\(\int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx\) [111]

Optimal result

Integrand size = 25, antiderivative size = 366 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e} \]

[Out]

1/2*a^2*e^(5/2)*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)-1/2*a^2*e^(5/2)*arctan(1+2^(1/2)*(e*t
an(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)-1/4*a^2*e^(5/2)*ln(e^(1/2)-2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c)
)/d*2^(1/2)+1/4*a^2*e^(5/2)*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/d*2^(1/2)-12/5*a^2*e^2
*cos(d*x+c)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticE(cos(c+1/4*Pi+d*x),2^(1/2))*(e*tan(d*x+c))^
(1/2)/d/sin(2*d*x+2*c)^(1/2)+2/3*a^2*e*(e*tan(d*x+c))^(3/2)/d-12/5*a^2*e*cos(d*x+c)*(e*tan(d*x+c))^(3/2)/d+4/5
*a^2*e*sec(d*x+c)*(e*tan(d*x+c))^(3/2)/d+2/7*a^2*(e*tan(d*x+c))^(7/2)/d/e

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3971, 3554, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2691, 2693, 2695, 2652, 2719, 2687, 32} \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d} \]

[In]

Int[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]

[Out]

(a^2*e^(5/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*ArcTan[1 + (Sqrt[2
]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d) - (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt
[e*Tan[c + d*x]]])/(2*Sqrt[2]*d) + (a^2*e^(5/2)*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*
x]]])/(2*Sqrt[2]*d) + (12*a^2*e^2*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2]*Sqrt[e*Tan[c + d*x]])/(5*d*Sqrt[Si
n[2*c + 2*d*x]]) + (2*a^2*e*(e*Tan[c + d*x])^(3/2))/(3*d) - (12*a^2*e*Cos[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*
d) + (4*a^2*e*Sec[c + d*x]*(e*Tan[c + d*x])^(3/2))/(5*d) + (2*a^2*(e*Tan[c + d*x])^(7/2))/(7*d*e)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 210

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

Rule 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2693

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^2*(a*Sec[e
+ f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Dist[a^2*((m - 2)/(m + n - 1)), Int[(a*Sec
[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[
n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 2695

Int[Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]]/sec[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[Sqrt[Cos[e + f*x]]*(Sqrt[b*
Tan[e + f*x]]/Sqrt[Sin[e + f*x]]), Int[Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]], x], x] /; FreeQ[{b, e, f}, x]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 (e \tan (c+d x))^{5/2}+2 a^2 \sec (c+d x) (e \tan (c+d x))^{5/2}+a^2 \sec ^2(c+d x) (e \tan (c+d x))^{5/2}\right ) \, dx \\ & = a^2 \int (e \tan (c+d x))^{5/2} \, dx+a^2 \int \sec ^2(c+d x) (e \tan (c+d x))^{5/2} \, dx+\left (2 a^2\right ) \int \sec (c+d x) (e \tan (c+d x))^{5/2} \, dx \\ & = \frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {a^2 \text {Subst}\left (\int (e x)^{5/2} \, dx,x,\tan (c+d x)\right )}{d}-\left (a^2 e^2\right ) \int \sqrt {e \tan (c+d x)} \, dx-\frac {1}{5} \left (6 a^2 e^2\right ) \int \sec (c+d x) \sqrt {e \tan (c+d x)} \, dx \\ & = \frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac {1}{5} \left (12 a^2 e^2\right ) \int \cos (c+d x) \sqrt {e \tan (c+d x)} \, dx-\frac {\left (a^2 e^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{e^2+x^2} \, dx,x,e \tan (c+d x)\right )}{d} \\ & = \frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}-\frac {\left (2 a^2 e^3\right ) \text {Subst}\left (\int \frac {x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}+\frac {\left (12 a^2 e^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{5 \sqrt {\sin (c+d x)}} \\ & = \frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}+\frac {\left (a^2 e^3\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}-\frac {\left (a^2 e^3\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{d}+\frac {\left (12 a^2 e^2 \cos (c+d x) \sqrt {e \tan (c+d x)}\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{5 \sqrt {\sin (2 c+2 d x)}} \\ & = \frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}-\frac {\left (a^2 e^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2 e^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2 e^3\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 d}-\frac {\left (a^2 e^3\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 d} \\ & = -\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e}-\frac {\left (a^2 e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}+\frac {\left (a^2 e^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d} \\ & = \frac {a^2 e^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d}-\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {a^2 e^{5/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {12 a^2 e^2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sqrt {e \tan (c+d x)}}{5 d \sqrt {\sin (2 c+2 d x)}}+\frac {2 a^2 e (e \tan (c+d x))^{3/2}}{3 d}-\frac {12 a^2 e \cos (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {4 a^2 e \sec (c+d x) (e \tan (c+d x))^{3/2}}{5 d}+\frac {2 a^2 (e \tan (c+d x))^{7/2}}{7 d e} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 12.89 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.32 \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\frac {2 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} \arctan (\tan (c+d x))\right ) (e \tan (c+d x))^{3/2} \left (35-42 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-35 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\tan ^2(c+d x)\right )+42 \sqrt {\sec ^2(c+d x)}+15 \tan ^2(c+d x)\right )}{105 d} \]

[In]

Integrate[(a + a*Sec[c + d*x])^2*(e*Tan[c + d*x])^(5/2),x]

[Out]

(2*a^2*e*Cos[(c + d*x)/2]^4*Sec[ArcTan[Tan[c + d*x]]/2]^4*(e*Tan[c + d*x])^(3/2)*(35 - 42*Hypergeometric2F1[1/
2, 3/4, 7/4, -Tan[c + d*x]^2] - 35*Hypergeometric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2] + 42*Sqrt[Sec[c + d*x]^2] +
15*Tan[c + d*x]^2))/(105*d)

Maple [A] (warning: unable to verify)

Time = 6.60 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.65

method	result	size
parts	\(\frac {2 a^{2} e \left (\frac {\left (e \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \sqrt {2}\, \left (\ln \left (\frac {e \tan \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \tan \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \tan \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}+\frac {2 a^{2} \left (e \tan \left (d x +c \right )\right )^{\frac {7}{2}}}{7 d e}-\frac {2 a^{2} \sqrt {2}\, \left (-6 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{3}+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{3}-6 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{2}+3 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )^{3} \sqrt {2}-4 \sqrt {2}\, \cos \left (d x +c \right )^{2}+\sqrt {2}\right ) e^{2} \sqrt {e \tan \left (d x +c \right )}\, \sec \left (d x +c \right ) \tan \left (d x +c \right )}{5 d \left (\cos \left (d x +c \right )^{2}-1\right )}\)	\(605\)
default	\(\text {Expression too large to display}\)	\(1167\)

[In]

int((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

2*a^2/d*e*(1/3*(e*tan(d*x+c))^(3/2)-1/8*e^2/(e^2)^(1/4)*2^(1/2)*(ln((e*tan(d*x+c)-(e^2)^(1/4)*(e*tan(d*x+c))^(
1/2)*2^(1/2)+(e^2)^(1/2))/(e*tan(d*x+c)+(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2
)/(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*tan(d*x+c))^(1/2)+1)))+2/7*a^2*(e*tan(d
*x+c))^(7/2)/d/e-2/5*a^2/d*2^(1/2)*(-6*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*
x+c)-csc(d*x+c))^(1/2)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^3+3*(csc(d*x+c)-cot(d
*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+
1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^3-6*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+
c)-csc(d*x+c))^(1/2)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^2+3*(csc(d*x+c)-cot(d*x
+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)
^(1/2),1/2*2^(1/2))*cos(d*x+c)^2+3*cos(d*x+c)^3*2^(1/2)-4*2^(1/2)*cos(d*x+c)^2+2^(1/2))*e^2*(e*tan(d*x+c))^(1/
2)/(cos(d*x+c)^2-1)*sec(d*x+c)*tan(d*x+c)

Fricas [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sec(d*x+c))**2*(e*tan(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \tan \left (d x + c\right )\right )^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^2*(e*tan(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^2*(e*tan(d*x + c))^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^2 (e \tan (c+d x))^{5/2} \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \]

[In]

int((e*tan(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2,x)

[Out]

int((e*tan(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2, x)