a - b)
)*a^3*b^5 + 19*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^6 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^7 - sqr
t(a^2 - a*b - sqrt(a*b)*(a - b))*b^8)*abs(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b - sqr
t(a*b)*(a - b))*sqrt(a*b)*a^9*b - 69*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^2 + 216*sqrt(a^2 - a*
b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^3 - 352*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^4 + 306*sqr
t(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^5 - 114*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^6
- 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^7 + 24*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)
*a^2*b^8 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^9 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*
b)*b^10)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*
b^3 - sqrt((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)^2 - (a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(a^4 - 4*a^3*b + 6*a^2*
b^2 - 4*a*b^3 + b^4)))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4))))/((3*a^11 - 30*a^10*b + 131*a^9*b^2 - 328
*a^8*b^3 + 518*a^7*b^4 - 532*a^6*b^5 + 350*a^5*b^6 - 136*a^4*b^7 + 23*a^3*b^8 + 2*a^2*b^9 - a*b^10)*abs(a^3 -
3*a^2*b + 3*a*b^2 - b^3)))/d

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Mupad [B]
time = 17.74, size = 2500, normalized size = 15.53 \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)^4*(a - b*sin(c + d*x)^4)),x)

[Out]

tan(c + d*x)^3/(3*d*(a - b)) - (tan(c + d*x)*((2*a)/(a - b)^2 - 3/(a - b)))/d + (atan(((((16*a*b^6 - 32*a^2*b^
5 + 32*a^4*b^3 - 16*a^5*b^2)/(3*a*b^2 - 3*a^2*b + a^3 - b^3) - (4*tan(c + d*x)*((5*a^2*(a^3*b^5)^(1/2) + b^2*(
a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a
^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*(16*a^7*b - 16*a^2*b^6 + 80*a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80
*a^6*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3
*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))
^(1/2) - (4*tan(c + d*x)*(15*a*b^5 + b^6 + 15*a^2*b^4 + a^3*b^3))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^
3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b -
a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*1i - (((16*a*b^6 - 32*a^2*b^5 + 32*a^4*b^3 - 16*
a^5*b^2)/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (4*tan(c + d*x)*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a
^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3
- 10*a^6*b^2)))^(1/2)*(16*a^7*b - 16*a^2*b^6 + 80*a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80*a^6*b^2))/(3*a*b^2
- 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*
a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2) + (4*tan(c +
d*x)*(15*a*b^5 + b^6 + 15*a^2*b^4 + a^3*b^3))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(
a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a
^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*1i)/((((16*a*b^6 - 32*a^2*b^5 + 32*a^4*b^3 - 16*a^5*b^2)/(3*a*b^2 -
3*a^2*b + a^3 - b^3) - (4*tan(c + d*x)*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3
+ a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2
)*(16*a^7*b - 16*a^2*b^6 + 80*a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80*a^6*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b^
3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))
/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2) - (4*tan(c + d*x)*(15*a*b^5 + b^6
+ 15*a^2*b^4 + a^3*b^3))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a
^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3
- 10*a^6*b^2)))^(1/2) - (2*(a*b^4 + 3*b^5))/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (((16*a*b^6 - 32*a^2*b^5 + 32*a^
4*b^3 - 16*a^5*b^2)/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (4*tan(c + d*x)*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^
(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 +
10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*(16*a^7*b - 16*a^2*b^6 + 80*a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80*a^6*b^2)
)/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^
4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2) +
(4*tan(c + d*x)*(15*a*b^5 + b^6 + 15*a^2*b^4 + a^3*b^3))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*((5*a^2*(a^3*b^5)^(1
/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^
3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)))*((5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) + 5*a^2*b
^4 + 10*a^3*b^3 + a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10
*a^6*b^2)))^(1/2)*2i)/d + (atan(((((16*a*b^6 - 32*a^2*b^5 + 32*a^4*b^3 - 16*a^5*b^2)/(3*a*b^2 - 3*a^2*b + a^3
- b^3) - (4*tan(c + d*x)*(-(5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) - 5*a^2*b^4 - 10*a^3*b^3 - a^4*b^2 + 1
0*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*(16*a^7*b -
16*a^2*b^6 + 80*a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80*a^6*b^2))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*(-(5*a^2*
(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) - 5*a^2*b^4 - 10*a^3*b^3 - a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*
b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2) - (4*tan(c + d*x)*(15*a*b^5 + b^6 + 15*a^2*b^
4 + a^3*b^3))/(3*a*b^2 - 3*a^2*b + a^3 - b^3))*(-(5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) - 5*a^2*b^4 - 10
*a^3*b^3 - a^4*b^2 + 10*a*b*(a^3*b^5)^(1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^
2)))^(1/2)*1i - (((16*a*b^6 - 32*a^2*b^5 + 32*a^4*b^3 - 16*a^5*b^2)/(3*a*b^2 - 3*a^2*b + a^3 - b^3) + (4*tan(c
+ d*x)*(-(5*a^2*(a^3*b^5)^(1/2) + b^2*(a^3*b^5)^(1/2) - 5*a^2*b^4 - 10*a^3*b^3 - a^4*b^2 + 10*a*b*(a^3*b^5)^(
1/2))/(16*(5*a^7*b - a^8 + a^3*b^5 - 5*a^4*b^4 + 10*a^5*b^3 - 10*a^6*b^2)))^(1/2)*(16*a^7*b - 16*a^2*b^6 + 80*
a^3*b^5 - 160*a^4*b^4 + 160*a^5*b^3 - 80*a^6*b^...

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