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

Optimal. Leaf size=193 \[ -\frac {\sqrt {b} \left (15 a^2+10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{3/2} f (a-b)^4}-\frac {b (11 a+b) \tan (e+f x)}{8 a f (a-b)^3 \left (a+b \tan ^2(e+f x)\right )}-\frac {3 b \tan (e+f x)}{4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac {x (a+5 b)}{2 (a-b)^4} \]

[Out]

1/2*(a+5*b)*x/(a-b)^4-1/8*(15*a^2+10*a*b-b^2)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*b^(1/2)/a^(3/2)/(a-b)^4/f-1/2
*cos(f*x+e)*sin(f*x+e)/(a-b)/f/(a+b*tan(f*x+e)^2)^2-3/4*b*tan(f*x+e)/(a-b)^2/f/(a+b*tan(f*x+e)^2)^2-1/8*b*(11*
a+b)*tan(f*x+e)/a/(a-b)^3/f/(a+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.25, antiderivative size = 193, 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 = {3663, 471, 527, 522, 203, 205} \[ -\frac {\sqrt {b} \left (15 a^2+10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{3/2} f (a-b)^4}-\frac {b (11 a+b) \tan (e+f x)}{8 a f (a-b)^3 \left (a+b \tan ^2(e+f x)\right )}-\frac {3 b \tan (e+f x)}{4 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )^2}-\frac {\sin (e+f x) \cos (e+f x)}{2 f (a-b) \left (a+b \tan ^2(e+f x)\right )^2}+\frac {x (a+5 b)}{2 (a-b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a + 5*b)*x)/(2*(a - b)^4) - (Sqrt[b]*(15*a^2 + 10*a*b - b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(8*a^(3
/2)*(a - b)^4*f) - (Cos[e + f*x]*Sin[e + f*x])/(2*(a - b)*f*(a + b*Tan[e + f*x]^2)^2) - (3*b*Tan[e + f*x])/(4*
(a - b)^2*f*(a + b*Tan[e + f*x]^2)^2) - (b*(11*a + b)*Tan[e + f*x])/(8*a*(a - b)^3*f*(a + 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 3663

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

Rubi steps

\begin {align*} \int \frac {\sin ^2(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^2\right )^2 \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {a-5 b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{2 (a-b) f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {2 a (2 a+b)-18 a b x^2}{\left (1+x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{8 a (a-b)^2 f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+b) \tan (e+f x)}{8 a (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {2 a \left (4 a^2+9 a b-b^2\right )-2 a b (11 a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{16 a^2 (a-b)^3 f}\\ &=-\frac {\cos (e+f x) \sin (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+b) \tan (e+f x)}{8 a (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}+\frac {(a+5 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{2 (a-b)^4 f}-\frac {\left (b \left (15 a^2+10 a b-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a (a-b)^4 f}\\ &=\frac {(a+5 b) x}{2 (a-b)^4}-\frac {\sqrt {b} \left (15 a^2+10 a b-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{8 a^{3/2} (a-b)^4 f}-\frac {\cos (e+f x) \sin (e+f x)}{2 (a-b) f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {3 b \tan (e+f x)}{4 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )^2}-\frac {b (11 a+b) \tan (e+f x)}{8 a (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 2.56, size = 164, normalized size = 0.85 \[ \frac {\frac {\sqrt {b} \left (-15 a^2-10 a b+b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{a^{3/2}}+\frac {4 b^2 (a-b) \sin (2 (e+f x))}{((a-b) \cos (2 (e+f x))+a+b)^2}+4 (a+5 b) (e+f x)-2 (a-b) \sin (2 (e+f x))-\frac {b (a-b) (9 a+b) \sin (2 (e+f x))}{a ((a-b) \cos (2 (e+f x))+a+b)}}{8 f (a-b)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(a + 5*b)*(e + f*x) + (Sqrt[b]*(-15*a^2 - 10*a*b + b^2)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/a^(3/2) - 2
*(a - b)*Sin[2*(e + f*x)] + (4*(a - b)*b^2*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)])^2 - ((a - b)*b
*(9*a + b)*Sin[2*(e + f*x)])/(a*(a + b + (a - b)*Cos[2*(e + f*x)])))/(8*(a - b)^4*f)

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fricas [B]  time = 0.88, size = 1076, normalized size = 5.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/32*(16*(a^4 + 3*a^3*b - 9*a^2*b^2 + 5*a*b^3)*f*x*cos(f*x + e)^4 + 32*(a^3*b + 4*a^2*b^2 - 5*a*b^3)*f*x*cos(
f*x + e)^2 + 16*(a^2*b^2 + 5*a*b^3)*f*x - ((15*a^4 - 20*a^3*b - 6*a^2*b^2 + 12*a*b^3 - b^4)*cos(f*x + e)^4 + 1
5*a^2*b^2 + 10*a*b^3 - b^4 + 2*(15*a^3*b - 5*a^2*b^2 - 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(-b/a)*log(((a^2 +
6*a*b + b^2)*cos(f*x + e)^4 - 2*(3*a*b + b^2)*cos(f*x + e)^2 - 4*((a^2 + a*b)*cos(f*x + e)^3 - a*b*cos(f*x + e
))*sqrt(-b/a)*sin(f*x + e) + b^2)/((a^2 - 2*a*b + b^2)*cos(f*x + e)^4 + 2*(a*b - b^2)*cos(f*x + e)^2 + b^2)) -
4*(4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + (17*a^3*b - 33*a^2*b^2 + 15*a*b^3 + b^4)*cos(f*x +
e)^3 + (11*a^2*b^2 - 10*a*b^3 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 1
5*a^3*b^4 - 6*a^2*b^5 + a*b^6)*f*cos(f*x + e)^4 + 2*(a^6*b - 5*a^5*b^2 + 10*a^4*b^3 - 10*a^3*b^4 + 5*a^2*b^5 -
a*b^6)*f*cos(f*x + e)^2 + (a^5*b^2 - 4*a^4*b^3 + 6*a^3*b^4 - 4*a^2*b^5 + a*b^6)*f), 1/16*(8*(a^4 + 3*a^3*b -
9*a^2*b^2 + 5*a*b^3)*f*x*cos(f*x + e)^4 + 16*(a^3*b + 4*a^2*b^2 - 5*a*b^3)*f*x*cos(f*x + e)^2 + 8*(a^2*b^2 + 5
*a*b^3)*f*x + ((15*a^4 - 20*a^3*b - 6*a^2*b^2 + 12*a*b^3 - b^4)*cos(f*x + e)^4 + 15*a^2*b^2 + 10*a*b^3 - b^4 +
2*(15*a^3*b - 5*a^2*b^2 - 11*a*b^3 + b^4)*cos(f*x + e)^2)*sqrt(b/a)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*s
qrt(b/a)/(b*cos(f*x + e)*sin(f*x + e))) - 2*(4*(a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*cos(f*x + e)^5 + (17*a^3*b
- 33*a^2*b^2 + 15*a*b^3 + b^4)*cos(f*x + e)^3 + (11*a^2*b^2 - 10*a*b^3 - b^4)*cos(f*x + e))*sin(f*x + e))/((a^
7 - 6*a^6*b + 15*a^5*b^2 - 20*a^4*b^3 + 15*a^3*b^4 - 6*a^2*b^5 + a*b^6)*f*cos(f*x + e)^4 + 2*(a^6*b - 5*a^5*b^
2 + 10*a^4*b^3 - 10*a^3*b^4 + 5*a^2*b^5 - a*b^6)*f*cos(f*x + e)^2 + (a^5*b^2 - 4*a^4*b^3 + 6*a^3*b^4 - 4*a^2*b
^5 + a*b^6)*f)]

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giac [A]  time = 2.93, size = 282, normalized size = 1.46 \[ \frac {\frac {4 \, {\left (f x + e\right )} {\left (a + 5 \, b\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (15 \, a^{2} b + 10 \, a b^{2} - b^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{{\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {a b}} - \frac {4 \, \tan \left (f x + e\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\tan \left (f x + e\right )^{2} + 1\right )}} - \frac {7 \, a b^{2} \tan \left (f x + e\right )^{3} + b^{3} \tan \left (f x + e\right )^{3} + 9 \, a^{2} b \tan \left (f x + e\right ) - a b^{2} \tan \left (f x + e\right )}{{\left (a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3}\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{2}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(f*x + e)*(a + 5*b)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (15*a^2*b + 10*a*b^2 - b^3)*(pi*floor
((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^
4)*sqrt(a*b)) - 4*tan(f*x + e)/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(tan(f*x + e)^2 + 1)) - (7*a*b^2*tan(f*x + e)^
3 + b^3*tan(f*x + e)^3 + 9*a^2*b*tan(f*x + e) - a*b^2*tan(f*x + e))/((a^4 - 3*a^3*b + 3*a^2*b^2 - a*b^3)*(b*ta
n(f*x + e)^2 + a)^2))/f

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maple [B]  time = 0.56, size = 430, normalized size = 2.23 \[ -\frac {7 b^{2} a \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {3 b^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{4 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {b^{4} \left (\tan ^{3}\left (f x +e \right )\right )}{8 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2} a}-\frac {9 b \tan \left (f x +e \right ) a^{2}}{8 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}+\frac {5 b^{2} \tan \left (f x +e \right ) a}{4 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {b^{3} \tan \left (f x +e \right )}{8 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{2}}-\frac {15 b a \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \left (a -b \right )^{4} \sqrt {a b}}-\frac {5 b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{4 f \left (a -b \right )^{4} \sqrt {a b}}+\frac {b^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{8 f \left (a -b \right )^{4} a \sqrt {a b}}-\frac {\tan \left (f x +e \right ) a}{2 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )}+\frac {\tan \left (f x +e \right ) b}{2 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) a}{2 f \left (a -b \right )^{4}}+\frac {5 \arctan \left (\tan \left (f x +e \right )\right ) b}{2 f \left (a -b \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 346, normalized size = 1.79 \[ \frac {\frac {4 \, {\left (f x + e\right )} {\left (a + 5 \, b\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (15 \, a^{2} b + 10 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{5} - 4 \, a^{4} b + 6 \, a^{3} b^{2} - 4 \, a^{2} b^{3} + a b^{4}\right )} \sqrt {a b}} - \frac {{\left (11 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{5} + {\left (17 \, a^{2} b + 6 \, a b^{2} + b^{3}\right )} \tan \left (f x + e\right )^{3} + {\left (4 \, a^{3} + 9 \, a^{2} b - a b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{4} b^{2} - 3 \, a^{3} b^{3} + 3 \, a^{2} b^{4} - a b^{5}\right )} \tan \left (f x + e\right )^{6} + a^{6} - 3 \, a^{5} b + 3 \, a^{4} b^{2} - a^{3} b^{3} + {\left (2 \, a^{5} b - 5 \, a^{4} b^{2} + 3 \, a^{3} b^{3} + a^{2} b^{4} - a b^{5}\right )} \tan \left (f x + e\right )^{4} + {\left (a^{6} - a^{5} b - 3 \, a^{4} b^{2} + 5 \, a^{3} b^{3} - 2 \, a^{2} b^{4}\right )} \tan \left (f x + e\right )^{2}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(f*x + e)*(a + 5*b)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (15*a^2*b + 10*a*b^2 - b^3)*arctan(b*
tan(f*x + e)/sqrt(a*b))/((a^5 - 4*a^4*b + 6*a^3*b^2 - 4*a^2*b^3 + a*b^4)*sqrt(a*b)) - ((11*a*b^2 + b^3)*tan(f*
x + e)^5 + (17*a^2*b + 6*a*b^2 + b^3)*tan(f*x + e)^3 + (4*a^3 + 9*a^2*b - a*b^2)*tan(f*x + e))/((a^4*b^2 - 3*a
^3*b^3 + 3*a^2*b^4 - a*b^5)*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 + a^2*b^4 - a*b^5)*tan(f*x + e)^4 + (a^6 - a^5*b - 3*a^4*b^2 + 5*a^3*b^3 - 2*a^2*b^4)*tan(f*x + e)^2))/f

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mupad [B]  time = 16.49, size = 4997, normalized size = 25.89 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((tan(e + f*x)^5*(11*a*b^2 + b^3))/(8*a*(3*a*b^2 - 3*a^2*b + a^3 - b^3)) + (tan(e + f*x)*(9*a*b + 4*a^2 - b^
2))/(8*(a - b)*(a^2 - 2*a*b + b^2)) + (b*tan(e + f*x)^3*(6*a*b + 17*a^2 + b^2))/(8*a*(a - b)*(a^2 - 2*a*b + b^
2)))/(f*(tan(e + f*x)^2*(2*a*b + a^2) + tan(e + f*x)^4*(2*a*b + b^2) + a^2 + b^2*tan(e + f*x)^6)) - (atan(((((
tan(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*
b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) - ((((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 2
31*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b
- a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2)
- (tan(e + f*x)*(a*1i + b*5i)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^7 + 35
84*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4
+ 6*a^2*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(a*1i + b*5i))/(4*
(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*1i + b*5i)*1i)/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)
) + (((tan(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 -
6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) + ((((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*
b^9 - 231*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9
*a^10*b - a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a
^9*b^2) + (tan(e + f*x)*(a*1i + b*5i)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b
^7 + 3584*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3
+ b^4 + 6*a^2*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(a*1i + b*5
i))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*1i + b*5i)*1i)/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a
^2*b^2)))/(((39*a^2*b^5)/4 - (5*b^7)/64 - (3*a*b^6)/32 + (475*a^3*b^4)/32 + (165*a^4*b^3)/64)/(9*a^10*b - a^11
+ a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) - (((t
an(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b
^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) - ((((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 23
1*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b
- a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2)
- (tan(e + f*x)*(a*1i + b*5i)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^7 + 358
4*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 +
6*a^2*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(a*1i + b*5i))/(4*(
a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*1i + b*5i))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) +
(((tan(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a
^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) + ((((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9
- 231*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^1
0*b - a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b
^2) + (tan(e + f*x)*(a*1i + b*5i)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^7 +
3584*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b
^4 + 6*a^2*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(a*1i + b*5i))/
(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*1i + b*5i))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)
)))*(a*1i + b*5i)*1i)/(2*f*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (atan(((((tan(e + f*x)*(b^7 - 20*a*b
^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*
b^3 + 15*a^6*b^2)) - ((-a^3*b)^(1/2)*(((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 231*a^5*b^8
+ 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b - a^11 + a
^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) - (tan(e +
f*x)*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^2)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 358
4*a^6*b^7 + 3584*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(512*(a^7 - 4*a^6*b +
a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2))
)*(10*a*b + 15*a^2 - b^2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))*(-a^3*b)^(1/2)*(10*a*b + 15
*a^2 - b^2)*1i)/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (((tan(e + f*x)*(b^7 - 20*a*b^6 + 470
*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15
*a^6*b^2)) + ((-a^3*b)^(1/2)*(((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 231*a^5*b^8 + 231*a^
6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b - a^11 + a^2*b^9 -
9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) + (tan(e + f*x)*(-
a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^2)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^
7 + 3584*a^7*b^6 - 7168*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(512*(a^7 - 4*a^6*b + a^3*b^4
- 4*a^4*b^3 + 6*a^5*b^2)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(10*a*
b + 15*a^2 - b^2))/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b
^2)*1i)/(16*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))/(((39*a^2*b^5)/4 - (5*b^7)/64 - (3*a*b^6)/32 +
(475*a^3*b^4)/32 + (165*a^4*b^3)/64)/(9*a^10*b - a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b^6 + 126*a
^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) - (((tan(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4
+ 241*a^4*b^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) - ((-a^3*b)^
(1/2)*(((17*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 231*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 1
8*a^8*b^5 + (39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b - a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7
- 84*a^5*b^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) - (tan(e + f*x)*(-a^3*b)^(1/2)*(10*a*b +
15*a^2 - b^2)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^7 + 3584*a^7*b^6 - 7168
*a^8*b^5 + 5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(512*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2
)*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(10*a*b + 15*a^2 - b^2))/(16*
(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^2))/(16*(a^7 - 4*a^6*b
+ a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)) + (((tan(e + f*x)*(b^7 - 20*a*b^6 + 470*a^2*b^5 + 460*a^3*b^4 + 241*a^4*b
^3))/(32*(a^8 - 6*a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)) + ((-a^3*b)^(1/2)*(((17
*a^2*b^11)/2 - (a*b^12)/2 - 48*a^3*b^10 + 138*a^4*b^9 - 231*a^5*b^8 + 231*a^6*b^7 - 126*a^7*b^6 + 18*a^8*b^5 +
(39*a^9*b^4)/2 - (23*a^10*b^3)/2 + 2*a^11*b^2)/(9*a^10*b - a^11 + a^2*b^9 - 9*a^3*b^8 + 36*a^4*b^7 - 84*a^5*b
^6 + 126*a^6*b^5 - 126*a^7*b^4 + 84*a^8*b^3 - 36*a^9*b^2) + (tan(e + f*x)*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^
2)*(256*a^2*b^11 - 1792*a^3*b^10 + 5120*a^4*b^9 - 7168*a^5*b^8 + 3584*a^6*b^7 + 3584*a^7*b^6 - 7168*a^8*b^5 +
5120*a^9*b^4 - 1792*a^10*b^3 + 256*a^11*b^2))/(512*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)*(a^8 - 6*
a^7*b + a^2*b^6 - 6*a^3*b^5 + 15*a^4*b^4 - 20*a^5*b^3 + 15*a^6*b^2)))*(10*a*b + 15*a^2 - b^2))/(16*(a^7 - 4*a^
6*b + a^3*b^4 - 4*a^4*b^3 + 6*a^5*b^2)))*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^2))/(16*(a^7 - 4*a^6*b + a^3*b^4
- 4*a^4*b^3 + 6*a^5*b^2))))*(-a^3*b)^(1/2)*(10*a*b + 15*a^2 - b^2)*1i)/(8*f*(a^7 - 4*a^6*b + a^3*b^4 - 4*a^4*b
^3 + 6*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*tan(f*x+e)**2)**3,x)

[Out]

Timed out

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