3.62 \(\int \frac {\sin ^2(e+f x)}{(a+b \sec ^2(e+f x))^3} \, dx\)

Optimal. Leaf size=184 \[ \frac {x (a+6 b)}{2 a^4}-\frac {b (11 a+12 b) \tan (e+f x)}{8 a^3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^4 f (a+b)^{3/2}}-\frac {\sin (e+f x) \cos (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]

[Out]

1/2*(a+6*b)*x/a^4-1/8*(15*a^2+40*a*b+24*b^2)*arctan(b^(1/2)*tan(f*x+e)/(a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(3/2)/f-
1/2*cos(f*x+e)*sin(f*x+e)/a/f/(a+b+b*tan(f*x+e)^2)^2-3/4*b*tan(f*x+e)/a^2/f/(a+b+b*tan(f*x+e)^2)^2-1/8*b*(11*a
+12*b)*tan(f*x+e)/a^3/(a+b)/f/(a+b+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.28, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4132, 471, 527, 522, 203, 205} \[ -\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^4 f (a+b)^{3/2}}-\frac {b (11 a+12 b) \tan (e+f x)}{8 a^3 f (a+b) \left (a+b \tan ^2(e+f x)+b\right )}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {x (a+6 b)}{2 a^4}-\frac {\sin (e+f x) \cos (e+f x)}{2 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

((a + 6*b)*x)/(2*a^4) - (Sqrt[b]*(15*a^2 + 40*a*b + 24*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(8*a^4
*(a + b)^(3/2)*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*a*f*(a + b + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*a
^2*f*(a + b + b*Tan[e + f*x]^2)^2) - (b*(11*a + 12*b)*Tan[e + f*x])/(8*a^3*(a + b)*f*(a + b + b*Tan[e + f*x]^2
))

Rule 203

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

Rule 205

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

Rule 471

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e^(n -
1)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(n*(b*c - a*d)*(p + 1)), x] - Dist[e^n/(n*(b*c -
a*d)*(p + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(m - n + 1) + d*(m + n*(p + q + 1)
+ 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && GeQ[n
, m - n + 1] && GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 522

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[(b*e - a*f
)/(b*c - a*d), Int[1/(a + b*x^n), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[1/(c + d*x^n), x], x] /; FreeQ[{a
, b, c, d, e, f, n}, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 4132

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = Fr
eeFactors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p)/(
1 + ff^2*x^2)^(m/2 + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && Integer
Q[n/2]

Rubi steps

\begin {align*} \int \frac {\sin ^2(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {a+b-5 b x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{2 a f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {2 (a+b) (2 a+3 b)-18 b (a+b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 a^2 (a+b) f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+12 b) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 (a+b) \left (4 a^2+17 a b+12 b^2\right )-2 b (a+b) (11 a+12 b) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{16 a^3 (a+b)^2 f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+12 b) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {(a+6 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{2 a^4 f}-\frac {\left (b \left (15 a^2+40 a b+24 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^4 (a+b) f}\\ &=\frac {(a+6 b) x}{2 a^4}-\frac {\sqrt {b} \left (15 a^2+40 a b+24 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^4 (a+b)^{3/2} f}-\frac {\cos (e+f x) \sin (e+f x)}{2 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+12 b) \tan (e+f x)}{8 a^3 (a+b) f \left (a+b+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [C]  time = 17.64, size = 1915, normalized size = 10.41 \[ \text {result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[e + f*x]^2/(a + b*Sec[e + f*x]^2)^3,x]

[Out]

(5*(a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(((3*a^2 + 8*a*b + 8*b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqr
t[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cos[2*(e + f*x)])*Sin[2*(e + f*
x)])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(8192*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) + ((a + 2*b + a*
Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*((-3*a*(a + 2*b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b]])/(a + b)^(5/2)
+ (Sqrt[b]*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 + a*(3*a^2 + 4*a*b + 4*b^2)*Cos[2*(e + f*x)])*Sin[2*(e + f*x)
])/((a + b)^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(2048*b^(5/2)*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Co
s[2*e + 2*f*x])^3*Sec[e + f*x]^6*((2*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*ArcTa
n[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e]
- I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (Sec[2*e]*(256*b^2*(a
+ b)^2*(3*a^2 + 8*a*b + 8*b^2)*f*x*Cos[2*e] + 512*a*b^2*(a + b)^2*(a + 2*b)*f*x*Cos[2*f*x] + 128*a^4*b^2*f*x*C
os[2*(e + 2*f*x)] + 256*a^3*b^3*f*x*Cos[2*(e + 2*f*x)] + 128*a^2*b^4*f*x*Cos[2*(e + 2*f*x)] + 512*a^4*b^2*f*x*
Cos[4*e + 2*f*x] + 2048*a^3*b^3*f*x*Cos[4*e + 2*f*x] + 2560*a^2*b^4*f*x*Cos[4*e + 2*f*x] + 1024*a*b^5*f*x*Cos[
4*e + 2*f*x] + 128*a^4*b^2*f*x*Cos[6*e + 4*f*x] + 256*a^3*b^3*f*x*Cos[6*e + 4*f*x] + 128*a^2*b^4*f*x*Cos[6*e +
4*f*x] - 9*a^6*Sin[2*e] + 12*a^5*b*Sin[2*e] + 684*a^4*b^2*Sin[2*e] + 2880*a^3*b^3*Sin[2*e] + 5280*a^2*b^4*Sin
[2*e] + 4608*a*b^5*Sin[2*e] + 1536*b^6*Sin[2*e] + 9*a^6*Sin[2*f*x] - 14*a^5*b*Sin[2*f*x] - 608*a^4*b^2*Sin[2*f
*x] - 2112*a^3*b^3*Sin[2*f*x] - 2560*a^2*b^4*Sin[2*f*x] - 1024*a*b^5*Sin[2*f*x] + 3*a^6*Sin[2*(e + 2*f*x)] - 1
2*a^5*b*Sin[2*(e + 2*f*x)] - 204*a^4*b^2*Sin[2*(e + 2*f*x)] - 384*a^3*b^3*Sin[2*(e + 2*f*x)] - 192*a^2*b^4*Sin
[2*(e + 2*f*x)] - 3*a^6*Sin[4*e + 2*f*x] + 10*a^5*b*Sin[4*e + 2*f*x] + 304*a^4*b^2*Sin[4*e + 2*f*x] + 1056*a^3
*b^3*Sin[4*e + 2*f*x] + 1280*a^2*b^4*Sin[4*e + 2*f*x] + 512*a*b^5*Sin[4*e + 2*f*x]))/(a + 2*b + a*Cos[2*(e + f
*x)])^2))/(4096*a^3*b^2*(a + b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]
^6*((-6*a^2*ArcTan[(Sec[f*x]*(Cos[2*e] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b
]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Cos[2*e] - I*Sin[2*e]))/(Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (a*Se
c[2*e]*((-9*a^4 - 16*a^3*b + 48*a^2*b^2 + 128*a*b^3 + 64*b^4)*Sin[2*f*x] + a*(-3*a^3 + 2*a^2*b + 24*a*b^2 + 16
*b^3)*Sin[2*(e + 2*f*x)] + (3*a^4 - 64*a^2*b^2 - 128*a*b^3 - 64*b^4)*Sin[4*e + 2*f*x]) + (9*a^5 + 18*a^4*b - 6
4*a^3*b^2 - 256*a^2*b^3 - 320*a*b^4 - 128*b^5)*Tan[2*e])/(a^2*(a + 2*b + a*Cos[2*(e + f*x)])^2)))/(4096*b^2*(a
+ b)^2*f*(a + b*Sec[e + f*x]^2)^3) - ((a + 2*b + a*Cos[2*e + 2*f*x])^3*Sec[e + f*x]^6*(-1536*(a + 2*b)*x - (3
*(a^6 - 8*a^5*b + 120*a^4*b^2 + 1280*a^3*b^3 + 3200*a^2*b^4 + 3072*a*b^5 + 1024*b^6)*ArcTan[(Sec[f*x]*(Cos[2*e
] - I*Sin[2*e])*(-((a + 2*b)*Sin[f*x]) + a*Sin[2*e + f*x]))/(2*Sqrt[a + b]*Sqrt[b*(Cos[e] - I*Sin[e])^4])]*(Co
s[2*e] - I*Sin[2*e]))/(b^2*(a + b)^(5/2)*f*Sqrt[b*(Cos[e] - I*Sin[e])^4]) + (4*(a^4 + 32*a^3*b + 160*a^2*b^2 +
256*a*b^3 + 128*b^4)*Sec[2*e]*((a + 2*b)*Sin[2*e] - a*Sin[2*f*x]))/(b*(a + b)*f*(a + 2*b + a*Cos[2*(e + f*x)]
)^2) + (256*a*Sin[2*(e + f*x)])/f + (a*(-3*a^5 + 26*a^4*b + 736*a^3*b^2 + 2624*a^2*b^3 + 3200*a*b^4 + 1280*b^5
)*Sec[2*e]*Sin[2*f*x] + (3*a^6 - 24*a^5*b - 920*a^4*b^2 - 4864*a^3*b^3 - 10112*a^2*b^4 - 9216*a*b^5 - 3072*b^6
)*Tan[2*e])/(b^2*(a + b)^2*f*(a + 2*b + a*Cos[2*(e + f*x)]))))/(8192*a^4*(a + b*Sec[e + f*x]^2)^3)

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fricas [B]  time = 0.70, size = 815, normalized size = 4.43 \[ \left [\frac {16 \, {\left (a^{4} + 7 \, a^{3} b + 6 \, a^{2} b^{2}\right )} f x \cos \left (f x + e\right )^{4} + 32 \, {\left (a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 16 \, {\left (a^{2} b^{2} + 7 \, a b^{3} + 6 \, b^{4}\right )} f x + {\left ({\left (15 \, a^{4} + 40 \, a^{3} b + 24 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 24 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 40 \, a^{2} b^{2} + 24 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-\frac {b}{a + b}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (a b + b^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-\frac {b}{a + b}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) - 4 \, {\left (4 \, {\left (a^{4} + a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (17 \, a^{3} b + 18 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (11 \, a^{2} b^{2} + 12 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{32 \, {\left ({\left (a^{7} + a^{6} b\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}, \frac {8 \, {\left (a^{4} + 7 \, a^{3} b + 6 \, a^{2} b^{2}\right )} f x \cos \left (f x + e\right )^{4} + 16 \, {\left (a^{3} b + 7 \, a^{2} b^{2} + 6 \, a b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 8 \, {\left (a^{2} b^{2} + 7 \, a b^{3} + 6 \, b^{4}\right )} f x + {\left ({\left (15 \, a^{4} + 40 \, a^{3} b + 24 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, a^{2} b^{2} + 40 \, a b^{3} + 24 \, b^{4} + 2 \, {\left (15 \, a^{3} b + 40 \, a^{2} b^{2} + 24 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {\frac {b}{a + b}} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {b}{a + b}}}{2 \, b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) - 2 \, {\left (4 \, {\left (a^{4} + a^{3} b\right )} \cos \left (f x + e\right )^{5} + {\left (17 \, a^{3} b + 18 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{3} + {\left (11 \, a^{2} b^{2} + 12 \, a b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{16 \, {\left ({\left (a^{7} + a^{6} b\right )} f \cos \left (f x + e\right )^{4} + 2 \, {\left (a^{6} b + a^{5} b^{2}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{5} b^{2} + a^{4} b^{3}\right )} f\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm="fricas")

[Out]

[1/32*(16*(a^4 + 7*a^3*b + 6*a^2*b^2)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 7*a^2*b^2 + 6*a*b^3)*f*x*cos(f*x + e)^2
+ 16*(a^2*b^2 + 7*a*b^3 + 6*b^4)*f*x + ((15*a^4 + 40*a^3*b + 24*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b
^3 + 24*b^4 + 2*(15*a^3*b + 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^2)*sqrt(-b/(a + b))*log(((a^2 + 8*a*b + 8*b^2)
*cos(f*x + e)^4 - 2*(3*a*b + 4*b^2)*cos(f*x + e)^2 + 4*((a^2 + 3*a*b + 2*b^2)*cos(f*x + e)^3 - (a*b + b^2)*cos
(f*x + e))*sqrt(-b/(a + b))*sin(f*x + e) + b^2)/(a^2*cos(f*x + e)^4 + 2*a*b*cos(f*x + e)^2 + b^2)) - 4*(4*(a^4
+ a^3*b)*cos(f*x + e)^5 + (17*a^3*b + 18*a^2*b^2)*cos(f*x + e)^3 + (11*a^2*b^2 + 12*a*b^3)*cos(f*x + e))*sin(
f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 + 2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f), 1/1
6*(8*(a^4 + 7*a^3*b + 6*a^2*b^2)*f*x*cos(f*x + e)^4 + 16*(a^3*b + 7*a^2*b^2 + 6*a*b^3)*f*x*cos(f*x + e)^2 + 8*
(a^2*b^2 + 7*a*b^3 + 6*b^4)*f*x + ((15*a^4 + 40*a^3*b + 24*a^2*b^2)*cos(f*x + e)^4 + 15*a^2*b^2 + 40*a*b^3 + 2
4*b^4 + 2*(15*a^3*b + 40*a^2*b^2 + 24*a*b^3)*cos(f*x + e)^2)*sqrt(b/(a + b))*arctan(1/2*((a + 2*b)*cos(f*x + e
)^2 - b)*sqrt(b/(a + b))/(b*cos(f*x + e)*sin(f*x + e))) - 2*(4*(a^4 + a^3*b)*cos(f*x + e)^5 + (17*a^3*b + 18*a
^2*b^2)*cos(f*x + e)^3 + (11*a^2*b^2 + 12*a*b^3)*cos(f*x + e))*sin(f*x + e))/((a^7 + a^6*b)*f*cos(f*x + e)^4 +
2*(a^6*b + a^5*b^2)*f*cos(f*x + e)^2 + (a^5*b^2 + a^4*b^3)*f)]

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giac [A]  time = 0.86, size = 219, normalized size = 1.19 \[ -\frac {\frac {{\left (15 \, a^{2} b + 40 \, a b^{2} + 24 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{{\left (a^{5} + a^{4} b\right )} \sqrt {a b + b^{2}}} + \frac {7 \, a b^{2} \tan \left (f x + e\right )^{3} + 8 \, b^{3} \tan \left (f x + e\right )^{3} + 9 \, a^{2} b \tan \left (f x + e\right ) + 17 \, a b^{2} \tan \left (f x + e\right ) + 8 \, b^{3} \tan \left (f x + e\right )}{{\left (a^{4} + a^{3} b\right )} {\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2}} - \frac {4 \, {\left (f x + e\right )} {\left (a + 6 \, b\right )}}{a^{4}} + \frac {4 \, \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )} a^{3}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm="giac")

[Out]

-1/8*((15*a^2*b + 40*a*b^2 + 24*b^3)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b + b
^2)))/((a^5 + a^4*b)*sqrt(a*b + b^2)) + (7*a*b^2*tan(f*x + e)^3 + 8*b^3*tan(f*x + e)^3 + 9*a^2*b*tan(f*x + e)
+ 17*a*b^2*tan(f*x + e) + 8*b^3*tan(f*x + e))/((a^4 + a^3*b)*(b*tan(f*x + e)^2 + a + b)^2) - 4*(f*x + e)*(a +
6*b)/a^4 + 4*tan(f*x + e)/((tan(f*x + e)^2 + 1)*a^3))/f

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maple [A]  time = 0.96, size = 314, normalized size = 1.71 \[ -\frac {7 b^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \,a^{2} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2} \left (a +b \right )}-\frac {b^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{f \,a^{3} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2} \left (a +b \right )}-\frac {9 b \tan \left (f x +e \right )}{8 a^{2} f \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {b^{2} \tan \left (f x +e \right )}{f \,a^{3} \left (a +b +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{8 f \,a^{2} \left (a +b \right ) \sqrt {\left (a +b \right ) b}}-\frac {5 b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,a^{3} \left (a +b \right ) \sqrt {\left (a +b \right ) b}}-\frac {3 b^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {\left (a +b \right ) b}}\right )}{f \,a^{4} \left (a +b \right ) \sqrt {\left (a +b \right ) b}}-\frac {\tan \left (f x +e \right )}{2 f \,a^{3} \left (1+\tan ^{2}\left (f x +e \right )\right )}+\frac {\arctan \left (\tan \left (f x +e \right )\right )}{2 f \,a^{3}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right ) b}{f \,a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x)

[Out]

-7/8/f/a^2*b^2/(a+b+b*tan(f*x+e)^2)^2/(a+b)*tan(f*x+e)^3-1/f/a^3*b^3/(a+b+b*tan(f*x+e)^2)^2/(a+b)*tan(f*x+e)^3
-9/8*b*tan(f*x+e)/a^2/f/(a+b+b*tan(f*x+e)^2)^2-1/f*b^2/a^3/(a+b+b*tan(f*x+e)^2)^2*tan(f*x+e)-15/8/f/a^2*b/(a+b
)/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))-5/f/a^3*b^2/(a+b)/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/(
(a+b)*b)^(1/2))-3/f/a^4*b^3/(a+b)/((a+b)*b)^(1/2)*arctan(tan(f*x+e)*b/((a+b)*b)^(1/2))-1/2/f/a^3*tan(f*x+e)/(t
an(f*x+e)^2+1)+1/2/f/a^3*arctan(tan(f*x+e))+3/f/a^4*arctan(tan(f*x+e))*b

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maxima [A]  time = 0.44, size = 272, normalized size = 1.48 \[ -\frac {\frac {{\left (15 \, a^{2} b + 40 \, a b^{2} + 24 \, b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{{\left (a^{5} + a^{4} b\right )} \sqrt {{\left (a + b\right )} b}} + \frac {{\left (11 \, a b^{2} + 12 \, b^{3}\right )} \tan \left (f x + e\right )^{5} + {\left (17 \, a^{2} b + 40 \, a b^{2} + 24 \, b^{3}\right )} \tan \left (f x + e\right )^{3} + {\left (4 \, a^{3} + 21 \, a^{2} b + 29 \, a b^{2} + 12 \, b^{3}\right )} \tan \left (f x + e\right )}{{\left (a^{4} b^{2} + a^{3} b^{3}\right )} \tan \left (f x + e\right )^{6} + a^{6} + 3 \, a^{5} b + 3 \, a^{4} b^{2} + a^{3} b^{3} + {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 3 \, a^{3} b^{3}\right )} \tan \left (f x + e\right )^{4} + {\left (a^{6} + 5 \, a^{5} b + 7 \, a^{4} b^{2} + 3 \, a^{3} b^{3}\right )} \tan \left (f x + e\right )^{2}} - \frac {4 \, {\left (f x + e\right )} {\left (a + 6 \, b\right )}}{a^{4}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^2/(a+b*sec(f*x+e)^2)^3,x, algorithm="maxima")

[Out]

-1/8*((15*a^2*b + 40*a*b^2 + 24*b^3)*arctan(b*tan(f*x + e)/sqrt((a + b)*b))/((a^5 + a^4*b)*sqrt((a + b)*b)) +
((11*a*b^2 + 12*b^3)*tan(f*x + e)^5 + (17*a^2*b + 40*a*b^2 + 24*b^3)*tan(f*x + e)^3 + (4*a^3 + 21*a^2*b + 29*a
*b^2 + 12*b^3)*tan(f*x + e))/((a^4*b^2 + a^3*b^3)*tan(f*x + e)^6 + a^6 + 3*a^5*b + 3*a^4*b^2 + a^3*b^3 + (2*a^
5*b + 5*a^4*b^2 + 3*a^3*b^3)*tan(f*x + e)^4 + (a^6 + 5*a^5*b + 7*a^4*b^2 + 3*a^3*b^3)*tan(f*x + e)^2) - 4*(f*x
+ e)*(a + 6*b)/a^4)/f

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mupad [B]  time = 7.83, size = 2628, normalized size = 14.28 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^2/(a + b/cos(e + f*x)^2)^3,x)

[Out]

(atan(((((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b + a^8
+ a^6*b^2)) - (((6*a^8*b^5 + (29*a^9*b^4)/2 + (21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) - (ta
n(e + f*x)*(a*1i + b*6i)*(512*a^8*b^5 + 1280*a^9*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(128*a^4*(2*a^7*b + a^8
+ a^6*b^2)))*(a*1i + b*6i))/(4*a^4))*(a*1i + b*6i)*1i)/(4*a^4) + (((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296
*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b + a^8 + a^6*b^2)) + (((6*a^8*b^5 + (29*a^9*b^4)/2 + (21*a
^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) + (tan(e + f*x)*(a*1i + b*6i)*(512*a^8*b^5 + 1280*a^9*b^4
+ 1024*a^10*b^3 + 256*a^11*b^2))/(128*a^4*(2*a^7*b + a^8 + a^6*b^2)))*(a*1i + b*6i))/(4*a^4))*(a*1i + b*6i)*1
i)/(4*a^4))/(((297*a*b^6)/4 + 27*b^7 + (279*a^2*b^5)/4 + (805*a^3*b^4)/32 + (165*a^4*b^3)/64)/(2*a^10*b + a^11
+ a^9*b^2) - (((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*
b + a^8 + a^6*b^2)) - (((6*a^8*b^5 + (29*a^9*b^4)/2 + (21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2
) - (tan(e + f*x)*(a*1i + b*6i)*(512*a^8*b^5 + 1280*a^9*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(128*a^4*(2*a^7*b
+ a^8 + a^6*b^2)))*(a*1i + b*6i))/(4*a^4))*(a*1i + b*6i))/(4*a^4) + (((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 +
3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b + a^8 + a^6*b^2)) + (((6*a^8*b^5 + (29*a^9*b^4)/2 + (
21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) + (tan(e + f*x)*(a*1i + b*6i)*(512*a^8*b^5 + 1280*a^9
*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(128*a^4*(2*a^7*b + a^8 + a^6*b^2)))*(a*1i + b*6i))/(4*a^4))*(a*1i + b*6
i))/(4*a^4)))*(a*1i + b*6i)*1i)/(2*a^4*f) - ((tan(e + f*x)*(17*a*b + 4*a^2 + 12*b^2))/(8*a^3) + (tan(e + f*x)^
5*(11*a*b^2 + 12*b^3))/(8*a^3*(a + b)) + (b*tan(e + f*x)^3*(40*a*b + 17*a^2 + 24*b^2))/(8*a^3*(a + b)))/(f*(2*
a*b + tan(e + f*x)^2*(4*a*b + a^2 + 3*b^2) + a^2 + b^2 + tan(e + f*x)^4*(2*a*b + 3*b^2) + b^2*tan(e + f*x)^6))
+ (atan((((-b*(a + b)^3)^(1/2)*((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*
b^3))/(32*(2*a^7*b + a^8 + a^6*b^2)) - ((-b*(a + b)^3)^(1/2)*((6*a^8*b^5 + (29*a^9*b^4)/2 + (21*a^10*b^3)/2 +
2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) - (tan(e + f*x)*(-b*(a + b)^3)^(1/2)*(40*a*b + 15*a^2 + 24*b^2)*(512*a
^8*b^5 + 1280*a^9*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(512*(2*a^7*b + a^8 + a^6*b^2)*(3*a^6*b + a^7 + a^4*b^3
+ 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*
b^2)*1i)/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)) + ((-b*(a + b)^3)^(1/2)*((tan(e + f*x)*(3264*a*b^6 + 1152*
b^7 + 3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b + a^8 + a^6*b^2)) + ((-b*(a + b)^3)^(1/2)*((6*a
^8*b^5 + (29*a^9*b^4)/2 + (21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) + (tan(e + f*x)*(-b*(a + b
)^3)^(1/2)*(40*a*b + 15*a^2 + 24*b^2)*(512*a^8*b^5 + 1280*a^9*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(512*(2*a^7
*b + a^8 + a^6*b^2)*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b + a^7 + a
^4*b^3 + 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*b^2)*1i)/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))/(((297*a*b^6)
/4 + 27*b^7 + (279*a^2*b^5)/4 + (805*a^3*b^4)/32 + (165*a^4*b^3)/64)/(2*a^10*b + a^11 + a^9*b^2) - ((-b*(a + b
)^3)^(1/2)*((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296*a^2*b^5 + 1424*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b +
a^8 + a^6*b^2)) - ((-b*(a + b)^3)^(1/2)*((6*a^8*b^5 + (29*a^9*b^4)/2 + (21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b
+ a^11 + a^9*b^2) - (tan(e + f*x)*(-b*(a + b)^3)^(1/2)*(40*a*b + 15*a^2 + 24*b^2)*(512*a^8*b^5 + 1280*a^9*b^4
+ 1024*a^10*b^3 + 256*a^11*b^2))/(512*(2*a^7*b + a^8 + a^6*b^2)*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(40*a
*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b +
a^7 + a^4*b^3 + 3*a^5*b^2)) + ((-b*(a + b)^3)^(1/2)*((tan(e + f*x)*(3264*a*b^6 + 1152*b^7 + 3296*a^2*b^5 + 142
4*a^3*b^4 + 241*a^4*b^3))/(32*(2*a^7*b + a^8 + a^6*b^2)) + ((-b*(a + b)^3)^(1/2)*((6*a^8*b^5 + (29*a^9*b^4)/2
+ (21*a^10*b^3)/2 + 2*a^11*b^2)/(2*a^10*b + a^11 + a^9*b^2) + (tan(e + f*x)*(-b*(a + b)^3)^(1/2)*(40*a*b + 15*
a^2 + 24*b^2)*(512*a^8*b^5 + 1280*a^9*b^4 + 1024*a^10*b^3 + 256*a^11*b^2))/(512*(2*a^7*b + a^8 + a^6*b^2)*(3*a
^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(40*a*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2)))*(4
0*a*b + 15*a^2 + 24*b^2))/(16*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2))))*(-b*(a + b)^3)^(1/2)*(40*a*b + 15*a^2 +
24*b^2)*1i)/(8*f*(3*a^6*b + a^7 + a^4*b^3 + 3*a^5*b^2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)**2/(a+b*sec(f*x+e)**2)**3,x)

[Out]

Timed out

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