3.61 \(\int \frac {\sin ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx\)

Optimal. Leaf size=178 \[ -\frac {a^{5/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{f (a-b)^4}-\frac {\left (11 a^2-4 a b+b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f (a-b)^3}+\frac {x \left (5 a^3+15 a^2 b-5 a b^2+b^3\right )}{16 (a-b)^4}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 f (a-b)}+\frac {(3 a-b) \sin (e+f x) \cos ^3(e+f x)}{8 f (a-b)^2} \]

[Out]

1/16*(5*a^3+15*a^2*b-5*a*b^2+b^3)*x/(a-b)^4-1/16*(11*a^2-4*a*b+b^2)*cos(f*x+e)*sin(f*x+e)/(a-b)^3/f+1/8*(3*a-b
)*cos(f*x+e)^3*sin(f*x+e)/(a-b)^2/f+1/6*cos(f*x+e)^3*sin(f*x+e)^3/(a-b)/f-a^(5/2)*arctan(b^(1/2)*tan(f*x+e)/a^
(1/2))*b^(1/2)/(a-b)^4/f

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Rubi [A]  time = 0.29, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3663, 470, 578, 527, 522, 203, 205} \[ -\frac {\left (11 a^2-4 a b+b^2\right ) \sin (e+f x) \cos (e+f x)}{16 f (a-b)^3}+\frac {x \left (15 a^2 b+5 a^3-5 a b^2+b^3\right )}{16 (a-b)^4}-\frac {a^{5/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{f (a-b)^4}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 f (a-b)}+\frac {(3 a-b) \sin (e+f x) \cos ^3(e+f x)}{8 f (a-b)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]

[Out]

((5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*x)/(16*(a - b)^4) - (a^(5/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/Sqrt[a]
])/((a - b)^4*f) - ((11*a^2 - 4*a*b + b^2)*Cos[e + f*x]*Sin[e + f*x])/(16*(a - b)^3*f) + ((3*a - b)*Cos[e + f*
x]^3*Sin[e + f*x])/(8*(a - b)^2*f) + (Cos[e + f*x]^3*Sin[e + f*x]^3)/(6*(a - b)*f)

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 578

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(g^(n - 1)*(b*e - a*f)*(g*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(b*n*(b*c -
a*d)*(p + 1)), x] - Dist[g^n/(b*n*(b*c - a*d)*(p + 1)), Int[(g*x)^(m - n)*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*S
imp[c*(b*e - a*f)*(m - n + 1) + (d*(b*e - a*f)*(m + n*q + 1) - b*n*(c*f - d*e)*(p + 1))*x^n, x], x], x] /; Fre
eQ[{a, b, c, d, e, f, g, q}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m - n + 1, 0]

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 ^6(e+f x)}{a+b \tan ^2(e+f x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^4 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 (a-b) f}-\frac {\operatorname {Subst}\left (\int \frac {x^2 \left (3 a-3 (2 a-b) x^2\right )}{\left (1+x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 (a-b) f}\\ &=\frac {(3 a-b) \cos ^3(e+f x) \sin (e+f x)}{8 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 (a-b) f}-\frac {\operatorname {Subst}\left (\int \frac {3 a (3 a-b)-3 \left (8 a^2-3 a b+b^2\right ) x^2}{\left (1+x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{24 (a-b)^2 f}\\ &=-\frac {\left (11 a^2-4 a b+b^2\right ) \cos (e+f x) \sin (e+f x)}{16 (a-b)^3 f}+\frac {(3 a-b) \cos ^3(e+f x) \sin (e+f x)}{8 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 (a-b) f}+\frac {\operatorname {Subst}\left (\int \frac {3 a (5 a-b) (a+b)-3 b \left (11 a^2-4 a b+b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{48 (a-b)^3 f}\\ &=-\frac {\left (11 a^2-4 a b+b^2\right ) \cos (e+f x) \sin (e+f x)}{16 (a-b)^3 f}+\frac {(3 a-b) \cos ^3(e+f x) \sin (e+f x)}{8 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 (a-b) f}-\frac {\left (a^3 b\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tan (e+f x)\right )}{(a-b)^4 f}+\frac {\left (5 a^3+15 a^2 b-5 a b^2+b^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 (a-b)^4 f}\\ &=\frac {\left (5 a^3+15 a^2 b-5 a b^2+b^3\right ) x}{16 (a-b)^4}-\frac {a^{5/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )}{(a-b)^4 f}-\frac {\left (11 a^2-4 a b+b^2\right ) \cos (e+f x) \sin (e+f x)}{16 (a-b)^3 f}+\frac {(3 a-b) \cos ^3(e+f x) \sin (e+f x)}{8 (a-b)^2 f}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 (a-b) f}\\ \end {align*}

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Mathematica [A]  time = 0.57, size = 140, normalized size = 0.79 \[ -\frac {192 a^{5/2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a}}\right )-12 \left (5 a^3+15 a^2 b-5 a b^2+b^3\right ) (e+f x)+(a-b)^3 \sin (6 (e+f x))-3 (3 a-b) (a-b)^2 \sin (4 (e+f x))+3 (5 a-b) (3 a+b) (a-b) \sin (2 (e+f x))}{192 f (a-b)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[e + f*x]^6/(a + b*Tan[e + f*x]^2),x]

[Out]

-1/192*(-12*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*(e + f*x) + 192*a^(5/2)*Sqrt[b]*ArcTan[(Sqrt[b]*Tan[e + f*x])/S
qrt[a]] + 3*(a - b)*(5*a - b)*(3*a + b)*Sin[2*(e + f*x)] - 3*(a - b)^2*(3*a - b)*Sin[4*(e + f*x)] + (a - b)^3*
Sin[6*(e + f*x)])/((a - b)^4*f)

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fricas [A]  time = 0.56, size = 521, normalized size = 2.93 \[ \left [\frac {12 \, \sqrt {-a b} a^{2} \log \left (\frac {{\left (a^{2} + 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b} \sin \left (f x + e\right ) + b^{2}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}\right ) + 3 \, {\left (5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right )} f x - {\left (8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (13 \, a^{3} - 33 \, a^{2} b + 27 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} - 15 \, a^{2} b + 5 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} f}, \frac {24 \, \sqrt {a b} a^{2} \arctan \left (\frac {{\left ({\left (a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {a b}}{2 \, a b \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) + 3 \, {\left (5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right )} f x - {\left (8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )^{5} - 2 \, {\left (13 \, a^{3} - 33 \, a^{2} b + 27 \, a b^{2} - 7 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (11 \, a^{3} - 15 \, a^{2} b + 5 \, a b^{2} - b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} f}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm="fricas")

[Out]

[1/48*(12*sqrt(-a*b)*a^2*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)) + 3*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*f*x - (8*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*
cos(f*x + e)^5 - 2*(13*a^3 - 33*a^2*b + 27*a*b^2 - 7*b^3)*cos(f*x + e)^3 + 3*(11*a^3 - 15*a^2*b + 5*a*b^2 - b^
3)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*f), 1/48*(24*sqrt(a*b)*a^2*arctan(
1/2*((a + b)*cos(f*x + e)^2 - b)*sqrt(a*b)/(a*b*cos(f*x + e)*sin(f*x + e))) + 3*(5*a^3 + 15*a^2*b - 5*a*b^2 +
b^3)*f*x - (8*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cos(f*x + e)^5 - 2*(13*a^3 - 33*a^2*b + 27*a*b^2 - 7*b^3)*cos(f*
x + e)^3 + 3*(11*a^3 - 15*a^2*b + 5*a*b^2 - b^3)*cos(f*x + e))*sin(f*x + e))/((a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a
*b^3 + b^4)*f)]

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giac [A]  time = 2.09, size = 292, normalized size = 1.64 \[ -\frac {\frac {48 \, {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (b) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )\right )} a^{3} b}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b}} - \frac {3 \, {\left (5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right )} {\left (f x + e\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {33 \, a^{2} \tan \left (f x + e\right )^{5} - 12 \, a b \tan \left (f x + e\right )^{5} + 3 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 16 \, a b \tan \left (f x + e\right )^{3} - 8 \, b^{2} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right ) + 12 \, a b \tan \left (f x + e\right ) - 3 \, b^{2} \tan \left (f x + e\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3}}}{48 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm="giac")

[Out]

-1/48*(48*(pi*floor((f*x + e)/pi + 1/2)*sgn(b) + arctan(b*tan(f*x + e)/sqrt(a*b)))*a^3*b/((a^4 - 4*a^3*b + 6*a
^2*b^2 - 4*a*b^3 + b^4)*sqrt(a*b)) - 3*(5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*(f*x + e)/(a^4 - 4*a^3*b + 6*a^2*b^2
- 4*a*b^3 + b^4) + (33*a^2*tan(f*x + e)^5 - 12*a*b*tan(f*x + e)^5 + 3*b^2*tan(f*x + e)^5 + 40*a^2*tan(f*x + e
)^3 + 16*a*b*tan(f*x + e)^3 - 8*b^2*tan(f*x + e)^3 + 15*a^2*tan(f*x + e) + 12*a*b*tan(f*x + e) - 3*b^2*tan(f*x
+ e))/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(tan(f*x + e)^2 + 1)^3))/f

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maple [B]  time = 0.52, size = 545, normalized size = 3.06 \[ -\frac {b \,a^{3} \arctan \left (\frac {\tan \left (f x +e \right ) b}{\sqrt {a b}}\right )}{f \left (a -b \right )^{4} \sqrt {a b}}-\frac {11 \left (\tan ^{5}\left (f x +e \right )\right ) a^{3}}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {15 \left (\tan ^{5}\left (f x +e \right )\right ) a^{2} b}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {5 \left (\tan ^{5}\left (f x +e \right )\right ) b^{2} a}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {\left (\tan ^{5}\left (f x +e \right )\right ) b^{3}}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {5 \left (\tan ^{3}\left (f x +e \right )\right ) a^{3}}{6 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {\left (\tan ^{3}\left (f x +e \right )\right ) a^{2} b}{2 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {\left (\tan ^{3}\left (f x +e \right )\right ) b^{2} a}{2 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {\left (\tan ^{3}\left (f x +e \right )\right ) b^{3}}{6 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {5 \tan \left (f x +e \right ) a^{3}}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {\tan \left (f x +e \right ) a^{2} b}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {5 \tan \left (f x +e \right ) b^{2} a}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}-\frac {\tan \left (f x +e \right ) b^{3}}{16 f \left (a -b \right )^{4} \left (1+\tan ^{2}\left (f x +e \right )\right )^{3}}+\frac {5 \arctan \left (\tan \left (f x +e \right )\right ) a^{3}}{16 f \left (a -b \right )^{4}}+\frac {15 \arctan \left (\tan \left (f x +e \right )\right ) a^{2} b}{16 f \left (a -b \right )^{4}}-\frac {5 \arctan \left (\tan \left (f x +e \right )\right ) b^{2} a}{16 f \left (a -b \right )^{4}}+\frac {\arctan \left (\tan \left (f x +e \right )\right ) b^{3}}{16 f \left (a -b \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(f*x+e)^6/(a+b*tan(f*x+e)^2),x)

[Out]

-1/f*b*a^3/(a-b)^4/(a*b)^(1/2)*arctan(tan(f*x+e)*b/(a*b)^(1/2))-11/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^
5*a^3+15/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^5*a^2*b-5/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^5*b^2
*a+1/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^5*b^3-5/6/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^3*a^3+1/2/f/
(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^3*a^2*b+1/2/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^3*b^2*a-1/6/f/(a-b)^
4/(1+tan(f*x+e)^2)^3*tan(f*x+e)^3*b^3-5/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)*a^3+1/16/f/(a-b)^4/(1+tan(f
*x+e)^2)^3*tan(f*x+e)*a^2*b+5/16/f/(a-b)^4/(1+tan(f*x+e)^2)^3*tan(f*x+e)*b^2*a-1/16/f/(a-b)^4/(1+tan(f*x+e)^2)
^3*tan(f*x+e)*b^3+5/16/f/(a-b)^4*arctan(tan(f*x+e))*a^3+15/16/f/(a-b)^4*arctan(tan(f*x+e))*a^2*b-5/16/f/(a-b)^
4*arctan(tan(f*x+e))*b^2*a+1/16/f/(a-b)^4*arctan(tan(f*x+e))*b^3

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maxima [A]  time = 0.81, size = 305, normalized size = 1.71 \[ -\frac {\frac {48 \, a^{3} b \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right )}{{\left (a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} \sqrt {a b}} - \frac {3 \, {\left (5 \, a^{3} + 15 \, a^{2} b - 5 \, a b^{2} + b^{3}\right )} {\left (f x + e\right )}}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (11 \, a^{2} - 4 \, a b + b^{2}\right )} \tan \left (f x + e\right )^{5} + 8 \, {\left (5 \, a^{2} + 2 \, a b - b^{2}\right )} \tan \left (f x + e\right )^{3} + 3 \, {\left (5 \, a^{2} + 4 \, a b - b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{6} + 3 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{4} + a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3} + 3 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} \tan \left (f x + e\right )^{2}}}{48 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(f*x+e)^6/(a+b*tan(f*x+e)^2),x, algorithm="maxima")

[Out]

-1/48*(48*a^3*b*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*(
5*a^3 + 15*a^2*b - 5*a*b^2 + b^3)*(f*x + e)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + (3*(11*a^2 - 4*a*b +
b^2)*tan(f*x + e)^5 + 8*(5*a^2 + 2*a*b - b^2)*tan(f*x + e)^3 + 3*(5*a^2 + 4*a*b - b^2)*tan(f*x + e))/((a^3 -
3*a^2*b + 3*a*b^2 - b^3)*tan(f*x + e)^6 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*tan(f*x + e)^4 + a^3 - 3*a^2*b + 3
*a*b^2 - b^3 + 3*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*tan(f*x + e)^2))/f

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mupad [B]  time = 17.02, size = 4910, normalized size = 27.58 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(e + f*x)^6/(a + b*tan(e + f*x)^2),x)

[Out]

(atan(-((((((3*a^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^
6)/2 - 40*a^8*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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) - (tan(e + f*x)*(a^2*b*15i - a*b^2*5i + a
^3*5i + b^3*1i)*(1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 286
72*a^6*b^5 + 20480*a^7*b^4 - 7168*a^8*b^3 + 1024*a^9*b^2))/(4096*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1
i))/(32*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^
6 + 175*a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))/(128*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i)*1i)/(32*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) -
(((((3*a^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 4
0*a^8*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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) + (tan(e + f*x)*(a^2*b*15i - a*b^2*5i + a^3*5i +
b^3*1i)*(1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b
^5 + 20480*a^7*b^4 - 7168*a^8*b^3 + 1024*a^9*b^2))/(4096*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i))/(32*
(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^6 + 175*
a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))/(128*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i)*1i)/(32*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)))/(((a^3*b^
8)/128 - (9*a^4*b^7)/128 + (23*a^5*b^6)/64 - (55*a^6*b^5)/64 + (145*a^7*b^4)/128 + (55*a^8*b^3)/128)/(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^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 40*a^8
*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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) - (tan(e + f*x)*(a^2*b*15i - a*b^2*5i + a^3*5i + b^3*1
i)*(1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 +
20480*a^7*b^4 - 7168*a^8*b^3 + 1024*a^9*b^2))/(4096*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i))/(32*(a^4
- 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^6 + 175*a^4*b
^5 + 150*a^5*b^4 + 281*a^6*b^3))/(128*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i))/(32*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (((((3*a^2*b^11
- (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 40*a^8*b^5 + (107
*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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) + (tan(e + f*x)*(a^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i)*(1024*b^
11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 + 20480*a^7*b
^4 - 7168*a^8*b^3 + 1024*a^9*b^2))/(4096*(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^2*b*15i - a*b^2*5i + a^3*5i + b^3*1i))/(32*(a^4 - 4*a^3*b -
4*a*b^3 + b^4 + 6*a^2*b^2)) + (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^6 + 175*a^4*b^5 + 150*a^
5*b^4 + 281*a^6*b^3))/(128*(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^2*b*15i
- a*b^2*5i + a^3*5i + b^3*1i))/(32*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))))*(a^2*b*15i - a*b^2*5i + a^3
*5i + b^3*1i)*1i)/(16*f*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - ((tan(e + f*x)*(4*a*b + 5*a^2 - b^2))/(
16*(3*a*b^2 - 3*a^2*b + a^3 - b^3)) + (tan(e + f*x)^5*(11*a^2 - 4*a*b + b^2))/(16*(3*a*b^2 - 3*a^2*b + a^3 - b
^3)) + (tan(e + f*x)^3*(2*a*b + 5*a^2 - b^2))/(6*(3*a*b^2 - 3*a^2*b + a^3 - b^3)))/(f*(3*tan(e + f*x)^2 + 3*ta
n(e + f*x)^4 + tan(e + f*x)^6 + 1)) - (atan((((-a^5*b)^(1/2)*((((3*a^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 3
2*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 40*a^8*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11
*b^2)/4)/(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)) - (tan(e + f*x)*(-a^5*b)^(1/2)*(1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336
*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 + 20480*a^7*b^4 - 7168*a^8*b^3 + 1024*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^5*b)^
(1/2))/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*
b^6 + 175*a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))/(256*(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)))*1i)/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2) - ((-a^5*b)^(1/2)*((((3*a^2*b^11 - (a*b^12)/4
- (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 40*a^8*b^5 + (107*a^9*b^4)/4 -
9*a^10*b^3 + (5*a^11*b^2)/4)/(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)) + (tan(e + f*x)*(-a^5*b)^(1/2)*(1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 -
28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 + 20480*a^7*b^4 - 7168*a^8*b^3 + 1024*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^5*b)^(1/2))/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) + (tan(e + f*x)*(b^9 - 10*a*b^8 +
55*a^2*b^7 - 140*a^3*b^6 + 175*a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))/(256*(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)))*1i)/(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2))/(((a^3*b^8)/128 - (9*a^4
*b^7)/128 + (23*a^5*b^6)/64 - (55*a^6*b^5)/64 + (145*a^7*b^4)/128 + (55*a^8*b^3)/128)/(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) + ((-a^5*b)^(1/2)*((((
3*a^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a^7*b^6)/2 - 40*a^8
*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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)) - (tan(e + f*x)*(-a^5*b)^(1/2)*(1024*b^11 - 7168*a
*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 + 20480*a^7*b^4 - 7168*a
^8*b^3 + 1024*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^5*b)^(1/2))/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2)) - (tan(e +
f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^6 + 175*a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))/(256*(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^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^2*b^2) + ((
-a^5*b)^(1/2)*((((3*a^2*b^11 - (a*b^12)/4 - (55*a^3*b^10)/4 + 32*a^4*b^9 - (77*a^5*b^8)/2 + 14*a^6*b^7 + (49*a
^7*b^6)/2 - 40*a^8*b^5 + (107*a^9*b^4)/4 - 9*a^10*b^3 + (5*a^11*b^2)/4)/(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)) + (tan(e + f*x)*(-a^5*b)^(1/2)*(
1024*b^11 - 7168*a*b^10 + 20480*a^2*b^9 - 28672*a^3*b^8 + 14336*a^4*b^7 + 14336*a^5*b^6 - 28672*a^6*b^5 + 2048
0*a^7*b^4 - 7168*a^8*b^3 + 1024*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^5*b)^(1/2))/(2*(a^4 - 4*a^3*b - 4*a*b^3 + b^4 + 6*a^
2*b^2)) + (tan(e + f*x)*(b^9 - 10*a*b^8 + 55*a^2*b^7 - 140*a^3*b^6 + 175*a^4*b^5 + 150*a^5*b^4 + 281*a^6*b^3))
/(256*(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^4 - 4*a^3*b - 4*a*b^3 + b^4
+ 6*a^2*b^2)))*(-a^5*b)^(1/2)*1i)/(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)**6/(a+b*tan(f*x+e)**2),x)

[Out]

Timed out

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