\(\int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [416]

Optimal result

Integrand size = 24, antiderivative size = 142 \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} d}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} d}+\frac {\tan (c+d x)}{(a-b) d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3303, 1184, 1180, 211} \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}+\frac {\tan (c+d x)}{d (a-b)} \]

[In]

Int[Sec[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]

[Out]

-1/2*(Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(a^(3/4)*(Sqrt[a] - Sqrt[b])^(3/2)*d) +
(Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + Tan
[c + d*x]/((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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{a-b}-\frac {b \left (1+2 x^2\right )}{(a-b) \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {\tan (c+d x)}{(a-b) d}-\frac {b \text {Subst}\left (\int \frac {1+2 x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{(a-b) d} \\ & = \frac {\tan (c+d x)}{(a-b) d}-\frac {\left (\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {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) d}-\frac {\left (b \left (2-\frac {a+b}{\sqrt {a} \sqrt {b}}\right )\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 (a-b) d} \\ & = -\frac {\sqrt {b} \arctan \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 )^{3/2} d}+\frac {\sqrt {b} \arctan \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 )^{3/2} d}+\frac {\tan (c+d x)}{(a-b) d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.82 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.23 \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {\left (\sqrt {a} \sqrt {b}-b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\left (\sqrt {a} \sqrt {b}+b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+2 \tan (c+d x)}{2 (a-b) d} \]

[In]

Integrate[Sec[c + d*x]^2/(a - b*Sin[c + d*x]^4),x]

[Out]

(((Sqrt[a]*Sqrt[b] - b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a
+ Sqrt[a]*Sqrt[b]]) + ((Sqrt[a]*Sqrt[b] + b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt
[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + 2*Tan[c + d*x])/(2*(a - b)*d)

Maple [A] (verified)

Time = 1.47 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.17

method	result	size
derivativedivides	\(\frac {\frac {\tan \left (d x +c \right )}{a -b}-b \left (\frac {\left (-a -b +2 \sqrt {a b}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a +b +2 \sqrt {a b}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d}\)	\(166\)
default	\(\frac {\frac {\tan \left (d x +c \right )}{a -b}-b \left (\frac {\left (-a -b +2 \sqrt {a b}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (a +b +2 \sqrt {a b}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{d}\)	\(166\)
risch	\(\frac {2 i}{d \left (a -b \right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+4 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (65536 a^{6} d^{4}-196608 a^{5} b \,d^{4}+196608 a^{4} b^{2} d^{4}-65536 a^{3} b^{3} d^{4}\right ) \textit {\_Z}^{4}+\left (512 a^{3} b \,d^{2}+1536 a^{2} b^{2} d^{2}\right ) \textit {\_Z}^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {16384 i d^{3} a^{6}}{3 a \,b^{2}+b^{3}}-\frac {49152 i a^{5} b \,d^{3}}{3 a \,b^{2}+b^{3}}+\frac {49152 i a^{4} b^{2} d^{3}}{3 a \,b^{2}+b^{3}}-\frac {16384 i a^{3} b^{3} d^{3}}{3 a \,b^{2}+b^{3}}\right ) \textit {\_R}^{3}+\left (-\frac {512 d^{2} a^{5}}{3 a \,b^{2}+b^{3}}+\frac {1536 a^{4} b \,d^{2}}{3 a \,b^{2}+b^{3}}-\frac {1536 a^{3} b^{2} d^{2}}{3 a \,b^{2}+b^{3}}+\frac {512 a^{2} b^{3} d^{2}}{3 a \,b^{2}+b^{3}}\right ) \textit {\_R}^{2}+\left (\frac {160 i d \,a^{3} b}{3 a \,b^{2}+b^{3}}+\frac {320 i a^{2} b^{2} d}{3 a \,b^{2}+b^{3}}+\frac {32 i a \,b^{3} d}{3 a \,b^{2}+b^{3}}\right ) \textit {\_R} -\frac {2 a^{2} b}{3 a \,b^{2}+b^{3}}-\frac {9 a \,b^{2}}{3 a \,b^{2}+b^{3}}-\frac {b^{3}}{3 a \,b^{2}+b^{3}}\right )\right )\)	\(426\)

[In]

int(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2589 vs. \(2 (102) = 204\).

Time = 0.67 (sec) , antiderivative size = 2589, normalized size of antiderivative = 18.23 \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="fricas")

[Out]

1/8*((a - b)*d*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b
+ 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2
- a*b^3)*d^2))*cos(d*x + c)*log(3/4*a*b^2 + 1/4*b^3 - 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 + 1/2*(2*(a^6 - 3*a^
5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15
*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)
*sin(d*x + c))*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b
+ 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2
- a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2
- a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a
^4*b^5 + a^3*b^6)*d^4))) - (a - b)*d*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 +
b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4
- 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(3/4*a*b^2 + 1/4*b^3 - 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^
2 - 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7
*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (3*a^3*b + 4*a^2*b^2 +
a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 +
b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4)) - a*b - 3*b^2)/((a^4
- 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5
- 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*
b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) + (a - b)*d*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sqrt
((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4
)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(-3/4*a*b^2 - 1/4*b^3 + 1/4*(3*a*
b^2 + b^3)*cos(d*x + c)^2 + 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/
((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*cos(d*x + c)*sin(d*x + c)
- (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2*sq
rt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d
^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*
d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8
*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) - (a - b)*d*sqrt(-((a^4 - 3*a^3*b + 3*
a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 -
6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2))*cos(d*x + c)*log(-3/4*a*
b^2 - 1/4*b^3 + 1/4*(3*a*b^2 + b^3)*cos(d*x + c)^2 - 1/2*(2*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^3*sqrt((9*
a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))*c
os(d*x + c)*sin(d*x + c) - (3*a^3*b + 4*a^2*b^2 + a*b^3)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^4 - 3*a^3*b +
3*a^2*b^2 - a*b^3)*d^2*sqrt((9*a^2*b^3 + 6*a*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4
- 6*a^4*b^5 + a^3*b^6)*d^4)) + a*b + 3*b^2)/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*d^2)) - 1/4*(2*(a^5 - 3*a^4*
b + 3*a^3*b^2 - a^2*b^3)*d^2*cos(d*x + c)^2 - (a^5 - 3*a^4*b + 3*a^3*b^2 - a^2*b^3)*d^2)*sqrt((9*a^2*b^3 + 6*a
*b^4 + b^5)/((a^9 - 6*a^8*b + 15*a^7*b^2 - 20*a^6*b^3 + 15*a^5*b^4 - 6*a^4*b^5 + a^3*b^6)*d^4))) + 8*sin(d*x +
c))/((a - b)*d*cos(d*x + c))

Sympy [F]

\[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx \]

[In]

integrate(sec(d*x+c)**2/(a-b*sin(d*x+c)**4),x)

[Out]

Integral(sec(c + d*x)**2/(a - b*sin(c + d*x)**4), x)

Maxima [F]

\[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\sec \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{4} - a} \,d x } \]

[In]

integrate(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="maxima")

[Out]

(((a - b)*d*cos(2*d*x + 2*c)^2 + (a - b)*d*sin(2*d*x + 2*c)^2 + 2*(a - b)*d*cos(2*d*x + 2*c) + (a - b)*d)*inte
grate(4*(4*b^2*cos(6*d*x + 6*c)^2 + 4*b^2*cos(2*d*x + 2*c)^2 + 4*b^2*sin(6*d*x + 6*c)^2 + 4*b^2*sin(2*d*x + 2*
c)^2 - 12*(8*a*b - 3*b^2)*cos(4*d*x + 4*c)^2 - b^2*cos(2*d*x + 2*c) - 12*(8*a*b - 3*b^2)*sin(4*d*x + 4*c)^2 +
2*(8*a*b - 15*b^2)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) - (b^2*cos(6*d*x + 6*c) - 6*b^2*cos(4*d*x + 4*c) + b^2*co
s(2*d*x + 2*c))*cos(8*d*x + 8*c) + (8*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*b - 15*b^2)*cos(4*d*x + 4*c))*cos(6*
d*x + 6*c) + 2*(3*b^2 + (8*a*b - 15*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (b^2*sin(6*d*x + 6*c) - 6*b^2*si
n(4*d*x + 4*c) + b^2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(4*b^2*sin(2*d*x + 2*c) + (8*a*b - 15*b^2)*sin(4*d
*x + 4*c))*sin(6*d*x + 6*c))/(a*b^2 - b^3 + (a*b^2 - b^3)*cos(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*cos(6*d*x + 6*
c)^2 + 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^3)*cos(4*d*x + 4*c)^2 + 16*(a*b^2 - b^3)*cos(2*d*x + 2*c)^2 + (a
*b^2 - b^3)*sin(8*d*x + 8*c)^2 + 16*(a*b^2 - b^3)*sin(6*d*x + 6*c)^2 + 4*(64*a^3 - 112*a^2*b + 57*a*b^2 - 9*b^
3)*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a*b^2 - b^3)*s
in(2*d*x + 2*c)^2 + 2*(a*b^2 - b^3 - 4*(a*b^2 - b^3)*cos(6*d*x + 6*c) - 2*(8*a^2*b - 11*a*b^2 + 3*b^3)*cos(4*d
*x + 4*c) - 4*(a*b^2 - b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a*b^2 - b^3 - 2*(8*a^2*b - 11*a*b^2 + 3*b^
3)*cos(4*d*x + 4*c) - 4*(a*b^2 - b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^2*b - 11*a*b^2 + 3*b^3 - 4*(
8*a^2*b - 11*a*b^2 + 3*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a*b^2 - b^3)*cos(2*d*x + 2*c) - 4*(2*(a*b^
2 - b^3)*sin(6*d*x + 6*c) + (8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*sin(2*d*x + 2*c))*
sin(8*d*x + 8*c) + 16*((8*a^2*b - 11*a*b^2 + 3*b^3)*sin(4*d*x + 4*c) + 2*(a*b^2 - b^3)*sin(2*d*x + 2*c))*sin(6
*d*x + 6*c)), x) + 2*sin(2*d*x + 2*c))/((a - b)*d*cos(2*d*x + 2*c)^2 + (a - b)*d*sin(2*d*x + 2*c)^2 + 2*(a - b
)*d*cos(2*d*x + 2*c) + (a - b)*d)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1211 vs. \(2 (102) = 204\).

Time = 0.84 (sec) , antiderivative size = 1211, normalized size of antiderivative = 8.53 \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(sec(d*x+c)^2/(a-b*sin(d*x+c)^4),x, algorithm="giac")

[Out]

-1/2*((3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a
^4*b + 2*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a
*b)*a^2*b^3 - 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt
(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*s
qrt(a*b)*a*b^2 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 + (3*sqrt(a^2 - a*b - sqrt(a*b)*
(a - b))*a^4*b - 12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b
^3 - 4*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^5)*abs(-a + b))*(pi*f
loor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b + sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2 - 2*a*b
+ b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8 - 21*a^7*b + 59*a^6*b^2 - 85*a^5*b^3 + 65*a^4*b^4 - 23*a^3*b^5 + a^2*b
^6 + a*b^7) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5 - 9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*a^4*b + 2*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^2 + 10*sqrt(a^2 - a*b + sqrt(a*b)*(a - b)
)*sqrt(a*b)*a^2*b^3 - 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*b^5 - 2*(3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b - 6*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*a*b^2 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a - b)^2 - (3*sqrt(a^2 - a*b + sq
rt(a*b)*(a - b))*a^4*b - 12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^2 + 14*sqrt(a^2 - a*b + sqrt(a*b)*(a - b
))*a^2*b^3 - 4*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^4 - sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*b^5)*abs(-a + b
))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^2 - a*b - sqrt((a^2 - a*b)^2 - (a^2 - a*b)*(a^2
- 2*a*b + b^2)))/(a^2 - 2*a*b + b^2))))/(3*a^8 - 21*a^7*b + 59*a^6*b^2 - 85*a^5*b^3 + 65*a^4*b^4 - 23*a^3*b^5
+ a^2*b^6 + a*b^7) - 2*tan(d*x + c)/(a - b))/d

Mupad [B] (verification not implemented)

Time = 15.54 (sec) , antiderivative size = 2832, normalized size of antiderivative = 19.94 \[ \int \frac {\sec ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^2*(a - b*sin(c + d*x)^4)),x)

[Out]

tan(c + d*x)/(d*(a - b)) + (atan(((((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b^2))/(a - b) - (4*tan(c + d*x)*((3*a*(a^
3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^
5*b - 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^
2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) - (4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))
*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1
/2)*1i - (((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b^2))/(a - b) + (4*tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3
)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b - 16*a^2*b^4 + 48*a^3
*b^3 - 48*a^4*b^2))/(a - b))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6
+ a^3*b^3 - 3*a^4*b^2)))^(1/2) + (4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*((3*a*(a^3*b^3)^(1/2) +
b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*1i)/((((2*(8*a*b^4 -
16*a^2*b^3 + 8*a^3*b^2))/(a - b) - (4*tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b
^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b - 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a -
b))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))
^(1/2) - (4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b
+ 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) - (4*b^3)/(a - b) + (((2*(8*a*b^4 - 16*a^2*b^3
+ 8*a^3*b^2))/(a - b) + (4*tan(c + d*x)*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3
*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b - 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*((3*a*
(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) + (
4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*a^2*b
^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)))*((3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) + a^3*b + 3*
a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*2i)/d + (atan(((((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b
^2))/(a - b) - (4*tan(c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b -
a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b - 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*(-(3*a*(a^3*b^3
)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) - (4*tan(c
+ d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16
*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*1i - (((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b^2))/(a - b) + (4*tan(
c + d*x)*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*
b^2)))^(1/2)*(16*a^5*b - 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^
(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) + (4*tan(c + d*x)*(6*a*b^3 + b^4
+ a^2*b^2))/(a - b))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*
b^3 - 3*a^4*b^2)))^(1/2)*1i)/((((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b^2))/(a - b) - (4*tan(c + d*x)*(-(3*a*(a^3*b
^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b
- 16*a^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*
b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) - (4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*(
-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/
2) - (4*b^3)/(a - b) + (((2*(8*a*b^4 - 16*a^2*b^3 + 8*a^3*b^2))/(a - b) + (4*tan(c + d*x)*(-(3*a*(a^3*b^3)^(1/
2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)*(16*a^5*b - 16*a
^2*b^4 + 48*a^3*b^3 - 48*a^4*b^2))/(a - b))*(-(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(1
6*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2) + (4*tan(c + d*x)*(6*a*b^3 + b^4 + a^2*b^2))/(a - b))*(-(3*a*(
a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2)))*(-
(3*a*(a^3*b^3)^(1/2) + b*(a^3*b^3)^(1/2) - a^3*b - 3*a^2*b^2)/(16*(3*a^5*b - a^6 + a^3*b^3 - 3*a^4*b^2)))^(1/2
)*2i)/d