3.680 \(\int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx\)
Optimal. Leaf size=719 \[ -\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{5 i a \cos ^2(c+d x)}{8 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}+\frac{i a}{3 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}} \]
[Out]
((I/3)*a)/(d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/8)*a*Cos[c + d*x]^2)/(d*(e*Cos[c + d
*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/8)*a^(3/2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*T
an[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])
^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/8)*a^(3/2)*e^(7/2)*ArcTan[1 + (Sqrt[2]
*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])
^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/16)*a^(3/2)*e^(
7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a
*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*
x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/16)*a^(3/2)*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Ta
n[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d
*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/12)*Cos[c +
d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*(e*Cos[c + d*x])^(7/2))
________________________________________________________________________________________
Rubi [A] time = 0.859819, antiderivative size = 719, normalized size of antiderivative =
1., number of steps used = 15, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.367, Rules used = {3515, 3498, 3501, 3499, 3495, 297, 1162, 617, 204, 1165, 628}
\[ -\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt{2} d \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{5 i a \cos ^2(c+d x)}{8 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}+\frac{i a}{3 d \sqrt{a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]
[Out]
((I/3)*a)/(d*(e*Cos[c + d*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/8)*a*Cos[c + d*x]^2)/(d*(e*Cos[c + d
*x])^(7/2)*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/8)*a^(3/2)*e^(7/2)*ArcTan[1 - (Sqrt[2]*Sqrt[e]*Sqrt[a - I*a*T
an[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])
^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/8)*a^(3/2)*e^(7/2)*ArcTan[1 + (Sqrt[2]
*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/(Sqrt[a]*Sqrt[e*Sec[c + d*x]])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])
^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) + (((5*I)/16)*a^(3/2)*e^(
7/2)*Log[a - (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Tan[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a
*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*
x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/16)*a^(3/2)*e^(7/2)*Log[a + (Sqrt[2]*Sqrt[a]*Sqrt[e]*Sqrt[a - I*a*Ta
n[c + d*x]])/Sqrt[e*Sec[c + d*x]] + Cos[c + d*x]*(a - I*a*Tan[c + d*x])]*Sec[c + d*x])/(Sqrt[2]*d*(e*Cos[c + d
*x])^(7/2)*(e*Sec[c + d*x])^(7/2)*Sqrt[a - I*a*Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]) - (((5*I)/12)*Cos[c +
d*x]^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*(e*Cos[c + d*x])^(7/2))
Rule 3515
Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(d*Co
s[e + f*x])^m*(d*Sec[e + f*x])^m, Int[(a + b*Tan[e + f*x])^n/(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e,
f, m, n}, x] && !IntegerQ[m]
Rule 3498
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] + Dist[(a*(m + 2*n - 2))/(m + n - 1), Int[(
d*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] &&
GtQ[n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Rule 3501
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Rule 3499
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(3/2)/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(d*Sec
[e + f*x])/(Sqrt[a - b*Tan[e + f*x]]*Sqrt[a + b*Tan[e + f*x]]), Int[Sqrt[d*Sec[e + f*x]]*Sqrt[a - b*Tan[e + f*
x]], x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0]
Rule 3495
Int[Sqrt[(d_.)*sec[(e_.) + (f_.)*(x_)]]*Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(-4*b*d^
2)/f, Subst[Int[x^2/(a^2 + d^2*x^4), x], x, Sqrt[a + b*Tan[e + f*x]]/Sqrt[d*Sec[e + f*x]]], x] /; FreeQ[{a, b,
d, e, f}, x] && EqQ[a^2 + b^2, 0]
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]
Rubi steps
\begin{align*} \int \frac{\sqrt{a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx &=\frac{\int (e \sec (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{(5 a) \int \frac{(e \sec (c+d x))^{7/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{6 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 e^2\right ) \int (e \sec (c+d x))^{3/2} \sqrt{a+i a \tan (c+d x)} \, dx}{8 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 a e^2\right ) \int \frac{(e \sec (c+d x))^{3/2}}{\sqrt{a+i a \tan (c+d x)}} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 a e^3 \sec (c+d x)\right ) \int \sqrt{e \sec (c+d x)} \sqrt{a-i a \tan (c+d x)} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 i a^2 e^5 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{4 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}-\frac{\left (5 i a^2 e^4 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (5 i a^2 e^4 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 i a^2 e^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (5 i a^2 e^3 \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}+x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}+2 x}{-\frac{a}{e}-\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt{a}}{\sqrt{e}}-2 x}{-\frac{a}{e}+\frac{\sqrt{2} \sqrt{a} x}{\sqrt{e}}-x^2} \, dx,x,\frac{\sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}\right )}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{3/2} e^{7/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{3/2} e^{7/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac{\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right )}{8 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}\\ &=\frac{i a}{3 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{3/2} e^{7/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{3/2} e^{7/2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{a} \sqrt{e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}+\frac{5 i a^{3/2} e^{7/2} \log \left (a-\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i a^{3/2} e^{7/2} \log \left (a+\frac{\sqrt{2} \sqrt{a} \sqrt{e} \sqrt{a-i a \tan (c+d x)}}{\sqrt{e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt{2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt{a-i a \tan (c+d x)} \sqrt{a+i a \tan (c+d x)}}-\frac{5 i \cos ^2(c+d x) \sqrt{a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 2.98095, size = 305, normalized size = 0.42 \[ \frac{\sqrt{\cos (c+d x)} \sqrt{a+i a \tan (c+d x)} \left (-\frac{40}{3} i \cos ^{\frac{3}{2}}(c+d x)+\frac{5}{8} i e^{-\frac{7}{2} i (c+d x)} \sqrt{e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (1+e^{2 i (c+d x)}\right )^3 \left (\log \left (-\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right )-\log \left (\sqrt{2} e^{\frac{1}{2} i (c+d x)}+e^{i (c+d x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right )-2 \tan ^{-1}\left (1+\sqrt{2} e^{\frac{1}{2} i (c+d x)}\right )\right )+20 \cos ^{\frac{5}{2}}(c+d x) (\sin (c+d x)+i \cos (c+d x))+\frac{32}{3} \sqrt{\cos (c+d x)} (\sin (c+d x)+i \cos (c+d x))\right )}{32 d (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + I*a*Tan[c + d*x]]/(e*Cos[c + d*x])^(7/2),x]
[Out]
(Sqrt[Cos[c + d*x]]*(((-40*I)/3)*Cos[c + d*x]^(3/2) + (((5*I)/8)*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[(1 + E^((2*I
)*(c + d*x)))/E^(I*(c + d*x))]*(2*ArcTan[1 - Sqrt[2]*E^((I/2)*(c + d*x))] - 2*ArcTan[1 + Sqrt[2]*E^((I/2)*(c +
d*x))] + Log[1 - Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*(c + d*x))] - Log[1 + Sqrt[2]*E^((I/2)*(c + d*x)) + E^(I*
(c + d*x))]))/E^(((7*I)/2)*(c + d*x)) + (32*Sqrt[Cos[c + d*x]]*(I*Cos[c + d*x] + Sin[c + d*x]))/3 + 20*Cos[c +
d*x]^(5/2)*(I*Cos[c + d*x] + Sin[c + d*x]))*Sqrt[a + I*a*Tan[c + d*x]])/(32*d*(e*Cos[c + d*x])^(7/2))
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Maple [A] time = 0.379, size = 417, normalized size = 0.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x)
[Out]
1/48/d*(a*(I*sin(d*x+c)+cos(d*x+c))/cos(d*x+c))^(1/2)*cos(d*x+c)*(cos(d*x+c)-1)^4*(30*I*sin(d*x+c)*cos(d*x+c)^
2*(1/(cos(d*x+c)+1))^(1/2)+15*I*cos(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(-cos(d*x+c)-1+sin(d*x+c)))+
15*I*cos(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))+20*I*sin(d*x+c)*cos(d*x+c)*(
1/(cos(d*x+c)+1))^(1/2)-15*cos(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(-cos(d*x+c)-1+sin(d*x+c)))+15*co
s(d*x+c)^3*arctanh(1/2*(1/(cos(d*x+c)+1))^(1/2)*(cos(d*x+c)+1+sin(d*x+c)))-30*cos(d*x+c)^3*(1/(cos(d*x+c)+1))^
(1/2)+16*I*sin(d*x+c)*(1/(cos(d*x+c)+1))^(1/2)-10*cos(d*x+c)^2*(1/(cos(d*x+c)+1))^(1/2)+4*cos(d*x+c)*(1/(cos(d
*x+c)+1))^(1/2)-16*(1/(cos(d*x+c)+1))^(1/2))/sin(d*x+c)^7/(I*sin(d*x+c)+cos(d*x+c)-1)/(1/(cos(d*x+c)+1))^(7/2)
/(e*cos(d*x+c))^(7/2)
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Maxima [B] time = 4.3269, size = 3606, normalized size = 5.02 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="maxima")
[Out]
-((5760*sqrt(2)*cos(6*d*x + 6*c) + 17280*sqrt(2)*cos(4*d*x + 4*c) + 17280*sqrt(2)*cos(2*d*x + 2*c) + 5760*I*sq
rt(2)*sin(6*d*x + 6*c) + 17280*I*sqrt(2)*sin(4*d*x + 4*c) + 17280*I*sqrt(2)*sin(2*d*x + 2*c) + 5760*sqrt(2))*a
rctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2
*c), cos(2*d*x + 2*c))) + 1) + (5760*sqrt(2)*cos(6*d*x + 6*c) + 17280*sqrt(2)*cos(4*d*x + 4*c) + 17280*sqrt(2)
*cos(2*d*x + 2*c) + 5760*I*sqrt(2)*sin(6*d*x + 6*c) + 17280*I*sqrt(2)*sin(4*d*x + 4*c) + 17280*I*sqrt(2)*sin(2
*d*x + 2*c) + 5760*sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1, -sqrt(2)
*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + (5760*sqrt(2)*cos(6*d*x + 6*c) + 17280*sqrt(2)*co
s(4*d*x + 4*c) + 17280*sqrt(2)*cos(2*d*x + 2*c) + 5760*I*sqrt(2)*sin(6*d*x + 6*c) + 17280*I*sqrt(2)*sin(4*d*x
+ 4*c) + 17280*I*sqrt(2)*sin(2*d*x + 2*c) + 5760*sqrt(2))*arctan2(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), co
s(2*d*x + 2*c))) - 1, sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + (5760*sqrt(2)*cos(6*
d*x + 6*c) + 17280*sqrt(2)*cos(4*d*x + 4*c) + 17280*sqrt(2)*cos(2*d*x + 2*c) + 5760*I*sqrt(2)*sin(6*d*x + 6*c)
+ 17280*I*sqrt(2)*sin(4*d*x + 4*c) + 17280*I*sqrt(2)*sin(2*d*x + 2*c) + 5760*sqrt(2))*arctan2(sqrt(2)*cos(1/4
*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 1, -sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)
)) + 1) + (5760*I*sqrt(2)*cos(6*d*x + 6*c) + 17280*I*sqrt(2)*cos(4*d*x + 4*c) + 17280*I*sqrt(2)*cos(2*d*x + 2*
c) - 5760*sqrt(2)*sin(6*d*x + 6*c) - 17280*sqrt(2)*sin(4*d*x + 4*c) - 17280*sqrt(2)*sin(2*d*x + 2*c) + 5760*I*
sqrt(2))*arctan2(sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + sin(1/2*arctan2(sin(2*d*x + 2*
c), cos(2*d*x + 2*c))), sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) + 1) + (-5760*I*sqrt(2)*cos(6*d*x + 6*c) - 17280*I*sqrt(2)*cos(4*d*x + 4*c) - 17
280*I*sqrt(2)*cos(2*d*x + 2*c) + 5760*sqrt(2)*sin(6*d*x + 6*c) + 17280*sqrt(2)*sin(4*d*x + 4*c) + 17280*sqrt(2
)*sin(2*d*x + 2*c) - 5760*I*sqrt(2))*arctan2(-sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + s
in(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))), -sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*
c))) + cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + (2880*sqrt(2)*cos(6*d*x + 6*c) + 8640*sqrt(
2)*cos(4*d*x + 4*c) + 8640*sqrt(2)*cos(2*d*x + 2*c) + 2880*I*sqrt(2)*sin(6*d*x + 6*c) + 8640*I*sqrt(2)*sin(4*d
*x + 4*c) + 8640*I*sqrt(2)*sin(2*d*x + 2*c) + 2880*sqrt(2))*log(2*sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), co
s(2*d*x + 2*c)))*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x +
2*c), cos(2*d*x + 2*c))) + 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*
x + 2*c), cos(2*d*x + 2*c)))^2 + 2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*co
s(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - (2880*sqrt(2)*cos(6*d*x + 6*c) + 8640*sqrt(2)*cos(4*
d*x + 4*c) + 8640*sqrt(2)*cos(2*d*x + 2*c) + 2880*I*sqrt(2)*sin(6*d*x + 6*c) + 8640*I*sqrt(2)*sin(4*d*x + 4*c)
+ 8640*I*sqrt(2)*sin(2*d*x + 2*c) + 2880*sqrt(2))*log(-2*sqrt(2)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c)))*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*(sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), c
os(2*d*x + 2*c))) - 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + cos(1/2*arctan2(sin(2*d*x + 2*c)
, cos(2*d*x + 2*c)))^2 + 2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) + (2880*I*sqrt(2)*cos(6*d*x + 6*c) + 8640*I*sqrt(2)*cos(4*d*x
+ 4*c) + 8640*I*sqrt(2)*cos(2*d*x + 2*c) - 2880*sqrt(2)*sin(6*d*x + 6*c) - 8640*sqrt(2)*sin(4*d*x + 4*c) - 864
0*sqrt(2)*sin(2*d*x + 2*c) + 2880*I*sqrt(2))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*
sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x
+ 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + (-2880*I*sqrt(2)*cos(6*d*x +
6*c) - 8640*I*sqrt(2)*cos(4*d*x + 4*c) - 8640*I*sqrt(2)*cos(2*d*x + 2*c) + 2880*sqrt(2)*sin(6*d*x + 6*c) + 86
40*sqrt(2)*sin(4*d*x + 4*c) + 8640*sqrt(2)*sin(2*d*x + 2*c) - 2880*I*sqrt(2))*log(2*cos(1/4*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sqrt(2)*cos(1/4*ar
ctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) +
2) + (2880*I*sqrt(2)*cos(6*d*x + 6*c) + 8640*I*sqrt(2)*cos(4*d*x + 4*c) + 8640*I*sqrt(2)*cos(2*d*x + 2*c) - 28
80*sqrt(2)*sin(6*d*x + 6*c) - 8640*sqrt(2)*sin(4*d*x + 4*c) - 8640*sqrt(2)*sin(2*d*x + 2*c) + 2880*I*sqrt(2))*
log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x +
2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2*sqrt(2)*sin(1/4*arctan2(sin(2*d
*x + 2*c), cos(2*d*x + 2*c))) + 2) + (-2880*I*sqrt(2)*cos(6*d*x + 6*c) - 8640*I*sqrt(2)*cos(4*d*x + 4*c) - 864
0*I*sqrt(2)*cos(2*d*x + 2*c) + 2880*sqrt(2)*sin(6*d*x + 6*c) + 8640*sqrt(2)*sin(4*d*x + 4*c) + 8640*sqrt(2)*si
n(2*d*x + 2*c) - 2880*I*sqrt(2))*log(2*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/4*arct
an2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sqrt(2)*cos(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) -
2*sqrt(2)*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 2) + 15360*cos(9/4*arctan2(sin(2*d*x + 2*c),
cos(2*d*x + 2*c))) - 129024*cos(5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 46080*cos(1/4*arctan2(sin(2
*d*x + 2*c), cos(2*d*x + 2*c))) + 15360*I*sin(9/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 129024*I*sin(
5/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 46080*I*sin(1/4*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))
))*sqrt(a)*sqrt(e)/((-36864*I*e^4*cos(6*d*x + 6*c) - 110592*I*e^4*cos(4*d*x + 4*c) - 110592*I*e^4*cos(2*d*x +
2*c) + 36864*e^4*sin(6*d*x + 6*c) + 110592*e^4*sin(4*d*x + 4*c) + 110592*e^4*sin(2*d*x + 2*c) - 36864*I*e^4)*d
)
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Fricas [A] time = 2.72937, size = 1883, normalized size = 2.62 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="fricas")
[Out]
1/12*(sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-5*I*e^(4*I*d*x + 4
*I*c) + 42*I*e^(2*I*d*x + 2*I*c) + 15*I)*e^(1/2*I*d*x + 1/2*I*c) + 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4
*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt(25/64*I*a/(d^2*e^7))*log(8/5*I*d*e^4*sqrt(25/64*I*
a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*
x + 1/2*I*c)) - 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e
^4)*sqrt(25/64*I*a/(d^2*e^7))*log(-8/5*I*d*e^4*sqrt(25/64*I*a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x
+ 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c)) + 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d
*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)*sqrt(-25/64*I*a/(d^2*e^7))*log(8/5*I*d*e^4*sqr
t(-25/64*I*a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*
e^(1/2*I*d*x + 1/2*I*c)) - 6*(d*e^4*e^(6*I*d*x + 6*I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2
*I*c) + d*e^4)*sqrt(-25/64*I*a/(d^2*e^7))*log(-8/5*I*d*e^4*sqrt(-25/64*I*a/(d^2*e^7)) + sqrt(2)*sqrt(1/2)*sqrt
(e*e^(2*I*d*x + 2*I*c) + e)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(1/2*I*d*x + 1/2*I*c)))/(d*e^4*e^(6*I*d*x + 6*
I*c) + 3*d*e^4*e^(4*I*d*x + 4*I*c) + 3*d*e^4*e^(2*I*d*x + 2*I*c) + d*e^4)
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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+I*a*tan(d*x+c))**(1/2)/(e*cos(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{\sqrt{i \, a \tan \left (d x + c\right ) + a}}{\left (e \cos \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+I*a*tan(d*x+c))^(1/2)/(e*cos(d*x+c))^(7/2),x, algorithm="giac")
[Out]
integrate(sqrt(I*a*tan(d*x + c) + a)/(e*cos(d*x + c))^(7/2), x)