\(\int (a+b \sec ^2(e+f x))^{3/2} \tan ^6(e+f x) \, dx\) [395]

Optimal result

Integrand size = 25, antiderivative size = 290 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}-\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f} \]

[Out]

-a^(3/2)*arctan(a^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))/f+1/128*(3*a^4+20*a^3*b+90*a^2*b^2-60*a*b^3-5*b
^4)*arctanh(b^(1/2)*tan(f*x+e)/(a+b+b*tan(f*x+e)^2)^(1/2))/b^(5/2)/f-1/128*(3*a^3+17*a^2*b-55*a*b^2-5*b^3)*(a+
b+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)/b^2/f+1/192*(3*a^2-50*a*b-5*b^2)*(a+b+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^3/b/
f+1/48*(9*a+b)*(a+b+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^5/f+1/8*b*(a+b+b*tan(f*x+e)^2)^(1/2)*tan(f*x+e)^7/f

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4226, 2000, 488, 596, 537, 223, 212, 385, 209} \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=-\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{192 b f}-\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{128 b^2 f}+\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)+b}}\right )}{128 b^{5/2} f}+\frac {b \tan ^7(e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{8 f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b \tan ^2(e+f x)+b}}{48 f} \]

[In]

Int[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^6,x]

[Out]

-((a^(3/2)*ArcTan[(Sqrt[a]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/f) + ((3*a^4 + 20*a^3*b + 90*a^2*b^2
- 60*a*b^3 - 5*b^4)*ArcTanh[(Sqrt[b]*Tan[e + f*x])/Sqrt[a + b + b*Tan[e + f*x]^2]])/(128*b^(5/2)*f) - ((3*a^3
+ 17*a^2*b - 55*a*b^2 - 5*b^3)*Tan[e + f*x]*Sqrt[a + b + b*Tan[e + f*x]^2])/(128*b^2*f) + ((3*a^2 - 50*a*b -
5*b^2)*Tan[e + f*x]^3*Sqrt[a + b + b*Tan[e + f*x]^2])/(192*b*f) + ((9*a + b)*Tan[e + f*x]^5*Sqrt[a + b + b*Tan
[e + f*x]^2])/(48*f) + (b*Tan[e + f*x]^7*Sqrt[a + b + b*Tan[e + f*x]^2])/(8*f)

Rule 209

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

Rule 212

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 488

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[d*(e*x)^
(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(b*e*(m + n*(p + q) + 1))), x] + Dist[1/(b*(m + n*(p + q) + 1
)), Int[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*((c*b - a*d)*(m + 1) + c*b*n*(p + q)) + (d*(c*b - a*d
)*(m + 1) + d*n*(q - 1)*(b*c - a*d) + c*b*d*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && N
eQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 537

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

Rule 596

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

Rule 2000

Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*ExpandToSum[u, x]^p*ExpandToSum[v, x]^q
, x] /; FreeQ[{e, m, p, q}, x] && BinomialQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0]
&&  !BinomialMatchQ[{u, v}, x]

Rule 4226

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6 \left (a+b \left (1+x^2\right )\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {x^6 \left (a+b+b x^2\right )^{3/2}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}+\frac {\text {Subst}\left (\int \frac {x^6 \left ((a+b) (8 a+b)+b (9 a+b) x^2\right )}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}-\frac {\text {Subst}\left (\int \frac {x^4 \left (5 b (a+b) (9 a+b)-b \left (3 a^2-50 a b-5 b^2\right ) x^2\right )}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{48 b f} \\ & = \frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}+\frac {\text {Subst}\left (\int \frac {x^2 \left (-3 b (a+b) \left (3 a^2-50 a b-5 b^2\right )-3 b \left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) x^2\right )}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{192 b^2 f} \\ & = -\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}-\frac {\text {Subst}\left (\int \frac {-3 b (a+b) \left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right )-3 b \left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{384 b^3 f} \\ & = -\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{f}+\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b+b x^2}} \, dx,x,\tan (e+f x)\right )}{128 b^2 f} \\ & = -\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{128 b^2 f} \\ & = -\frac {a^{3/2} \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{f}+\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b+b \tan ^2(e+f x)}}\right )}{128 b^{5/2} f}-\frac {\left (3 a^3+17 a^2 b-55 a b^2-5 b^3\right ) \tan (e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{128 b^2 f}+\frac {\left (3 a^2-50 a b-5 b^2\right ) \tan ^3(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{192 b f}+\frac {(9 a+b) \tan ^5(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{48 f}+\frac {b \tan ^7(e+f x) \sqrt {a+b+b \tan ^2(e+f x)}}{8 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.88 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.22 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=-\frac {\left (128 a^{3/2} b^2 \arctan \left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+b-a \sin ^2(e+f x)}}\right )-\frac {\left (3 a^4+20 a^3 b+90 a^2 b^2-60 a b^3-5 b^4\right ) \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b-a \sin ^2(e+f x)}}\right )}{\sqrt {b}}\right ) \cos ^3(e+f x) \left (a+b \sec ^2(e+f x)\right )^{3/2}}{32 \sqrt {2} b^2 f (a+2 b+a \cos (2 e+2 f x))^{3/2}}-\frac {\left (90 a^3+498 a^2 b-1594 a b^2-626 b^3+\left (135 a^3+759 a^2 b-2303 a b^2+513 b^3\right ) \cos (2 (e+f x))+2 \left (27 a^3+159 a^2 b-523 a b^2-191 b^3\right ) \cos (4 (e+f x))+9 a^3 \cos (6 (e+f x))+57 a^2 b \cos (6 (e+f x))-337 a b^2 \cos (6 (e+f x))+15 b^3 \cos (6 (e+f x))\right ) \sec ^6(e+f x) \sqrt {a+b \sec ^2(e+f x)} \tan (e+f x)}{12288 b^2 f} \]

[In]

Integrate[(a + b*Sec[e + f*x]^2)^(3/2)*Tan[e + f*x]^6,x]

[Out]

-1/32*((128*a^(3/2)*b^2*ArcTan[(Sqrt[a]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]] - ((3*a^4 + 20*a^3*b + 9
0*a^2*b^2 - 60*a*b^3 - 5*b^4)*ArcTanh[(Sqrt[b]*Sin[e + f*x])/Sqrt[a + b - a*Sin[e + f*x]^2]])/Sqrt[b])*Cos[e +
f*x]^3*(a + b*Sec[e + f*x]^2)^(3/2))/(Sqrt[2]*b^2*f*(a + 2*b + a*Cos[2*e + 2*f*x])^(3/2)) - ((90*a^3 + 498*a^
2*b - 1594*a*b^2 - 626*b^3 + (135*a^3 + 759*a^2*b - 2303*a*b^2 + 513*b^3)*Cos[2*(e + f*x)] + 2*(27*a^3 + 159*a
^2*b - 523*a*b^2 - 191*b^3)*Cos[4*(e + f*x)] + 9*a^3*Cos[6*(e + f*x)] + 57*a^2*b*Cos[6*(e + f*x)] - 337*a*b^2*
Cos[6*(e + f*x)] + 15*b^3*Cos[6*(e + f*x)])*Sec[e + f*x]^6*Sqrt[a + b*Sec[e + f*x]^2]*Tan[e + f*x])/(12288*b^2
*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2228\) vs. \(2(260)=520\).

Time = 30.40 (sec) , antiderivative size = 2229, normalized size of antiderivative = 7.69

method	result	size
default	\(\text {Expression too large to display}\)	\(2229\)

[In]

int((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^6,x,method=_RETURNVERBOSE)

[Out]

-1/768/f/(-a)^(1/2)/b^(13/2)*(a+b*sec(f*x+e)^2)^(3/2)/((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)/(b+a*cos(f*x
+e)^2)/(1+cos(f*x+e))*(-674*cos(f*x+e)*sin(f*x+e)*(-a)^(1/2)*b^(13/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1
/2)*a-9*cos(f*x+e)^3*(-a)^(1/2)*ln(4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*(
(b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a-a-b)/(sin(f*x+e)+1))*a^4*b^4-60*cos(f*x+e)^3*(-a)^(1/2
)*ln(4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+
e))^2)^(1/2)-sin(f*x+e)*a-a-b)/(sin(f*x+e)+1))*a^3*b^5-270*cos(f*x+e)^3*(-a)^(1/2)*ln(4*(((b+a*cos(f*x+e)^2)/(
1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a-a-b
)/(sin(f*x+e)+1))*a^2*b^6+180*cos(f*x+e)^3*(-a)^(1/2)*ln(4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2
)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a-a-b)/(sin(f*x+e)+1))*a*b^7-9*cos
(f*x+e)^3*(-a)^(1/2)*ln(-4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f
*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a+a+b)/(sin(f*x+e)-1))*a^4*b^4-60*cos(f*x+e)^3*(-a)^(1/2)*ln(-4*((
(b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1
/2)-sin(f*x+e)*a+a+b)/(sin(f*x+e)-1))*a^3*b^5-270*cos(f*x+e)^3*(-a)^(1/2)*ln(-4*(((b+a*cos(f*x+e)^2)/(1+cos(f*
x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a+a+b)/(sin(f
*x+e)-1))*a^2*b^6+180*cos(f*x+e)^3*(-a)^(1/2)*ln(-4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f
*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a+a+b)/(sin(f*x+e)-1))*a*b^7+488*(-a)^(1/
2)*b^(13/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a*sin(f*x+e)+488*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)
^(1/2)*(-a)^(1/2)*b^(13/2)*a*tan(f*x+e)-96*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(15/2)*tan
(f*x+e)*sec(f*x+e)^4-96*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(15/2)*tan(f*x+e)*sec(f*x+e)^
3-12*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(11/2)*a^2*tan(f*x+e)+30*cos(f*x+e)^2*sin(f*x+e)
*(-a)^(1/2)*b^(15/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+30*cos(f*x+e)*sin(f*x+e)*(-a)^(1/2)*b^(15/2)*
((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)+272*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(15/2
)*tan(f*x+e)*sec(f*x+e)+272*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(15/2)*tan(f*x+e)*sec(f*x
+e)^2-12*(-a)^(1/2)*b^(11/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^2*sin(f*x+e)+18*cos(f*x+e)^2*sin(f*
x+e)*(-a)^(1/2)*b^(9/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^3+18*cos(f*x+e)*sin(f*x+e)*(-a)^(1/2)*b^
(9/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^3-674*cos(f*x+e)^2*sin(f*x+e)*(-a)^(1/2)*b^(13/2)*((b+a*co
s(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a-144*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(13/2)*a*ta
n(f*x+e)*sec(f*x+e)^2+114*cos(f*x+e)^2*sin(f*x+e)*(-a)^(1/2)*b^(11/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1
/2)*a^2+114*cos(f*x+e)*sin(f*x+e)*(-a)^(1/2)*b^(11/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*a^2-144*((b+
a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b^(13/2)*a*tan(f*x+e)*sec(f*x+e)+768*cos(f*x+e)^3*b^(13/2)*
ln(4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+4*(-a)^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos
(f*x+e))^2)^(1/2)-4*sin(f*x+e)*a)*a^2-236*(-a)^(1/2)*b^(15/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*sin(
f*x+e)+15*cos(f*x+e)^3*(-a)^(1/2)*ln(4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)
*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)-sin(f*x+e)*a-a-b)/(sin(f*x+e)+1))*b^8+15*cos(f*x+e)^3*(-a)^(1/2)*
ln(-4*(((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*b^(1/2)*cos(f*x+e)+b^(1/2)*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e
))^2)^(1/2)-sin(f*x+e)*a+a+b)/(sin(f*x+e)-1))*b^8-236*((b+a*cos(f*x+e)^2)/(1+cos(f*x+e))^2)^(1/2)*(-a)^(1/2)*b
^(15/2)*tan(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 14.98 (sec) , antiderivative size = 1973, normalized size of antiderivative = 6.80 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^6,x, algorithm="fricas")

[Out]

[1/1536*(192*sqrt(-a)*a*b^3*cos(f*x + e)^7*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*b)*cos(f*x + e)^6 + 32*
(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3 + b^4 - 32*(a^4 - 7*a^3
*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*cos(f*x + e)^5 + 2*(5*a^3
- 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*sqrt(-a)*sqrt((a*cos(f*x
+ e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) - 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(b)*cos(
f*x + e)^7*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*x + e)^3
+ 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x + e)^4)
- 4*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x +
e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)
)/(b^3*f*cos(f*x + e)^7), 1/768*(96*sqrt(-a)*a*b^3*cos(f*x + e)^7*log(128*a^4*cos(f*x + e)^8 - 256*(a^4 - a^3*
b)*cos(f*x + e)^6 + 32*(5*a^4 - 14*a^3*b + 5*a^2*b^2)*cos(f*x + e)^4 + a^4 - 28*a^3*b + 70*a^2*b^2 - 28*a*b^3
+ b^4 - 32*(a^4 - 7*a^3*b + 7*a^2*b^2 - a*b^3)*cos(f*x + e)^2 + 8*(16*a^3*cos(f*x + e)^7 - 24*(a^3 - a^2*b)*co
s(f*x + e)^5 + 2*(5*a^3 - 14*a^2*b + 5*a*b^2)*cos(f*x + e)^3 - (a^3 - 7*a^2*b + 7*a*b^2 - b^3)*cos(f*x + e))*s
qrt(-a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e)) + 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^
3 - 5*b^4)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b)*sqrt((a*cos(f*x + e)^2 +
b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^7 - 2*((9*a^3*b + 57*a^2*b^2 - 337*
a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*
b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/1536*
(384*a^(3/2)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x
+ e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*cos(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a
^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^7 - 3*(3*a^4 + 20*a^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqr
t(b)*cos(f*x + e)^7*log(((a^2 - 6*a*b + b^2)*cos(f*x + e)^4 + 8*(a*b - b^2)*cos(f*x + e)^2 - 4*((a - b)*cos(f*
x + e)^3 + 2*b*cos(f*x + e))*sqrt(b)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e) + 8*b^2)/cos(f*x
+ e)^4) - 4*((9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*
cos(f*x + e)^4 - 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin
(f*x + e))/(b^3*f*cos(f*x + e)^7), 1/768*(192*a^(3/2)*b^3*arctan(1/4*(8*a^2*cos(f*x + e)^5 - 8*(a^2 - a*b)*cos
(f*x + e)^3 + (a^2 - 6*a*b + b^2)*cos(f*x + e))*sqrt(a)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((2*a^3*co
s(f*x + e)^4 - a^2*b + a*b^2 - (a^3 - 3*a^2*b)*cos(f*x + e)^2)*sin(f*x + e)))*cos(f*x + e)^7 + 3*(3*a^4 + 20*a
^3*b + 90*a^2*b^2 - 60*a*b^3 - 5*b^4)*sqrt(-b)*arctan(-1/2*((a - b)*cos(f*x + e)^3 + 2*b*cos(f*x + e))*sqrt(-b
)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)/((a*b*cos(f*x + e)^2 + b^2)*sin(f*x + e)))*cos(f*x + e)^7 - 2*((
9*a^3*b + 57*a^2*b^2 - 337*a*b^3 + 15*b^4)*cos(f*x + e)^6 - 2*(3*a^2*b^2 - 122*a*b^3 + 59*b^4)*cos(f*x + e)^4
- 48*b^4 - 8*(9*a*b^3 - 17*b^4)*cos(f*x + e)^2)*sqrt((a*cos(f*x + e)^2 + b)/cos(f*x + e)^2)*sin(f*x + e))/(b^3
*f*cos(f*x + e)^7)]

Sympy [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{6}{\left (e + f x \right )}\, dx \]

[In]

integrate((a+b*sec(f*x+e)**2)**(3/2)*tan(f*x+e)**6,x)

[Out]

Integral((a + b*sec(e + f*x)**2)**(3/2)*tan(e + f*x)**6, x)

Maxima [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{6} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^6,x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e)^2 + a)^(3/2)*tan(f*x + e)^6, x)

Giac [F]

\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{6} \,d x } \]

[In]

integrate((a+b*sec(f*x+e)^2)^(3/2)*tan(f*x+e)^6,x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e)^2 + a)^(3/2)*tan(f*x + e)^6, x)

Mupad [F(-1)]

Timed out. \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^6(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^6\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2} \,d x \]

[In]

int(tan(e + f*x)^6*(a + b/cos(e + f*x)^2)^(3/2),x)

[Out]

int(tan(e + f*x)^6*(a + b/cos(e + f*x)^2)^(3/2), x)