3.208 \(\int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx\)

Optimal. Leaf size=202 \[ \frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}+\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d} \]

[Out]

((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a
+ I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (((6*I)/7)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(5
/2)) + (32*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(3/2)) + (((104*I)/21)*a^2*Sqrt[a + I*a*Tan[c +
d*x]])/(d*Sqrt[Tan[c + d*x]])

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Rubi [A]  time = 0.570611, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3553, 3598, 12, 3544, 205} \[ \frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}+\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(9/2),x]

[Out]

((4 - 4*I)*a^(5/2)*ArcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*a^2*Sqrt[a
+ I*a*Tan[c + d*x]])/(7*d*Tan[c + d*x]^(7/2)) - (((6*I)/7)*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(d*Tan[c + d*x]^(5
/2)) + (32*a^2*Sqrt[a + I*a*Tan[c + d*x]])/(21*d*Tan[c + d*x]^(3/2)) + (((104*I)/21)*a^2*Sqrt[a + I*a*Tan[c +
d*x]])/(d*Sqrt[Tan[c + d*x]])

Rule 3553

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Si
mp[(a^2*(b*c - a*d)*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(b*c + a*d)*(n + 1)), x] +
Dist[a/(d*(b*c + a*d)*(n + 1)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1)*Simp[b*(b*c*(m
- 2) - a*d*(m - 2*n - 4)) + (a*b*c*(m - 2) + b^2*d*(n + 1) - a^2*d*(m + n - 1))*Tan[e + f*x], x], x], x] /; Fr
eeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && LtQ[
n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3598

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((A*d - B*c)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(f
*(n + 1)*(c^2 + d^2)), x] - Dist[1/(a*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n
+ 1)*Simp[A*(b*d*m - a*c*(n + 1)) - B*(b*c*m + a*d*(n + 1)) - a*(B*c - A*d)*(m + n + 1)*Tan[e + f*x], x], x],
x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3544

Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
(-2*a*b)/f, Subst[Int[1/(a*c - b*d - 2*a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 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]

Rubi steps

\begin{align*} \int \frac{(a+i a \tan (c+d x))^{5/2}}{\tan ^{\frac{9}{2}}(c+d x)} \, dx &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{2}{7} \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{15 i a^2}{2}+\frac{13}{2} a^2 \tan (c+d x)\right )}{\tan ^{\frac{7}{2}}(c+d x)} \, dx\\ &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}-\frac{4 \int \frac{\sqrt{a+i a \tan (c+d x)} \left (20 a^3+15 i a^3 \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{35 a}\\ &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}-\frac{8 \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{65 i a^4}{2}-20 a^4 \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{105 a^2}\\ &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}-\frac{16 \int -\frac{105 a^5 \sqrt{a+i a \tan (c+d x)}}{4 \sqrt{\tan (c+d x)}} \, dx}{105 a^3}\\ &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}+\left (4 a^2\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx\\ &=-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}-\frac{\left (8 i a^4\right ) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}\\ &=\frac{(4-4 i) a^{5/2} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{d}-\frac{2 a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{7}{2}}(c+d x)}-\frac{6 i a^2 \sqrt{a+i a \tan (c+d x)}}{7 d \tan ^{\frac{5}{2}}(c+d x)}+\frac{32 a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \tan ^{\frac{3}{2}}(c+d x)}+\frac{104 i a^2 \sqrt{a+i a \tan (c+d x)}}{21 d \sqrt{\tan (c+d x)}}\\ \end{align*}

Mathematica [A]  time = 2.74844, size = 188, normalized size = 0.93 \[ \frac{4 i \sqrt{2} a^2 e^{-i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \left (e^{i (c+d x)} \left (70 e^{2 i (c+d x)}-77 e^{4 i (c+d x)}+40 e^{6 i (c+d x)}-21\right )-21 \left (-1+e^{2 i (c+d x)}\right )^{7/2} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )\right )}{21 d \left (-1+e^{2 i (c+d x)}\right )^3 \sqrt{\tan (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[c + d*x])^(5/2)/Tan[c + d*x]^(9/2),x]

[Out]

(((4*I)/21)*Sqrt[2]*a^2*Sqrt[(a*E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]*(E^(I*(c + d*x))*(-21 + 70*E^(
(2*I)*(c + d*x)) - 77*E^((4*I)*(c + d*x)) + 40*E^((6*I)*(c + d*x))) - 21*(-1 + E^((2*I)*(c + d*x)))^(7/2)*ArcT
anh[E^(I*(c + d*x))/Sqrt[-1 + E^((2*I)*(c + d*x))]]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*(c + d*x)))^3*Sqrt[Tan
[c + d*x]])

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Maple [B]  time = 0.036, size = 457, normalized size = 2.3 \begin{align*}{\frac{{a}^{2}}{21\,d}\sqrt{a \left ( 1+i\tan \left ( dx+c \right ) \right ) } \left ( 21\,i\sqrt{ia}\sqrt{2}\ln \left ({\frac{1}{\tan \left ( dx+c \right ) +i} \left ( 2\,\sqrt{2}\sqrt{-ia}\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }-ia+3\,a\tan \left ( dx+c \right ) \right ) } \right ) \left ( \tan \left ( dx+c \right ) \right ) ^{4}a-21\,\sqrt{ia}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{-ia}\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }-ia+3\,a\tan \left ( dx+c \right ) }{\tan \left ( dx+c \right ) +i}} \right ) \left ( \tan \left ( dx+c \right ) \right ) ^{4}a+84\,i\ln \left ({\frac{1}{2} \left ( 2\,ia\tan \left ( dx+c \right ) +2\,\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }\sqrt{ia}+a \right ){\frac{1}{\sqrt{ia}}}} \right ) \sqrt{-ia} \left ( \tan \left ( dx+c \right ) \right ) ^{4}a+32\,\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }\sqrt{-ia}\sqrt{ia} \left ( \tan \left ( dx+c \right ) \right ) ^{2}+104\,i\sqrt{ia}\sqrt{-ia} \left ( \tan \left ( dx+c \right ) \right ) ^{3}\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }-18\,i\tan \left ( dx+c \right ) \sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }\sqrt{ia}\sqrt{-ia}-6\,\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }\sqrt{-ia}\sqrt{ia} \right ) \left ( \tan \left ( dx+c \right ) \right ) ^{-{\frac{7}{2}}}{\frac{1}{\sqrt{a\tan \left ( dx+c \right ) \left ( 1+i\tan \left ( dx+c \right ) \right ) }}}{\frac{1}{\sqrt{ia}}}{\frac{1}{\sqrt{-ia}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(d*x+c))^(5/2)/tan(d*x+c)^(9/2),x)

[Out]

1/21/d*(a*(1+I*tan(d*x+c)))^(1/2)*a^2/tan(d*x+c)^(7/2)*(21*I*(I*a)^(1/2)*2^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a
*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*tan(d*x+c)^4*a-21*(I*a)^(1/2)*2^(1/2)*
ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*tan(d*x+c
)^4*a+84*I*ln(1/2*(2*I*a*tan(d*x+c)+2*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(I*a)^(1/2)+a)/(I*a)^(1/2))*(-I*a)
^(1/2)*tan(d*x+c)^4*a+32*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*(I*a)^(1/2)*tan(d*x+c)^2+104*I*(I*
a)^(1/2)*(-I*a)^(1/2)*tan(d*x+c)^3*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-18*I*tan(d*x+c)*(a*tan(d*x+c)*(1+I*ta
n(d*x+c)))^(1/2)*(I*a)^(1/2)*(-I*a)^(1/2)-6*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)*(-I*a)^(1/2)*(I*a)^(1/2))/(a
*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(I*a)^(1/2)/(-I*a)^(1/2)

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Maxima [B]  time = 3.8658, size = 4239, normalized size = 20.99 \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))^(5/2)/tan(d*x+c)^(9/2),x, algorithm="maxima")

[Out]

1/11025*(sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*((-(44100*I + 44100)*a^2*cos(7
*d*x + 7*c) + (44100*I + 44100)*a^2*cos(5*d*x + 5*c) - (26460*I + 26460)*a^2*cos(3*d*x + 3*c) + (1260*I + 1260
)*a^2*cos(d*x + c) - (44100*I - 44100)*a^2*sin(7*d*x + 7*c) + (44100*I - 44100)*a^2*sin(5*d*x + 5*c) - (26460*
I - 26460)*a^2*sin(3*d*x + 3*c) + (1260*I - 1260)*a^2*sin(d*x + c))*cos(7/2*arctan2(sin(2*d*x + 2*c), -cos(2*d
*x + 2*c) + 1)) + (((27300*I + 27300)*a^2*cos(d*x + c) + (27300*I - 27300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^
2 + (27300*I + 27300)*a^2*cos(d*x + c) + ((27300*I + 27300)*a^2*cos(d*x + c) + (27300*I - 27300)*a^2*sin(d*x +
c))*sin(2*d*x + 2*c)^2 + (27300*I - 27300)*a^2*sin(d*x + c) + (-(44100*I + 44100)*a^2*cos(2*d*x + 2*c)^2 - (4
4100*I + 44100)*a^2*sin(2*d*x + 2*c)^2 + (88200*I + 88200)*a^2*cos(2*d*x + 2*c) - (44100*I + 44100)*a^2)*cos(3
*d*x + 3*c) + (-(54600*I + 54600)*a^2*cos(d*x + c) - (54600*I - 54600)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + (-
(44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2 - (44100*I - 44100)*a^2*sin(2*d*x + 2*c)^2 + (88200*I - 88200)*a^2*co
s(2*d*x + 2*c) - (44100*I - 44100)*a^2)*sin(3*d*x + 3*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c)
+ 1)) + (-(44100*I - 44100)*a^2*cos(7*d*x + 7*c) + (44100*I - 44100)*a^2*cos(5*d*x + 5*c) - (26460*I - 26460)*
a^2*cos(3*d*x + 3*c) + (1260*I - 1260)*a^2*cos(d*x + c) + (44100*I + 44100)*a^2*sin(7*d*x + 7*c) - (44100*I +
44100)*a^2*sin(5*d*x + 5*c) + (26460*I + 26460)*a^2*sin(3*d*x + 3*c) - (1260*I + 1260)*a^2*sin(d*x + c))*sin(7
/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)) + (((27300*I - 27300)*a^2*cos(d*x + c) - (27300*I + 27300
)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (27300*I - 27300)*a^2*cos(d*x + c) + ((27300*I - 27300)*a^2*cos(d*x +
c) - (27300*I + 27300)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 - (27300*I + 27300)*a^2*sin(d*x + c) + (-(44100*I
- 44100)*a^2*cos(2*d*x + 2*c)^2 - (44100*I - 44100)*a^2*sin(2*d*x + 2*c)^2 + (88200*I - 88200)*a^2*cos(2*d*x
+ 2*c) - (44100*I - 44100)*a^2)*cos(3*d*x + 3*c) + (-(54600*I - 54600)*a^2*cos(d*x + c) + (54600*I + 54600)*a^
2*sin(d*x + c))*cos(2*d*x + 2*c) + ((44100*I + 44100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I + 44100)*a^2*sin(2*d*x
+ 2*c)^2 - (88200*I + 88200)*a^2*cos(2*d*x + 2*c) + (44100*I + 44100)*a^2)*sin(3*d*x + 3*c))*sin(3/2*arctan2(
sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)))*sqrt(a) + (((44100*I + 44100)*a^2*cos(2*d*x + 2*c)^4 + (44100*I + 4
4100)*a^2*sin(2*d*x + 2*c)^4 - (176400*I + 176400)*a^2*cos(2*d*x + 2*c)^3 + (264600*I + 264600)*a^2*cos(2*d*x
+ 2*c)^2 - (176400*I + 176400)*a^2*cos(2*d*x + 2*c) + ((88200*I + 88200)*a^2*cos(2*d*x + 2*c)^2 - (176400*I +
176400)*a^2*cos(2*d*x + 2*c) + (88200*I + 88200)*a^2)*sin(2*d*x + 2*c)^2 + (44100*I + 44100)*a^2)*arctan2((cos
(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), -cos(2*
d*x + 2*c) + 1)) - cos(d*x + c), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(
1/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)) - sin(d*x + c)) + ((22050*I - 22050)*a^2*cos(2*d*x + 2*c
)^4 + (22050*I - 22050)*a^2*sin(2*d*x + 2*c)^4 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c)^3 + (132300*I - 132300
)*a^2*cos(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + ((44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2
- (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (44100*I - 44100)*a^2)*sin(2*d*x + 2*c)^2 + (22050*I - 22050)*a^2)*
log(cos(d*x + c)^2 + sin(d*x + c)^2 + sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(
cos(1/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2
*c) + 1))^2) - 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(1/2*arctan2(sin
(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1))*sin(d*x + c) + cos(d*x + c)*sin(1/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*
x + 2*c) + 1)))))*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sqrt(a) + ((((63840
*I + 63840)*a^2*cos(d*x + c) + (63840*I - 63840)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 + (63840*I + 63840)*a^2*
cos(d*x + c) + ((63840*I + 63840)*a^2*cos(d*x + c) + (63840*I - 63840)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 +
(63840*I - 63840)*a^2*sin(d*x + c) + ((44100*I + 44100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I + 44100)*a^2*sin(2*d
*x + 2*c)^2 - (88200*I + 88200)*a^2*cos(2*d*x + 2*c) + (44100*I + 44100)*a^2)*cos(5*d*x + 5*c) + (-(102900*I +
102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I + 102900)*a^2*sin(2*d*x + 2*c)^2 + (205800*I + 205800)*a^2*cos(2*d
*x + 2*c) - (102900*I + 102900)*a^2)*cos(3*d*x + 3*c) + (-(127680*I + 127680)*a^2*cos(d*x + c) - (127680*I - 1
27680)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + ((44100*I - 44100)*a^2*cos(2*d*x + 2*c)^2 + (44100*I - 44100)*a^2*
sin(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (44100*I - 44100)*a^2)*sin(5*d*x + 5*c) + (-(102
900*I - 102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I - 102900)*a^2*sin(2*d*x + 2*c)^2 + (205800*I - 205800)*a^2*
cos(2*d*x + 2*c) - (102900*I - 102900)*a^2)*sin(3*d*x + 3*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2
*c) + 1)) + ((-(48300*I + 48300)*a^2*cos(d*x + c) - (48300*I - 48300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + (
-(48300*I + 48300)*a^2*cos(d*x + c) - (48300*I - 48300)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((193200*I + 19
3200)*a^2*cos(d*x + c) + (193200*I - 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^3 + (-(289800*I + 289800)*a^2*
cos(d*x + c) - (289800*I - 289800)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (48300*I + 48300)*a^2*cos(d*x + c) +
((-(96600*I + 96600)*a^2*cos(d*x + c) - (96600*I - 96600)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (96600*I + 9
6600)*a^2*cos(d*x + c) - (96600*I - 96600)*a^2*sin(d*x + c) + ((193200*I + 193200)*a^2*cos(d*x + c) + (193200*
I - 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x + 2*c)^2 - (48300*I - 48300)*a^2*sin(d*x + c) + ((19
3200*I + 193200)*a^2*cos(d*x + c) + (193200*I - 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*cos(1/2*arctan2(si
n(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)) + (((63840*I - 63840)*a^2*cos(d*x + c) - (63840*I + 63840)*a^2*sin(d*x
+ c))*cos(2*d*x + 2*c)^2 + (63840*I - 63840)*a^2*cos(d*x + c) + ((63840*I - 63840)*a^2*cos(d*x + c) - (63840*
I + 63840)*a^2*sin(d*x + c))*sin(2*d*x + 2*c)^2 - (63840*I + 63840)*a^2*sin(d*x + c) + ((44100*I - 44100)*a^2*
cos(2*d*x + 2*c)^2 + (44100*I - 44100)*a^2*sin(2*d*x + 2*c)^2 - (88200*I - 88200)*a^2*cos(2*d*x + 2*c) + (4410
0*I - 44100)*a^2)*cos(5*d*x + 5*c) + (-(102900*I - 102900)*a^2*cos(2*d*x + 2*c)^2 - (102900*I - 102900)*a^2*si
n(2*d*x + 2*c)^2 + (205800*I - 205800)*a^2*cos(2*d*x + 2*c) - (102900*I - 102900)*a^2)*cos(3*d*x + 3*c) + (-(1
27680*I - 127680)*a^2*cos(d*x + c) + (127680*I + 127680)*a^2*sin(d*x + c))*cos(2*d*x + 2*c) + (-(44100*I + 441
00)*a^2*cos(2*d*x + 2*c)^2 - (44100*I + 44100)*a^2*sin(2*d*x + 2*c)^2 + (88200*I + 88200)*a^2*cos(2*d*x + 2*c)
- (44100*I + 44100)*a^2)*sin(5*d*x + 5*c) + ((102900*I + 102900)*a^2*cos(2*d*x + 2*c)^2 + (102900*I + 102900)
*a^2*sin(2*d*x + 2*c)^2 - (205800*I + 205800)*a^2*cos(2*d*x + 2*c) + (102900*I + 102900)*a^2)*sin(3*d*x + 3*c)
)*sin(5/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)) + ((-(48300*I - 48300)*a^2*cos(d*x + c) + (48300*I
+ 48300)*a^2*sin(d*x + c))*cos(2*d*x + 2*c)^4 + (-(48300*I - 48300)*a^2*cos(d*x + c) + (48300*I + 48300)*a^2*
sin(d*x + c))*sin(2*d*x + 2*c)^4 + ((193200*I - 193200)*a^2*cos(d*x + c) - (193200*I + 193200)*a^2*sin(d*x + c
))*cos(2*d*x + 2*c)^3 + (-(289800*I - 289800)*a^2*cos(d*x + c) + (289800*I + 289800)*a^2*sin(d*x + c))*cos(2*d
*x + 2*c)^2 - (48300*I - 48300)*a^2*cos(d*x + c) + ((-(96600*I - 96600)*a^2*cos(d*x + c) + (96600*I + 96600)*a
^2*sin(d*x + c))*cos(2*d*x + 2*c)^2 - (96600*I - 96600)*a^2*cos(d*x + c) + (96600*I + 96600)*a^2*sin(d*x + c)
+ ((193200*I - 193200)*a^2*cos(d*x + c) - (193200*I + 193200)*a^2*sin(d*x + c))*cos(2*d*x + 2*c))*sin(2*d*x +
2*c)^2 + (48300*I + 48300)*a^2*sin(d*x + c) + ((193200*I - 193200)*a^2*cos(d*x + c) - (193200*I + 193200)*a^2*
sin(d*x + c))*cos(2*d*x + 2*c))*sin(1/2*arctan2(sin(2*d*x + 2*c), -cos(2*d*x + 2*c) + 1)))*sqrt(a))/((cos(2*d*
x + 2*c)^4 + sin(2*d*x + 2*c)^4 - 4*cos(2*d*x + 2*c)^3 + 2*(cos(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*sin(2
*d*x + 2*c)^2 + 6*cos(2*d*x + 2*c)^2 - 4*cos(2*d*x + 2*c) + 1)*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*co
s(2*d*x + 2*c) + 1)^(1/4)*d)

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Fricas [B]  time = 2.46674, size = 1469, normalized size = 7.27 \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))^(5/2)/tan(d*x+c)^(9/2),x, algorithm="fricas")

[Out]

-1/42*(8*sqrt(2)*(40*a^2*e^(8*I*d*x + 8*I*c) - 37*a^2*e^(6*I*d*x + 6*I*c) - 7*a^2*e^(4*I*d*x + 4*I*c) + 49*a^2
*e^(2*I*d*x + 2*I*c) - 21*a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x
+ 2*I*c) + 1))*e^(I*d*x + I*c) - 21*sqrt(-32*I*a^5/d^2)*(d*e^(8*I*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*
d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(1/4*(4*sqrt(2)*(a^2*e^(2*I*d*x + 2*I*c) + a^2)*sqrt(a
/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*e^(I*d*x + I*c) + sqr
t(-32*I*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/a^2) + 21*sqrt(-32*I*a^5/d^2)*(d*e^(8*I*d*x + 8*I
*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)*log(1/4*(4*sqrt(2)*(a^2
*e^(2*I*d*x + 2*I*c) + a^2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x +
2*I*c) + 1))*e^(I*d*x + I*c) - sqrt(-32*I*a^5/d^2)*d*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/a^2))/(d*e^(8*I
*d*x + 8*I*c) - 4*d*e^(6*I*d*x + 6*I*c) + 6*d*e^(4*I*d*x + 4*I*c) - 4*d*e^(2*I*d*x + 2*I*c) + d)

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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))**(5/2)/tan(d*x+c)**(9/2),x)

[Out]

Timed out

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Giac [A]  time = 1.40097, size = 220, normalized size = 1.09 \begin{align*} \frac{\left (i - 1\right ) \, \sqrt{-2 \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 2 \, a^{2}}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5} \log \left (\sqrt{i \, a \tan \left (d x + c\right ) + a}\right )}{-i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6} + 7 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} a - 20 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a^{2} + 30 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{3} - 25 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{4} + 11 i \,{\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{5} - 2 i \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(d*x+c))^(5/2)/tan(d*x+c)^(9/2),x, algorithm="giac")

[Out]

(I - 1)*sqrt(-2*(I*a*tan(d*x + c) + a)*a + 2*a^2)*(I*a*tan(d*x + c) + a)^2*a^5*log(sqrt(I*a*tan(d*x + c) + a))
/(-I*(I*a*tan(d*x + c) + a)^6 + 7*I*(I*a*tan(d*x + c) + a)^5*a - 20*I*(I*a*tan(d*x + c) + a)^4*a^2 + 30*I*(I*a
*tan(d*x + c) + a)^3*a^3 - 25*I*(I*a*tan(d*x + c) + a)^2*a^4 + 11*I*(I*a*tan(d*x + c) + a)*a^5 - 2*I*a^6)