3.117 \(\int \frac{(a+a \sec (c+d x))^2}{(e \tan (c+d x))^{7/2}} \, dx\)

Optimal. Leaf size=370 \[ -\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}+\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{d e^3 \sqrt{e \tan (c+d x)}}+\frac{a^2 \log \left (\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}-\frac{a^2 \log \left (\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{12 a^2 \cos (c+d x)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{12 a^2 \cos (c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right ) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac{4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.455875, antiderivative size = 370, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 17, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.68, Rules used = {3886, 3474, 3476, 329, 297, 1162, 617, 204, 1165, 628, 2609, 2608, 2615, 2572, 2639, 2607, 32} \[ -\frac{a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}\right )}{\sqrt{2} d e^{7/2}}+\frac{a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{e \tan (c+d x)}}{\sqrt{e}}+1\right )}{\sqrt{2} d e^{7/2}}+\frac{2 a^2}{d e^3 \sqrt{e \tan (c+d x)}}+\frac{a^2 \log \left (\sqrt{e} \tan (c+d x)-\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}-\frac{a^2 \log \left (\sqrt{e} \tan (c+d x)+\sqrt{2} \sqrt{e \tan (c+d x)}+\sqrt{e}\right )}{2 \sqrt{2} d e^{7/2}}+\frac{12 a^2 \cos (c+d x)}{5 d e^3 \sqrt{e \tan (c+d x)}}+\frac{12 a^2 \cos (c+d x) E\left (\left .c+d x-\frac{\pi }{4}\right |2\right ) \sqrt{e \tan (c+d x)}}{5 d e^4 \sqrt{\sin (2 c+2 d x)}}-\frac{4 a^2}{5 d e (e \tan (c+d x))^{5/2}}-\frac{4 a^2 \sec (c+d x)}{5 d e (e \tan (c+d x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2),x]

[Out]

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

Rule 3886

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]

Rule 3474

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

Rule 3476

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 329

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 297

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 1162

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 617

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 204

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

Rule 1165

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 628

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 2609

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((a*Sec[e +
f*x])^m*(b*Tan[e + f*x])^(n + 1))/(b*f*(n + 1)), x] - Dist[(m + n + 1)/(b^2*(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] && LtQ[n, -1] && IntegersQ[2*m, 2*n]

Rule 2608

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*(n + 1)), x] - Dist[(a^2*(m - 2))/(b^2*(n + 1)), Int[(a*Sec[e
+ f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || (Eq
Q[m, 1] && EqQ[n, -3/2])) && IntegersQ[2*m, 2*n]

Rule 2615

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 2572

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 2639

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

Rule 2607

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 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]

Rubi steps

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

Mathematica [C]  time = 13.9828, size = 2820, normalized size = 7.62 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + a*Sec[c + d*x])^2/(e*Tan[c + d*x])^(7/2),x]

[Out]

(Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*((7*Cot[c/2])/(10*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^2)/(20*d) - (
3*(4*Cos[c/2] - Cos[(3*c)/2] + Cos[(5*c)/2])*Cos[d*x]*Sec[2*c]*Sin[c/2])/(10*d*(-1 + 2*Cos[c])) - (7*Csc[c/2]*
Csc[c/2 + (d*x)/2]*Sin[(d*x)/2])/(10*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^3*Sin[(d*x)/2])/(20*d) - (3*(2 - 5*Cos[
c] + 6*Cos[2*c] + Cos[3*c])*Sec[2*c]*Sin[d*x])/(20*d*(-1 + 2*Cos[c])))*Sin[c + d*x]^2*Tan[c + d*x]^2)/(e*Tan[c
+ d*x])^(7/2) + ((E^((2*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*Sqrt[
-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*
(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E
^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos
[c])*(e*Tan[c + d*x])^(7/2)) + ((-(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d
*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)
))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*
x]^(7/2))/(16*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c
+ d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - ((-(E^((6*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt
[-1 + E^((4*I)*(c + d*x))]]) + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1
+ E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c
+ d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((3*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x
)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + ((Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcT
an[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((2*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))
]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)
/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^(I*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^(
(2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - ((Sqrt[-1 + E^((4*I)*(c
+ d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2
*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c + d*x]^2*Sec[2*c]*S
ec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((2*I)*c)*Sqrt[((-I)*(-1 + E^((2*I)*(c
+ d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) + ((Sqr
t[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] - 2*E^((6*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x
))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c +
d*x]^2*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(16*d*E^((3*I)*c)*Sqrt[((-I)*
(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*
x])^(7/2)) - (Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4
*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Se
c[c + d*x])^2*Tan[c + d*x]^(7/2))/(20*d*E^(I*(2*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)
*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(3 - 3*E^((4
*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2,
3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(20*d
*E^(I*d*x)*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2
*Cos[c])*(e*Tan[c + d*x])^(7/2)) + (E^(I*(c - d*x))*Cos[c + d*x]^2*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*d*x)*
(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c
]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(8*d*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/
(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2
*(3 - 3*E^((4*I)*(c + d*x)) + E^((4*I)*(c + d*x))*(1 + E^((4*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometr
ic2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(
7/2))/(40*d*E^(I*(3*c + d*x))*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*
(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c + d*x])^(7/2)) - (Cos[c + d*x]^2*(-3*E^((2*I)*c)*(-1 + E^((4*I)*(c + d*x)
)) + E^((4*I)*d*x)*(1 + E^((6*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(
c + d*x))])*Sec[2*c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*Tan[c + d*x]^(7/2))/(10*d*E^(I*d*x)*Sqrt[((-I
)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*(-1 + 2*Cos[c])*(e*Tan[c +
d*x])^(7/2))

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Maple [C]  time = 0.291, size = 1429, normalized size = 3.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(7/2),x)

[Out]

1/10*a^2/d*2^(1/2)*(5*I*cos(d*x+c)^2*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/
sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1
/2-1/2*I,1/2*2^(1/2))-5*I*cos(d*x+c)^2*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c)
)/sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)
,1/2+1/2*I,1/2*2^(1/2))-24*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((
1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticE(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))
*cos(d*x+c)^2+12*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+
c)+sin(d*x+c))/sin(d*x+c))^(1/2)*EllipticF(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*cos(d*x+c
)^2-5*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+
c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c
)^2-5*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+
c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*cos(d*x+c
)^2-5*I*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*
x+c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))+5*I*((1
-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d
*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))+24*((1-cos(d*x+c)+
sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)
*EllipticE(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))-12*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))
^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*EllipticF(((1-cos(d*x+
c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))+5*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+s
in(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+
c))^(1/2),1/2-1/2*I,1/2*2^(1/2))+5*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/si
n(d*x+c))^(1/2)*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2
+1/2*I,1/2*2^(1/2))+26*cos(d*x+c)^2*2^(1/2)-22*cos(d*x+c)*2^(1/2))*sin(d*x+c)^3/(-1+cos(d*x+c))/cos(d*x+c)^4/(
e*sin(d*x+c)/cos(d*x+c))^(7/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(7/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))**2/(e*tan(d*x+c))**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \tan \left (d x + c\right )\right )^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sec(d*x+c))^2/(e*tan(d*x+c))^(7/2),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^2/(e*tan(d*x + c))^(7/2), x)