3.74 \(\int \frac {\sin ^4(e+f x)}{(a+b \tan ^2(e+f x))^2} \, dx\)

Optimal. Leaf size=196 \[ \frac {3 x \left (a^2+6 a b+b^2\right )}{8 (a-b)^4}-\frac {3 \sqrt {a} \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 f (a-b)^4}-\frac {3 b (3 a+b) \tan (e+f x)}{8 f (a-b)^3 \left (a+b \tan ^2(e+f x)\right )}+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {(5 a+b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )} \]

[Out]

3/8*(a^2+6*a*b+b^2)*x/(a-b)^4-3/2*(a+b)*arctan(b^(1/2)*tan(f*x+e)/a^(1/2))*a^(1/2)*b^(1/2)/(a-b)^4/f-1/8*(5*a+
b)*cos(f*x+e)*sin(f*x+e)/(a-b)^2/f/(a+b*tan(f*x+e)^2)+1/4*cos(f*x+e)^3*sin(f*x+e)/(a-b)/f/(a+b*tan(f*x+e)^2)-3
/8*b*(3*a+b)*tan(f*x+e)/(a-b)^3/f/(a+b*tan(f*x+e)^2)

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Rubi [A]  time = 0.25, antiderivative size = 196, 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, 470, 527, 522, 203, 205} \[ \frac {3 x \left (a^2+6 a b+b^2\right )}{8 (a-b)^4}-\frac {3 \sqrt {a} \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 f (a-b)^4}-\frac {3 b (3 a+b) \tan (e+f x)}{8 f (a-b)^3 \left (a+b \tan ^2(e+f x)\right )}+\frac {\sin (e+f x) \cos ^3(e+f x)}{4 f (a-b) \left (a+b \tan ^2(e+f x)\right )}-\frac {(5 a+b) \sin (e+f x) \cos (e+f x)}{8 f (a-b)^2 \left (a+b \tan ^2(e+f x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

(3*(a^2 + 6*a*b + b^2)*x)/(8*(a - b)^4) - (3*Sqrt[a]*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]])/(
2*(a - b)^4*f) - ((5*a + b)*Cos[e + f*x]*Sin[e + f*x])/(8*(a - b)^2*f*(a + b*Tan[e + f*x]^2)) + (Cos[e + f*x]^
3*Sin[e + f*x])/(4*(a - b)*f*(a + b*Tan[e + f*x]^2)) - (3*b*(3*a + b)*Tan[e + f*x])/(8*(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 470

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(a*e^(2
*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c - a*d)*(p + 1)), x] + Dist[e^(2
*n)/(b*n*(b*c - a*d)*(p + 1)), Int[(e*x)^(m - 2*n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[a*c*(m - 2*n + 1) +
(a*d*(m - n + n*q + 1) + b*c*n*(p + 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] && GtQ[m - n + 1, n] && 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 ^4(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^3 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {a+(-4 a-b) x^2}{\left (1+x^2\right )^2 \left (a+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{4 (a-b) f}\\ &=-\frac {(5 a+b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {3 a (a+b)-3 b (5 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-b)^2 f}\\ &=-\frac {(5 a+b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {3 b (3 a+b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {6 a^2 (a+3 b)-6 a b (3 a+b) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{16 a (a-b)^3 f}\\ &=-\frac {(5 a+b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {3 b (3 a+b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}-\frac {(3 a b (a+b)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{2 (a-b)^4 f}+\frac {\left (3 \left (a^2+6 a b+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{8 (a-b)^4 f}\\ &=\frac {3 \left (a^2+6 a b+b^2\right ) x}{8 (a-b)^4}-\frac {3 \sqrt {a} \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{2 (a-b)^4 f}-\frac {(5 a+b) \cos (e+f x) \sin (e+f x)}{8 (a-b)^2 f \left (a+b \tan ^2(e+f x)\right )}+\frac {\cos ^3(e+f x) \sin (e+f x)}{4 (a-b) f \left (a+b \tan ^2(e+f x)\right )}-\frac {3 b (3 a+b) \tan (e+f x)}{8 (a-b)^3 f \left (a+b \tan ^2(e+f x)\right )}\\ \end {align*}

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Mathematica [A]  time = 1.65, size = 136, normalized size = 0.69 \[ \frac {12 \left (a^2+6 a b+b^2\right ) (e+f x)+(a-b)^2 \sin (4 (e+f x))-8 (a+b) (a-b) \sin (2 (e+f x))-48 \sqrt {a} \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )-\frac {16 a b (a-b) \sin (2 (e+f x))}{(a-b) \cos (2 (e+f x))+a+b}}{32 f (a-b)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^4/(a + b*Tan[e + f*x]^2)^2,x]

[Out]

(12*(a^2 + 6*a*b + b^2)*(e + f*x) - 48*Sqrt[a]*Sqrt[b]*(a + b)*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]] - 8*(a -
b)*(a + b)*Sin[2*(e + f*x)] - (16*a*(a - b)*b*Sin[2*(e + f*x)])/(a + b + (a - b)*Cos[2*(e + f*x)]) + (a - b)^
2*Sin[4*(e + f*x)])/(32*(a - b)^4*f)

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fricas [A]  time = 0.64, size = 705, normalized size = 3.60 \[ \left [\frac {3 \, {\left (a^{3} + 5 \, a^{2} b - 5 \, a b^{2} - b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (a^{2} b + 6 \, a b^{2} + b^{3}\right )} f x + 3 \, {\left ({\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {-a b} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) + {\left (2 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, {\left ({\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )} f\right )}}, \frac {3 \, {\left (a^{3} + 5 \, a^{2} b - 5 \, a b^{2} - b^{3}\right )} f x \cos \left (f x + e\right )^{2} + 3 \, {\left (a^{2} b + 6 \, a b^{2} + b^{3}\right )} f x + 6 \, {\left ({\left (a^{2} - b^{2}\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {a b} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {a b}}{2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + {\left (2 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{5} - {\left (5 \, a^{3} - 9 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (3 \, a^{2} b - 2 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8 \, {\left ({\left (a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}\right )} f \cos \left (f x + e\right )^{2} + {\left (a^{4} b - 4 \, a^{3} b^{2} + 6 \, a^{2} b^{3} - 4 \, a b^{4} + b^{5}\right )} f\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="fricas")

[Out]

[1/8*(3*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 6*a*b^2 + b^3)*f*x + 3*((a^2 - b^2)*co
s(f*x + e)^2 + a*b + b^2)*sqrt(-a*b)*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 + b)*cos(f*x + e)^3 - b*cos(f*x + e))*sqrt(-a*b)*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)) + (2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - (5*a^3 - 9*a^2
*b + 3*a*b^2 + b^3)*cos(f*x + e)^3 - 3*(3*a^2*b - 2*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - 5*a^4*b +
10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f
), 1/8*(3*(a^3 + 5*a^2*b - 5*a*b^2 - b^3)*f*x*cos(f*x + e)^2 + 3*(a^2*b + 6*a*b^2 + b^3)*f*x + 6*((a^2 - b^2)*
cos(f*x + e)^2 + a*b + b^2)*sqrt(a*b)*arctan(1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(
f*x + e))) + (2*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - (5*a^3 - 9*a^2*b + 3*a*b^2 + b^3)*cos(f*x + e
)^3 - 3*(3*a^2*b - 2*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*
b^4 - b^5)*f*cos(f*x + e)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*f)]

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giac [A]  time = 2.70, size = 266, normalized size = 1.36 \[ \frac {\frac {3 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} {\left (f x + e\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {4 \, a b \tan \left (f x + e\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (b \tan \left (f x + e\right )^{2} + a\right )}} - \frac {12 \, {\left (a^{2} b + a b^{2}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )}}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b}} - \frac {5 \, a \tan \left (f x + e\right )^{3} + 3 \, b \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right ) + 5 \, b \tan \left (f x + e\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\tan \left (f x + e\right )^{2} + 1\right )}^{2}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="giac")

[Out]

1/8*(3*(a^2 + 6*a*b + b^2)*(f*x + e)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - 4*a*b*tan(f*x + e)/((a^3 -
3*a^2*b + 3*a*b^2 - b^3)*(b*tan(f*x + e)^2 + a)) - 12*(a^2*b + a*b^2)*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + a
rctan(b*tan(f*x + e)/sqrt(a*b)))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*sqrt(a*b)) - (5*a*tan(f*x + e)^3
+ 3*b*tan(f*x + e)^3 + 3*a*tan(f*x + e) + 5*b*tan(f*x + e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(tan(f*x + e)^2
+ 1)^2))/f

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maple [B]  time = 0.59, size = 411, normalized size = 2.10 \[ -\frac {a^{2} b \tan \left (f x +e \right )}{2 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}+\frac {a \,b^{2} \tan \left (f x +e \right )}{2 f \left (a -b \right )^{4} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )}-\frac {3 a^{2} b \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f \left (a -b \right )^{4} \sqrt {a b}}-\frac {3 a \,b^{2} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{2 f \left (a -b \right )^{4} \sqrt {a b}}-\frac {5 \left (\tan ^{3}\left (f x +e \right )\right ) a^{2}}{8 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {\left (\tan ^{3}\left (f x +e \right )\right ) a b}{4 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {3 \left (\tan ^{3}\left (f x +e \right )\right ) b^{2}}{8 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}-\frac {3 \tan \left (f x +e \right ) a^{2}}{8 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {5 \tan \left (f x +e \right ) b^{2}}{8 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}-\frac {\tan \left (f x +e \right ) a b}{4 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {9 \arctan \left (\tan \left (f x +e \right )\right ) a b}{4 f \left (a -b \right )^{4}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right ) b^{2}}{8 f \left (a -b \right )^{4}}+\frac {3 \arctan \left (\tan \left (f x +e \right )\right ) a^{2}}{8 f \left (a -b \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x)

[Out]

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

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maxima [A]  time = 1.02, size = 312, normalized size = 1.59 \[ \frac {\frac {3 \, {\left (a^{2} + 6 \, a b + b^{2}\right )} {\left (f x + e\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {12 \, {\left (a^{2} b + a b^{2}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b}} - \frac {3 \, {\left (3 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + {\left (5 \, a^{2} + 14 \, a b + 5 \, b^{2}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (a^{2} + 3 \, a b\right )} \tan \left (f x + e\right )}{{\left (a^{3} b - 3 \, a^{2} b^{2} + 3 \, a b^{3} - b^{4}\right )} \tan \left (f x + e\right )^{6} + {\left (a^{4} - a^{3} b - 3 \, a^{2} b^{2} + 5 \, a b^{3} - 2 \, b^{4}\right )} \tan \left (f x + e\right )^{4} + a^{4} - 3 \, a^{3} b + 3 \, a^{2} b^{2} - a b^{3} + {\left (2 \, a^{4} - 5 \, a^{3} b + 3 \, a^{2} b^{2} + a b^{3} - b^{4}\right )} \tan \left (f x + e\right )^{2}}}{8 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^4/(a+b*tan(f*x+e)^2)^2,x, algorithm="maxima")

[Out]

1/8*(3*(a^2 + 6*a*b + b^2)*(f*x + e)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - 12*(a^2*b + a*b^2)*arctan(b
*tan(f*x + e)/sqrt(a*b))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*sqrt(a*b)) - (3*(3*a*b + b^2)*tan(f*x +
e)^5 + (5*a^2 + 14*a*b + 5*b^2)*tan(f*x + e)^3 + 3*(a^2 + 3*a*b)*tan(f*x + e))/((a^3*b - 3*a^2*b^2 + 3*a*b^3 -
b^4)*tan(f*x + e)^6 + (a^4 - a^3*b - 3*a^2*b^2 + 5*a*b^3 - 2*b^4)*tan(f*x + e)^4 + a^4 - 3*a^3*b + 3*a^2*b^2
- a*b^3 + (2*a^4 - 5*a^3*b + 3*a^2*b^2 + a*b^3 - b^4)*tan(f*x + e)^2))/f

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mupad [B]  time = 16.14, size = 4616, normalized size = 23.55 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^4/(a + b*tan(e + f*x)^2)^2,x)

[Out]

(atan(((((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 - 6*a^5*b - 6*a
*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) - (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a^3*b^9 - 210*
a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(9*a*b^8 - 9*a
^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) - (3*tan(e +
f*x)*(a*b*6i + a^2*1i + b^2*1i)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4*b^7 + 3584*a
^5*b^6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(512*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(a*b*6i + a^2*1i + b^2*1i))/(1
6*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*b*6i + a^2*1i + b^2*1i)*3i)/(16*(a^4 - 4*a^3*b - 4*a*b^3 +
b^4 + 6*a^2*b^2)) + (((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 -
6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) + (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a
^3*b^9 - 210*a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(
9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2)
+ (3*tan(e + f*x)*(a*b*6i + a^2*1i + b^2*1i)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4
*b^7 + 3584*a^5*b^6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(512*(a^4 - 4*a^3*b - 4*a*b^3
+ b^4 + 6*a^2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(a*b*6i + a^2*1i
+ b^2*1i))/(16*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*b*6i + a^2*1i + b^2*1i)*3i)/(16*(a^4 - 4*a^3*b
- 4*a*b^3 + b^4 + 6*a^2*b^2)))/(((27*a*b^7)/64 + (135*a^2*b^6)/32 + (189*a^3*b^5)/16 + (297*a^4*b^4)/32 + (81
*a^5*b^3)/64)/(9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^
3 + 36*a^7*b^2) - (3*((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 -
6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) - (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a
^3*b^9 - 210*a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(
9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2)
- (3*tan(e + f*x)*(a*b*6i + a^2*1i + b^2*1i)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4
*b^7 + 3584*a^5*b^6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(512*(a^4 - 4*a^3*b - 4*a*b^3
+ b^4 + 6*a^2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(a*b*6i + a^2*1i
+ b^2*1i))/(16*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*b*6i + a^2*1i + b^2*1i))/(16*(a^4 - 4*a^3*b -
4*a*b^3 + b^4 + 6*a^2*b^2)) + (3*((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))
/(32*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) + (3*(((9*a*b^11)/2 - (69*a^2*b^1
0)/2 + 114*a^3*b^9 - 210*a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^
10*b^2)/2)/(9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 +
36*a^7*b^2) + (3*tan(e + f*x)*(a*b*6i + a^2*1i + b^2*1i)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^
8 + 3584*a^4*b^7 + 3584*a^5*b^6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(512*(a^4 - 4*a^3
*b - 4*a*b^3 + b^4 + 6*a^2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(a*b*
6i + a^2*1i + b^2*1i))/(16*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*(a*b*6i + a^2*1i + b^2*1i))/(16*(a^4
- 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))))*(a*b*6i + a^2*1i + b^2*1i)*3i)/(8*f*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 +
6*a^2*b^2)) - ((3*tan(e + f*x)^5*(3*a*b + b^2))/(8*(3*a*b^2 - 3*a^2*b + a^3 - b^3)) + (tan(e + f*x)^3*(14*a*b
+ 5*a^2 + 5*b^2))/(8*(a - b)*(a^2 - 2*a*b + b^2)) + (3*tan(e + f*x)*(3*a*b + a^2))/(8*(a - b)*(a^2 - 2*a*b +
b^2)))/(f*(a + b*tan(e + f*x)^6 + tan(e + f*x)^2*(2*a + b) + tan(e + f*x)^4*(a + 2*b))) + (atan((((-a*b)^(1/2)
*(a + b)*((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 - 6*a^5*b - 6*
a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) - (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a^3*b^9 - 210
*a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(9*a*b^8 - 9*
a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) - (3*tan(e
+ f*x)*(-a*b)^(1/2)*(a + b)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4*b^7 + 3584*a^5*b^
6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2
)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(-a*b)^(1/2)*(a + b))/(4*(a^4 - 4*a
^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))*3i)/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + ((-a*b)^(1/2)*(a + b
)*((tan(e + f*x)*(108*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 - 6*a^5*b - 6*a*b^5 +
b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)) + (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a^3*b^9 - 210*a^4*b^
8 + 231*a^5*b^7 - 147*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(9*a*b^8 - 9*a^8*b +
a^9 - b^9 - 36*a^2*b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) + (3*tan(e + f*x)*
(-a*b)^(1/2)*(a + b)*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4*b^7 + 3584*a^5*b^6 - 716
8*a^6*b^5 + 5120*a^7*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)*(a^6
- 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(-a*b)^(1/2)*(a + b))/(4*(a^4 - 4*a^3*b -
4*a*b^3 + b^4 + 6*a^2*b^2)))*3i)/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))/(((27*a*b^7)/64 + (135*a^2*b
^6)/32 + (189*a^3*b^5)/16 + (297*a^4*b^4)/32 + (81*a^5*b^3)/64)/(9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 +
84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) - (3*(-a*b)^(1/2)*(a + b)*((tan(e + f*x)*(10
8*a*b^6 + 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 -
20*a^3*b^3 + 15*a^4*b^2)) - (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a^3*b^9 - 210*a^4*b^8 + 231*a^5*b^7 - 14
7*a^6*b^6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*
b^7 + 84*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) - (3*tan(e + f*x)*(-a*b)^(1/2)*(a + b)
*(256*b^11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4*b^7 + 3584*a^5*b^6 - 7168*a^6*b^5 + 5120*a^7
*b^4 - 1792*a^8*b^3 + 256*a^9*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5
+ b^6 + 15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(-a*b)^(1/2)*(a + b))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2))))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (3*(-a*b)^(1/2)*(a + b)*((tan(e + f*x)*(108*a*b^6
+ 9*b^7 + 486*a^2*b^5 + 396*a^3*b^4 + 153*a^4*b^3))/(32*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 + 15*a^2*b^4 - 20*a^3*b
^3 + 15*a^4*b^2)) + (3*(((9*a*b^11)/2 - (69*a^2*b^10)/2 + 114*a^3*b^9 - 210*a^4*b^8 + 231*a^5*b^7 - 147*a^6*b^
6 + 42*a^7*b^5 + 6*a^8*b^4 - (15*a^9*b^3)/2 + (3*a^10*b^2)/2)/(9*a*b^8 - 9*a^8*b + a^9 - b^9 - 36*a^2*b^7 + 84
*a^3*b^6 - 126*a^4*b^5 + 126*a^5*b^4 - 84*a^6*b^3 + 36*a^7*b^2) + (3*tan(e + f*x)*(-a*b)^(1/2)*(a + b)*(256*b^
11 - 1792*a*b^10 + 5120*a^2*b^9 - 7168*a^3*b^8 + 3584*a^4*b^7 + 3584*a^5*b^6 - 7168*a^6*b^5 + 5120*a^7*b^4 - 1
792*a^8*b^3 + 256*a^9*b^2))/(128*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)*(a^6 - 6*a^5*b - 6*a*b^5 + b^6 +
15*a^2*b^4 - 20*a^3*b^3 + 15*a^4*b^2)))*(-a*b)^(1/2)*(a + b))/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))
)/(4*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))))*(-a*b)^(1/2)*(a + b)*3i)/(2*f*(a^4 - 4*a^3*b - 4*a*b^3 + b
^4 + 6*a^2*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)**4/(a+b*tan(f*x+e)**2)**2,x)

[Out]

Timed out

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