3.11.25 \(\int \frac {(a+i a \tan (e+f x))^{7/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx\) [1025]

Optimal. Leaf size=204 \[ -\frac {10 i a^{7/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f} \]

[Out]

-10*I*a^(7/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f*x+e))^(1/2))/c^(3/2)/f+5*I*a^3*(a+I
*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)/c^2/f+10/3*I*a^2*(a+I*a*tan(f*x+e))^(3/2)/c/f/(c-I*c*tan(f*x+e))
^(1/2)-2/3*I*a*(a+I*a*tan(f*x+e))^(5/2)/f/(c-I*c*tan(f*x+e))^(3/2)

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Rubi [A]
time = 0.13, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3604, 49, 52, 65, 223, 209} \begin {gather*} -\frac {10 i a^{7/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(3/2),x]

[Out]

((-10*I)*a^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^(3/2)*f
) - (((2*I)/3)*a*(a + I*a*Tan[e + f*x])^(5/2))/(f*(c - I*c*Tan[e + f*x])^(3/2)) + (((10*I)/3)*a^2*(a + I*a*Tan
[e + f*x])^(3/2))/(c*f*Sqrt[c - I*c*Tan[e + f*x]]) + ((5*I)*a^3*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e
+ f*x]])/(c^2*f)

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 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 3604

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

Rubi steps

\begin {align*} \int \frac {(a+i a \tan (e+f x))^{7/2}}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {\left (5 a^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{c f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}+\frac {\left (5 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{c f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}-\frac {\left (10 i a^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{c f}\\ &=-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}-\frac {\left (10 i a^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{c f}\\ &=-\frac {10 i a^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {2 i a (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {10 i a^2 (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {5 i a^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}\\ \end {align*}

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Mathematica [A]
time = 8.96, size = 348, normalized size = 1.71 \begin {gather*} -\frac {10 i e^{-i (4 e+f x)} \sqrt {e^{i f x}} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \text {ArcTan}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2}}{c \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} f \sec ^{\frac {7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2}}+\frac {\cos ^3(e+f x) \left (\frac {5 i \cos (3 e)}{c^2}+\cos (4 f x) \left (-\frac {2 i \cos (e)}{3 c^2}+\frac {2 \sin (e)}{3 c^2}\right )+\cos (2 f x) \left (\frac {10 i \cos (e)}{3 c^2}+\frac {10 \sin (e)}{3 c^2}\right )+\frac {5 \sin (3 e)}{c^2}+\left (-\frac {10 \cos (e)}{3 c^2}+\frac {10 i \sin (e)}{3 c^2}\right ) \sin (2 f x)+\left (\frac {2 \cos (e)}{3 c^2}+\frac {2 i \sin (e)}{3 c^2}\right ) \sin (4 f x)\right ) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} (a+i a \tan (e+f x))^{7/2}}{f (\cos (f x)+i \sin (f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^(7/2)/(c - I*c*Tan[e + f*x])^(3/2),x]

[Out]

((-10*I)*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e + f*x))]*(a + I*a*Tan[
e + f*x])^(7/2))/(c*E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))]*f*Sec[e + f*x]^(7/2)*(Cos[f*x] + I*Sin
[f*x])^(7/2)) + (Cos[e + f*x]^3*(((5*I)*Cos[3*e])/c^2 + Cos[4*f*x]*((((-2*I)/3)*Cos[e])/c^2 + (2*Sin[e])/(3*c^
2)) + Cos[2*f*x]*((((10*I)/3)*Cos[e])/c^2 + (10*Sin[e])/(3*c^2)) + (5*Sin[3*e])/c^2 + ((-10*Cos[e])/(3*c^2) +
(((10*I)/3)*Sin[e])/c^2)*Sin[2*f*x] + ((2*Cos[e])/(3*c^2) + (((2*I)/3)*Sin[e])/c^2)*Sin[4*f*x])*Sqrt[Sec[e + f
*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(a + I*a*Tan[e + f*x])^(7/2))/(f*(Cos[f*x] + I*Sin[f*x])^3)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (164 ) = 328\).
time = 0.36, size = 379, normalized size = 1.86

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (45 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+15 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )+3 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )-15 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -45 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-57 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-37 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+23 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\)	\(379\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (45 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{2}\left (f x +e \right )\right )+15 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \left (\tan ^{3}\left (f x +e \right )\right )+3 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )-15 i \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c -45 \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right ) a c \tan \left (f x +e \right )-57 i \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-37 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+23 \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan \left (f x +e \right )+i\right )^{3} \sqrt {a c}}\)	\(379\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^3/c^2*(45*I*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+
e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^2+15*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*
c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)^3+3*I*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^3-15*I*ln((c*a
*tan(f*x+e)+(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c-45*ln((c*a*tan(f*x+e)+(a*c*(1+tan(f*x+e
)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2))*a*c*tan(f*x+e)-57*I*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)-
37*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+23*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1+
tan(f*x+e)^2))^(1/2)/(tan(f*x+e)+I)^3/(a*c)^(1/2)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 706 vs. \(2 (160) = 320\).
time = 0.58, size = 706, normalized size = 3.46 \begin {gather*} -\frac {3 \, {\left (30 \, {\left (a^{3} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + a^{3}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 30 \, {\left (a^{3} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + a^{3}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 8 \, {\left (a^{3} \cos \left (2 \, f x + 2 \, e\right ) + i \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + a^{3}\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 12 \, {\left (4 \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 i \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) + 5 \, a^{3}\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 15 \, {\left (i \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) - a^{3} \sin \left (2 \, f x + 2 \, e\right ) + i \, a^{3}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 15 \, {\left (-i \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) + a^{3} \sin \left (2 \, f x + 2 \, e\right ) - i \, a^{3}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 8 \, {\left (i \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) - a^{3} \sin \left (2 \, f x + 2 \, e\right ) + i \, a^{3}\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 \, {\left (-4 i \, a^{3} \cos \left (2 \, f x + 2 \, e\right ) + 4 \, a^{3} \sin \left (2 \, f x + 2 \, e\right ) - 5 i \, a^{3}\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{-18 \, {\left (i \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) - c^{2} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{2}\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

-3*(30*(a^3*cos(2*f*x + 2*e) + I*a^3*sin(2*f*x + 2*e) + a^3)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f
*x + 2*e))), sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 30*(a^3*cos(2*f*x + 2*e) + I*a^3*sin(
2*f*x + 2*e) + a^3)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) + 1) + 8*(a^3*cos(2*f*x + 2*e) + I*a^3*sin(2*f*x + 2*e) + a^3)*cos(3/2*arctan2(sin(2
*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(4*a^3*cos(2*f*x + 2*e) + 4*I*a^3*sin(2*f*x + 2*e) + 5*a^3)*cos(1/2*arcta
n2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 15*(I*a^3*cos(2*f*x + 2*e) - a^3*sin(2*f*x + 2*e) + I*a^3)*log(cos(1
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2
*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 15*(-I*a^3*cos(2*f*x + 2*e) + a^3*sin(2*f*x + 2*e
) - I*a^3)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(
2*f*x + 2*e)))^2 - 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 8*(I*a^3*cos(2*f*x + 2*e) - a
^3*sin(2*f*x + 2*e) + I*a^3)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12*(-4*I*a^3*cos(2*f*x + 2
*e) + 4*a^3*sin(2*f*x + 2*e) - 5*I*a^3)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)*sqrt(c)/
((-18*I*c^2*cos(2*f*x + 2*e) + 18*c^2*sin(2*f*x + 2*e) - 18*I*c^2)*f)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (160) = 320\).
time = 0.81, size = 385, normalized size = 1.89 \begin {gather*} \frac {15 \, \sqrt {\frac {a^{7}}{c^{3} f^{2}}} c^{2} f \log \left (\frac {4 \, {\left (2 \, {\left (a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c^{2} f\right )} \sqrt {\frac {a^{7}}{c^{3} f^{2}}}\right )}}{a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3}}\right ) - 15 \, \sqrt {\frac {a^{7}}{c^{3} f^{2}}} c^{2} f \log \left (\frac {4 \, {\left (2 \, {\left (a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left (-i \, c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c^{2} f\right )} \sqrt {\frac {a^{7}}{c^{3} f^{2}}}\right )}}{a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + a^{3}}\right ) - 4 \, {\left (2 i \, a^{3} e^{\left (5 i \, f x + 5 i \, e\right )} - 10 i \, a^{3} e^{\left (3 i \, f x + 3 i \, e\right )} - 15 i \, a^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{6 \, c^{2} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

1/6*(15*sqrt(a^7/(c^3*f^2))*c^2*f*log(4*(2*(a^3*e^(3*I*f*x + 3*I*e) + a^3*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x
+ 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - (I*c^2*f*e^(2*I*f*x + 2*I*e) - I*c^2*f)*sqrt(a^7/(c^3*f^2))
)/(a^3*e^(2*I*f*x + 2*I*e) + a^3)) - 15*sqrt(a^7/(c^3*f^2))*c^2*f*log(4*(2*(a^3*e^(3*I*f*x + 3*I*e) + a^3*e^(I
*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - (-I*c^2*f*e^(2*I*f*x + 2*I*
e) + I*c^2*f)*sqrt(a^7/(c^3*f^2)))/(a^3*e^(2*I*f*x + 2*I*e) + a^3)) - 4*(2*I*a^3*e^(5*I*f*x + 5*I*e) - 10*I*a^
3*e^(3*I*f*x + 3*I*e) - 15*I*a^3*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e
) + 1)))/(c^2*f)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**(7/2)/(c-I*c*tan(f*x+e))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((I*a*tan(f*x + e) + a)^(7/2)/(-I*c*tan(f*x + e) + c)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}}{{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(e + f*x)*1i)^(7/2)/(c - c*tan(e + f*x)*1i)^(3/2),x)

[Out]

int((a + a*tan(e + f*x)*1i)^(7/2)/(c - c*tan(e + f*x)*1i)^(3/2), x)

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