3.12.53 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx\) [1153]
Optimal. Leaf size=257 \[ \frac {2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
[Out]
2*(-1)^(1/4)*d^(5/2)*arctanh((-1)^(3/4)*d^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c+d*tan(f*x+e))^(1/2))/a^(3/
2)/f-1/4*I*(c-I*d)^(5/2)*arctanh(2^(1/2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2)
)/a^(3/2)/f*2^(1/2)+1/2*(c+I*d)*(I*c+3*d)*(c+d*tan(f*x+e))^(1/2)/a/f/(a+I*a*tan(f*x+e))^(1/2)+1/3*(I*c-d)*(c+d
*tan(f*x+e))^(3/2)/f/(a+I*a*tan(f*x+e))^(3/2)
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Rubi [A]
time = 0.65, antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {3639, 3676,
3682, 3625, 214, 3680, 65, 223, 212} \begin {gather*} \frac {2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {(c+i d) (3 d+i c) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]
[Out]
(2*(-1)^(1/4)*d^(5/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]
])])/(a^(3/2)*f) - ((I/2)*(c - I*d)^(5/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sq
rt[a + I*a*Tan[e + f*x]])])/(Sqrt[2]*a^(3/2)*f) + ((c + I*d)*(I*c + 3*d)*Sqrt[c + d*Tan[e + f*x]])/(2*a*f*Sqrt
[a + I*a*Tan[e + f*x]]) + ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(3*f*(a + I*a*Tan[e + f*x])^(3/2))
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 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
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 3625
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 3639
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Dist[1/(2*a^2*m), Int[(
a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)
) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Tan[e + f*x], x], 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] && LtQ[m, 0] && GtQ[n, 1] && (IntegerQ[m] || IntegersQ[2*m
, 2*n])
Rule 3676
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*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f
*m)), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d
*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]
Rule 3680
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b*(B/f), Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x
]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && EqQ[A*b + a*B,
0]
Rule 3682
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(A*b + a*B)/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x]
, x] - Dist[B/b, Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[e + f*x]), x], x] /; FreeQ[{a, b
, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\frac {3}{2} a \left (c^2-2 i c d+d^2\right )+3 i a d^2 \tan (e+f x)\right )}{\sqrt {a+i a \tan (e+f x)}} \, dx}{3 a^2}\\ &=\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (\frac {3}{4} a^2 \left (c^3-3 i c^2 d-3 c d^2-3 i d^3\right )-3 a^2 d^3 \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 a^4}\\ &=\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {(c-i d)^3 \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a^2}-\frac {\left (i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx}{a^3}\\ &=\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (i d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{a f}+\frac {(i c+d)^3 \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{2 f}\\ &=-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+i d-\frac {i d x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{a^2 f}\\ &=-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}-\frac {\left (2 d^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {i d x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{a^2 f}\\ &=\frac {2 \sqrt [4]{-1} d^{5/2} \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{a^{3/2} f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{2 \sqrt {2} a^{3/2} f}+\frac {(c+i d) (i c+3 d) \sqrt {c+d \tan (e+f x)}}{2 a f \sqrt {a+i a \tan (e+f x)}}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(560\) vs. \(2(257)=514\).
time = 8.69, size = 560, normalized size = 2.18 \begin {gather*} \frac {\sec (e+f x) (\cos (f x)+i \sin (f x))^2 \left (\frac {\left ((2+2 i) d^{5/2} \log \left (\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {i e}{2}} \left (c+i d-i c e^{i (e+f x)}-d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{d^{7/2} \left (i+e^{i (e+f x)}\right )}\right )-(2+2 i) d^{5/2} \log \left (-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {i e}{2}} \left (c+i d+i c e^{i (e+f x)}+d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{d^{7/2} \left (-i+e^{i (e+f x)}\right )}\right )-i (c-i d)^{5/2} \log \left (2 \left (\sqrt {c-i d} \cos (e+f x)+i \sqrt {c-i d} \sin (e+f x)+\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))} \sqrt {c+d \tan (e+f x)}\right )\right )\right ) (\cos (2 e)+i \sin (2 e))}{\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}+\frac {1}{3} (c+i d) (i \cos (2 f x)+\sin (2 f x)) ((5 c-9 i d) \cos (e+f x)+(3 i c+11 d) \sin (e+f x)) \sqrt {c+d \tan (e+f x)}\right )}{2 f (a+i a \tan (e+f x))^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x])^(3/2),x]
[Out]
(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])^2*((((2 + 2*I)*d^(5/2)*Log[((1/4 + I/4)*E^((I/2)*e)*(c + I*d - I*c*E^(I*
(e + f*x)) - d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e
+ f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(7/2)*(I + E^(I*(e + f*x))))] - (2 + 2*I)*d^(5/2)*Log[((-1/4 - I/4)
*E^((I/2)*e)*(c + I*d + I*c*E^(I*(e + f*x)) + d*E^(I*(e + f*x)) + (1 + I)*Sqrt[d]*Sqrt[1 + E^((2*I)*(e + f*x))
]*Sqrt[c - (I*d*(-1 + E^((2*I)*(e + f*x))))/(1 + E^((2*I)*(e + f*x)))]))/(d^(7/2)*(-I + E^(I*(e + f*x))))] - I
*(c - I*d)^(5/2)*Log[2*(Sqrt[c - I*d]*Cos[e + f*x] + I*Sqrt[c - I*d]*Sin[e + f*x] + Sqrt[1 + Cos[2*(e + f*x)]
+ I*Sin[2*(e + f*x)]]*Sqrt[c + d*Tan[e + f*x]])])*(Cos[2*e] + I*Sin[2*e]))/Sqrt[1 + Cos[2*(e + f*x)] + I*Sin[2
*(e + f*x)]] + ((c + I*d)*(I*Cos[2*f*x] + Sin[2*f*x])*((5*c - (9*I)*d)*Cos[e + f*x] + ((3*I)*c + 11*d)*Sin[e +
f*x])*Sqrt[c + d*Tan[e + f*x]])/3))/(2*f*(a + I*a*Tan[e + f*x])^(3/2))
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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. 3160 vs. \(2 (201 ) = 402\).
time = 0.51, size = 3161, normalized size = 12.30
| | |
| method |
result |
size |
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| derivativedivides |
\(\text {Expression too large to display}\) |
\(3161\) |
| default |
\(\text {Expression too large to display}\) |
\(3161\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
1/24*I/f*(c+d*tan(f*x+e))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^2*(-9*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*
ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*
x+e)))^(1/2))/(tan(f*x+e)+I))*c*d^2*tan(f*x+e)+3*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(
f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan
(f*x+e)+I))*c*d^2*tan(f*x+e)^3-24*I*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(
1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^4+36*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*d^
3+20*c^3*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)-24*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c
+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^4*tan(f*x+e)^3-12*c^3*(a*(c+d*tan
(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+e)^2+52*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*
a*d)^(1/2)*c*d^2-80*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*d^3*tan(f*x+e)+72*ln(1/2*(2*I*a*
d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^4*tan(f
*x+e)-24*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*
a*d)^(1/2))*a*c*d^3-3*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+
e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^3+72*I*ln(1/2*(
2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^4
*tan(f*x+e)^2-44*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*d^3*tan(f*x+e)^2+32*I*c^3*(a*(c+d
*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*tan(f*x+e)+4*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*
(I*a*d)^(1/2)*c^2*d+9*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+
e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c*d^2*tan(f*x+e)^
2-9*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(
I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^2*d*tan(f*x+e)-9*I*2^(1/2)*(-a*(I
*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(
c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^3*tan(f*x+e)+3*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)
^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I
*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^2*d+3*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)
*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e
)+I))*c^2*d*tan(f*x+e)^3+3*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*t
an(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^3*tan(f*
x+e)^3-9*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2
)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*d^3*tan(f*x+e)^2+24*I*ln(1/2
*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c
*d^3*tan(f*x+e)^3+3*I*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+
e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*d^3-72*I*ln(1/2*(
2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*c*d
^3*tan(f*x+e)+20*I*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*c^2*d*tan(f*x+e)^2+128*I*(a*(c+d*
tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)*c*d^2*tan(f*x+e)-9*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*
ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*
x+e)))^(1/2))/(tan(f*x+e)+I))*d^3*tan(f*x+e)-3*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+
e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x
+e)+I))*c*d^2+9*2^(1/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2
^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^3*tan(f*x+e)^2+3*2^(1
/2)*(-a*(I*d-c))^(1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^
(1/2)*(a*(c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*d^3*tan(f*x+e)^3-9*I*2^(1/2)*(-a*(I*d-c))^(
1/2)*(I*a*d)^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(c+d*tan(
f*x+e))*(1+I*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c^2*d*tan(f*x+e)^2+72*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(
c+d*tan(f*x+e))*(1+I*tan(f*x+e)))^(1/2)*(I*a*d)...
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.
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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. 1105 vs. \(2 (197) = 394\).
time = 1.22, size = 1105, normalized size = 4.30 \begin {gather*} -\frac {{\left (3 \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c^{5} - 5 i \, c^{4} d - 10 \, c^{3} d^{2} + 10 i \, c^{2} d^{3} + 5 \, c d^{4} - i \, d^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (-\frac {2 i \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c^{5} - 5 i \, c^{4} d - 10 \, c^{3} d^{2} + 10 i \, c^{2} d^{3} + 5 \, c d^{4} - i \, d^{5}}{a^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (c^{2} - 2 i \, c d - d^{2} + {\left (c^{2} - 2 i \, c d - d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{c^{2} - 2 i \, c d - d^{2}}\right ) - 3 \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c^{5} - 5 i \, c^{4} d - 10 \, c^{3} d^{2} + 10 i \, c^{2} d^{3} + 5 \, c d^{4} - i \, d^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (-\frac {-2 i \, \sqrt {\frac {1}{2}} a^{2} f \sqrt {-\frac {c^{5} - 5 i \, c^{4} d - 10 \, c^{3} d^{2} + 10 i \, c^{2} d^{3} + 5 \, c d^{4} - i \, d^{5}}{a^{3} f^{2}}} e^{\left (i \, f x + i \, e\right )} - \sqrt {2} {\left (c^{2} - 2 i \, c d - d^{2} + {\left (c^{2} - 2 i \, c d - d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{c^{2} - 2 i \, c d - d^{2}}\right ) + 3 \, a^{2} f \sqrt {\frac {4 i \, d^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (\frac {2 \, {\left (4 \, \sqrt {2} {\left (d^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + d^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + {\left ({\left (a^{2} c - 3 i \, a^{2} d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (a^{2} c + i \, a^{2} d\right )} f\right )} \sqrt {\frac {4 i \, d^{5}}{a^{3} f^{2}}}\right )}}{i \, c^{3} + c^{2} d + i \, c d^{2} + d^{3} + {\left (i \, c^{3} + c^{2} d + i \, c d^{2} + d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}\right ) - 3 \, a^{2} f \sqrt {\frac {4 i \, d^{5}}{a^{3} f^{2}}} e^{\left (3 i \, f x + 3 i \, e\right )} \log \left (\frac {2 \, {\left (4 \, \sqrt {2} {\left (d^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + d^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - {\left ({\left (a^{2} c - 3 i \, a^{2} d\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (a^{2} c + i \, a^{2} d\right )} f\right )} \sqrt {\frac {4 i \, d^{5}}{a^{3} f^{2}}}\right )}}{i \, c^{3} + c^{2} d + i \, c d^{2} + d^{3} + {\left (i \, c^{3} + c^{2} d + i \, c d^{2} + d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}\right ) - \sqrt {2} {\left (i \, c^{2} - 2 \, c d - i \, d^{2} - 2 \, {\left (-2 i \, c^{2} - 3 \, c d - 5 i \, d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (5 i \, c^{2} + 4 \, c d + 9 i \, d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{12 \, a^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
-1/12*(3*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^3*f^2))*e^(3
*I*f*x + 3*I*e)*log(-(2*I*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5
)/(a^3*f^2))*e^(I*f*x + I*e) - sqrt(2)*(c^2 - 2*I*c*d - d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x + 2*I*e))*sqrt(
((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))/(c^2 -
2*I*c*d - d^2)) - 3*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*c*d^4 - I*d^5)/(a^
3*f^2))*e^(3*I*f*x + 3*I*e)*log(-(-2*I*sqrt(1/2)*a^2*f*sqrt(-(c^5 - 5*I*c^4*d - 10*c^3*d^2 + 10*I*c^2*d^3 + 5*
c*d^4 - I*d^5)/(a^3*f^2))*e^(I*f*x + I*e) - sqrt(2)*(c^2 - 2*I*c*d - d^2 + (c^2 - 2*I*c*d - d^2)*e^(2*I*f*x +
2*I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e)
+ 1)))/(c^2 - 2*I*c*d - d^2)) + 3*a^2*f*sqrt(4*I*d^5/(a^3*f^2))*e^(3*I*f*x + 3*I*e)*log(2*(4*sqrt(2)*(d^3*e^(3
*I*f*x + 3*I*e) + d^3*e^(I*f*x + I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1
))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) + ((a^2*c - 3*I*a^2*d)*f*e^(2*I*f*x + 2*I*e) + (a^2*c + I*a^2*d)*f)*sqrt(
4*I*d^5/(a^3*f^2)))/(I*c^3 + c^2*d + I*c*d^2 + d^3 + (I*c^3 + c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) - 3
*a^2*f*sqrt(4*I*d^5/(a^3*f^2))*e^(3*I*f*x + 3*I*e)*log(2*(4*sqrt(2)*(d^3*e^(3*I*f*x + 3*I*e) + d^3*e^(I*f*x +
I*e))*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) +
1)) - ((a^2*c - 3*I*a^2*d)*f*e^(2*I*f*x + 2*I*e) + (a^2*c + I*a^2*d)*f)*sqrt(4*I*d^5/(a^3*f^2)))/(I*c^3 + c^2*
d + I*c*d^2 + d^3 + (I*c^3 + c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) - sqrt(2)*(I*c^2 - 2*c*d - I*d^2 - 2
*(-2*I*c^2 - 3*c*d - 5*I*d^2)*e^(4*I*f*x + 4*I*e) + (5*I*c^2 + 4*c*d + 9*I*d^2)*e^(2*I*f*x + 2*I*e))*sqrt(((c
- I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)))*e^(-3*I*f*
x - 3*I*e)/(a^2*f)
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))**(5/2)/(a+I*a*tan(f*x+e))**(3/2),x)
[Out]
Integral((c + d*tan(e + f*x))**(5/2)/(I*a*(tan(e + f*x) - I))**(3/2), x)
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((c+d*tan(f*x+e))^(5/2)/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.Non regu
lar value [
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{{\left (a+a\,\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((c + d*tan(e + f*x))^(5/2)/(a + a*tan(e + f*x)*1i)^(3/2),x)
[Out]
int((c + d*tan(e + f*x))^(5/2)/(a + a*tan(e + f*x)*1i)^(3/2), x)
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