3.1116 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx\)
Optimal. Leaf size=185 \[ \frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}-\frac {d (c+5 i d) \sqrt {c+d \tan (e+f x)}}{2 a f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f}+\frac {(c+i d)^{3/2} (4 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f} \]
[Out]
-1/2*I*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/a/f+1/2*(c+I*d)^(3/2)*(I*c+4*d)*arctanh((c+
d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/a/f-1/2*(c+5*I*d)*d*(c+d*tan(f*x+e))^(1/2)/a/f+1/2*(I*c-d)*(c+d*tan(f*x+e))
^(3/2)/f/(a+I*a*tan(f*x+e))
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Rubi [A] time = 0.43, antiderivative size = 185, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used =
{3550, 3528, 3539, 3537, 63, 208} \[ \frac {(-d+i c) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}-\frac {d (c+5 i d) \sqrt {c+d \tan (e+f x)}}{2 a f}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f}+\frac {(c+i d)^{3/2} (4 d+i c) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f} \]
Antiderivative was successfully verified.
[In]
Int[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]
[Out]
((-I/2)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(a*f) + ((c + I*d)^(3/2)*(I*c + 4*d)*
ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(2*a*f) - ((c + (5*I)*d)*d*Sqrt[c + d*Tan[e + f*x]])/(2*a*f)
+ ((I*c - d)*(c + d*Tan[e + f*x])^(3/2))/(2*f*(a + I*a*Tan[e + f*x]))
Rule 63
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 3528
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Rule 3537
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c*
d)/f, Subst[Int[(a + (b*x)/d)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &&
NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Rule 3539
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(c
+ I*d)/2, Int[(a + b*Tan[e + f*x])^m*(1 - I*Tan[e + f*x]), x], x] + Dist[(c - I*d)/2, Int[(a + b*Tan[e + f*x]
)^m*(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0]
&& NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Rule 3550
Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((b
*c - a*d)*(c + d*Tan[e + f*x])^(n - 1))/(2*a*f*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a^2), Int[(c + d*Tan[e +
f*x])^(n - 2)*Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*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] && GtQ[n, 1]
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{a+i a \tan (e+f x)} \, dx &=\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \sqrt {c+d \tan (e+f x)} \left (\frac {1}{2} a (2 c+i d) (c-3 i d)-\frac {1}{2} a (c+5 i d) d \tan (e+f x)\right ) \, dx}{2 a^2}\\ &=-\frac {(c+5 i d) d \sqrt {c+d \tan (e+f x)}}{2 a f}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {\int \frac {\frac {1}{2} a \left (2 c^3-5 i c^2 d+4 c d^2+5 i d^3\right )+\frac {1}{2} a d \left (c^2-10 i c d+3 d^2\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 a^2}\\ &=-\frac {(c+5 i d) d \sqrt {c+d \tan (e+f x)}}{2 a f}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}+\frac {(c-i d)^3 \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a}+\frac {\left ((c+i d)^2 (c-4 i d)\right ) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{4 a}\\ &=-\frac {(c+5 i d) d \sqrt {c+d \tan (e+f x)}}{2 a f}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}-\frac {(i c+d)^3 \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{4 a f}-\frac {\left ((c+i d)^2 (i c+4 d)\right ) \operatorname {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{4 a f}\\ &=-\frac {(c+5 i d) d \sqrt {c+d \tan (e+f x)}}{2 a f}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}-\frac {(c-i d)^3 \operatorname {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{2 a d f}-\frac {\left ((c+i d)^2 (c-4 i d)\right ) \operatorname {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{2 a d f}\\ &=-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{2 a f}+\frac {(c+i d)^{3/2} (i c+4 d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{2 a f}-\frac {(c+5 i d) d \sqrt {c+d \tan (e+f x)}}{2 a f}+\frac {(i c-d) (c+d \tan (e+f x))^{3/2}}{2 f (a+i a \tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 2.04, size = 260, normalized size = 1.41 \[ \frac {\sec (e+f x) (\cos (f x)+i \sin (f x)) \left (2 (\sin (f x)+i \cos (f x)) \sqrt {c+d \tan (e+f x)} \left (\left (c^2+2 i c d-5 d^2\right ) \cos (e+f x)-4 i d^2 \sin (e+f x)\right )+\frac {2 (\cos (e)+i \sin (e)) \left (-\sqrt {-c-i d} (d+i c)^3 \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c+i d}}\right )-i \sqrt {-c+i d} (c+i d)^2 (c-4 i d) \tan ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {-c-i d}}\right )\right )}{\sqrt {-c-i d} \sqrt {-c+i d}}\right )}{4 f (a+i a \tan (e+f x))} \]
Antiderivative was successfully verified.
[In]
Integrate[(c + d*Tan[e + f*x])^(5/2)/(a + I*a*Tan[e + f*x]),x]
[Out]
(Sec[e + f*x]*(Cos[f*x] + I*Sin[f*x])*((2*((-I)*Sqrt[-c + I*d]*(c + I*d)^2*(c - (4*I)*d)*ArcTan[Sqrt[c + d*Tan
[e + f*x]]/Sqrt[-c - I*d]] - Sqrt[-c - I*d]*(I*c + d)^3*ArcTan[Sqrt[c + d*Tan[e + f*x]]/Sqrt[-c + I*d]])*(Cos[
e] + I*Sin[e]))/(Sqrt[-c - I*d]*Sqrt[-c + I*d]) + 2*(I*Cos[f*x] + Sin[f*x])*((c^2 + (2*I)*c*d - 5*d^2)*Cos[e +
f*x] - (4*I)*d^2*Sin[e + f*x])*Sqrt[c + d*Tan[e + f*x]]))/(4*f*(a + I*a*Tan[e + f*x]))
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fricas [B] time = 0.83, size = 1037, normalized size = 5.61 \[ \text {result too large to display} \]
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)),x, algorithm="fricas")
[Out]
-1/8*(a*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^2*f^2))*e^(2*I*f*x + 2*I*e)
*log(1/2*(4*c^3 - 8*I*c^2*d - 4*c*d^2 - (4*I*a*f*e^(2*I*f*x + 2*I*e) + 4*I*a*f)*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(-(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^2*f^2)) + 2*(2*c^3 - 6*I*c^2*d - 6*c*d^2 + 2*I*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(c^2 - 2*
I*c*d - d^2)) - a*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^2*f^2))*e^(2*I*f*
x + 2*I*e)*log(1/2*(4*c^3 - 8*I*c^2*d - 4*c*d^2 - (-4*I*a*f*e^(2*I*f*x + 2*I*e) - 4*I*a*f)*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(-(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^2*f^2)) + 2*(2*c^3 - 6*I*c^2*d - 6*c*d^2 + 2*I*d^3)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e
)/(c^2 - 2*I*c*d - d^2)) - a*f*sqrt(-(c^5 - 5*I*c^4*d + 5*c^3*d^2 - 25*I*c^2*d^3 + 40*c*d^4 + 16*I*d^5)/(a^2*f
^2))*e^(2*I*f*x + 2*I*e)*log(1/2*(I*c^3 + 2*c^2*d + 7*I*c*d^2 - 4*d^3 + (a*f*e^(2*I*f*x + 2*I*e) + a*f)*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(-(c^5 - 5*I*c^4*d + 5*c^3*d^2 - 25*I*
c^2*d^3 + 40*c*d^4 + 16*I*d^5)/(a^2*f^2)) + (I*c^3 + 3*c^2*d + 4*I*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2
*I*e)/(a*f)) + a*f*sqrt(-(c^5 - 5*I*c^4*d + 5*c^3*d^2 - 25*I*c^2*d^3 + 40*c*d^4 + 16*I*d^5)/(a^2*f^2))*e^(2*I*
f*x + 2*I*e)*log(1/2*(I*c^3 + 2*c^2*d + 7*I*c*d^2 - 4*d^3 - (a*f*e^(2*I*f*x + 2*I*e) + a*f)*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(-(c^5 - 5*I*c^4*d + 5*c^3*d^2 - 25*I*c^2*d^3 + 40
*c*d^4 + 16*I*d^5)/(a^2*f^2)) + (I*c^3 + 3*c^2*d + 4*I*c*d^2)*e^(2*I*f*x + 2*I*e))*e^(-2*I*f*x - 2*I*e)/(a*f))
- 2*(I*c^2 - 2*c*d - I*d^2 + (I*c^2 - 2*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)))*e^(-2*I*f*x - 2*I*e)/(a*f)
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giac [B] time = 0.86, size = 462, normalized size = 2.50 \[ -\frac {2 i \, \sqrt {d \tan \left (f x + e\right ) + c} d^{2}}{a f} + \frac {\sqrt {2} {\left (c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}\right )} \arctan \left (\frac {-16 i \, \sqrt {d \tan \left (f x + e\right ) + c} c - 16 i \, \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}}{8 \, {\left (\sqrt {2} \sqrt {c + \sqrt {c^{2} + d^{2}}} c - i \, \sqrt {2} \sqrt {c + \sqrt {c^{2} + d^{2}}} d + \sqrt {2} \sqrt {c^{2} + d^{2}} \sqrt {c + \sqrt {c^{2} + d^{2}}}\right )}}\right )}{2 \, a \sqrt {c + \sqrt {c^{2} + d^{2}}} f {\left (-\frac {i \, d}{c + \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {2 \, {\left (-i \, c^{3} - 2 \, c^{2} d - 7 i \, c d^{2} + 4 \, d^{3}\right )} \arctan \left (\frac {4 \, {\left (\sqrt {d \tan \left (f x + e\right ) + c} c - \sqrt {c^{2} + d^{2}} \sqrt {d \tan \left (f x + e\right ) + c}\right )}}{c \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} + i \, \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} d - \sqrt {c^{2} + d^{2}} \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}}}\right )}{a \sqrt {-8 \, c + 8 \, \sqrt {c^{2} + d^{2}}} f {\left (\frac {i \, d}{c - \sqrt {c^{2} + d^{2}}} + 1\right )}} + \frac {\sqrt {d \tan \left (f x + e\right ) + c} c^{2} d + 2 i \, \sqrt {d \tan \left (f x + e\right ) + c} c d^{2} - \sqrt {d \tan \left (f x + e\right ) + c} d^{3}}{2 \, {\left (d \tan \left (f x + e\right ) - i \, d\right )} a f} \]
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)),x, algorithm="giac")
[Out]
-2*I*sqrt(d*tan(f*x + e) + c)*d^2/(a*f) + 1/2*sqrt(2)*(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)*arctan(1/8*(-16*I*sq
rt(d*tan(f*x + e) + c)*c - 16*I*sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(sqrt(2)*sqrt(c + sqrt(c^2 + d^2))*c
- I*sqrt(2)*sqrt(c + sqrt(c^2 + d^2))*d + sqrt(2)*sqrt(c^2 + d^2)*sqrt(c + sqrt(c^2 + d^2))))/(a*sqrt(c + sqr
t(c^2 + d^2))*f*(-I*d/(c + sqrt(c^2 + d^2)) + 1)) + 2*(-I*c^3 - 2*c^2*d - 7*I*c*d^2 + 4*d^3)*arctan(4*(sqrt(d*
tan(f*x + e) + c)*c - sqrt(c^2 + d^2)*sqrt(d*tan(f*x + e) + c))/(c*sqrt(-8*c + 8*sqrt(c^2 + d^2)) + I*sqrt(-8*
c + 8*sqrt(c^2 + d^2))*d - sqrt(c^2 + d^2)*sqrt(-8*c + 8*sqrt(c^2 + d^2))))/(a*sqrt(-8*c + 8*sqrt(c^2 + d^2))*
f*(I*d/(c - sqrt(c^2 + d^2)) + 1)) + 1/2*(sqrt(d*tan(f*x + e) + c)*c^2*d + 2*I*sqrt(d*tan(f*x + e) + c)*c*d^2
- sqrt(d*tan(f*x + e) + c)*d^3)/((d*tan(f*x + e) - I*d)*a*f)
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maple [B] time = 0.38, size = 674, normalized size = 3.64 \[ -\frac {2 i d^{2} \sqrt {c +d \tan \left (f x +e \right )}}{f a}+\frac {3 i d^{2} \sqrt {c +d \tan \left (f x +e \right )}\, c^{2}}{2 f a \left (i d +c \right ) \left (d \tan \left (f x +e \right )-i d \right )}-\frac {i d^{4} \sqrt {c +d \tan \left (f x +e \right )}}{2 f a \left (i d +c \right ) \left (d \tan \left (f x +e \right )-i d \right )}+\frac {d \sqrt {c +d \tan \left (f x +e \right )}\, c^{3}}{2 f a \left (i d +c \right ) \left (d \tan \left (f x +e \right )-i d \right )}-\frac {3 d^{3} \sqrt {c +d \tan \left (f x +e \right )}\, c}{2 f a \left (i d +c \right ) \left (d \tan \left (f x +e \right )-i d \right )}-\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{4}}{2 f a \left (i d +c \right ) \sqrt {-i d -c}}-\frac {9 i d^{2} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{2}}{2 f a \left (i d +c \right ) \sqrt {-i d -c}}+\frac {2 i d^{4} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right )}{f a \left (i d +c \right ) \sqrt {-i d -c}}-\frac {d \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c^{3}}{2 f a \left (i d +c \right ) \sqrt {-i d -c}}+\frac {11 d^{3} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {-i d -c}}\right ) c}{2 f a \left (i d +c \right ) \sqrt {-i d -c}}+\frac {i \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right ) c^{3}}{2 f a \sqrt {i d -c}}-\frac {3 i d^{2} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right ) c}{2 f a \sqrt {i d -c}}+\frac {3 d \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right ) c^{2}}{2 f a \sqrt {i d -c}}-\frac {d^{3} \arctan \left (\frac {\sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {i d -c}}\right )}{2 f a \sqrt {i d -c}} \]
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)),x)
[Out]
-2*I/f/a*d^2*(c+d*tan(f*x+e))^(1/2)+3/2*I/f/a*d^2/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(d*tan(f*x+e)-I*d)*c^2-1/2*I/
f/a*d^4/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(d*tan(f*x+e)-I*d)+1/2/f/a*d/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(d*tan(f*x+
e)-I*d)*c^3-3/2/f/a*d^3/(c+I*d)*(c+d*tan(f*x+e))^(1/2)/(d*tan(f*x+e)-I*d)*c-1/2*I/f/a/(c+I*d)/(-I*d-c)^(1/2)*a
rctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^4-9/2*I/f/a*d^2/(c+I*d)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^
(1/2)/(-I*d-c)^(1/2))*c^2+2*I/f/a*d^4/(c+I*d)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))-1/2
/f/a*d/(c+I*d)/(-I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c^3+11/2/f/a*d^3/(c+I*d)/(-I*d-c)^
(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(-I*d-c)^(1/2))*c+1/2*I/f/a/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I
*d-c)^(1/2))*c^3-3/2*I/f/a*d^2/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c+3/2/f/a*d/(I*d-c)^
(1/2)*arctan((c+d*tan(f*x+e))^(1/2)/(I*d-c)^(1/2))*c^2-1/2/f/a*d^3/(I*d-c)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)
/(I*d-c)^(1/2))
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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)),x, algorithm="maxima")
[Out]
Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.
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mupad [B] time = 9.36, size = 4857, normalized size = 26.25 \[ \text {result too large to display} \]
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),x)
[Out]
log((a*f*(12*d^11 - c*d^10*25i + 75*c^2*d^9 - c^3*d^8*35i + 113*c^4*d^7 + c^5*d^6*5i + 49*c^6*d^5 + c^7*d^4*15
i - c^8*d^3))/2 - (((a*f*(a^2*d^6*f^2*80i + 16*a^2*c*d^5*f^2 + a^2*c^2*d^4*f^2*80i + 16*a^2*c^3*d^3*f^2))/2 +
64*a^4*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^
6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((((400*c^4*d^7 - 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10
+ 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13
+ 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^1
0 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(720*c*d^
10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((((400*c^4*d^7
- 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 -
4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^
4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(
1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2) + 2*a^2*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*50i - 17*d^8 + 80*c^2
*d^6 - 5*c^4*d^4 - c^5*d^3*10i + 2*c^6*d^2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d
^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((((400*c^4*d^7 - 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10
+ 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13
+ 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^
10 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(720*c*d
^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((((400*c^4*d^
7 - 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2
- 4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f
^4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^
(1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2) + log((a*f*(12*d^11 - c*d^10*25i + 75*c^2*d^9 - c^3*d^8*35i + 113*
c^4*d^7 + c^5*d^6*5i + 49*c^6*d^5 + c^7*d^4*15i - c^8*d^3))/2 - (((a*f*(a^2*d^6*f^2*80i + 16*a^2*c*d^5*f^2 + a
^2*c^2*d^4*f^2*80i + 16*a^2*c^3*d^3*f^2))/2 + 64*a^4*c*d^2*f^4*(c + d*tan(e + f*x))^(1/2)*(-(720*c*d^10 + d^11
*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 - a^2*f^2*((((400*c^4*d^7 - 240*d^
11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*d
^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35*
c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(51
2*a^2*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d
^5*160i + 32*c^7*d^4 - a^2*f^2*((((400*c^4*d^7 - 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^3
*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^5
*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*c
^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2) + 2*a^2*f^2*(c + d*ta
n(e + f*x))^(1/2)*(c*d^7*50i - 17*d^8 + 80*c^2*d^6 - 5*c^4*d^4 - c^5*d^3*10i + 2*c^6*d^2))*(-(720*c*d^10 + d^1
1*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 - a^2*f^2*((((400*c^4*d^7 - 240*d
^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*
d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35
*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(5
12*a^2*f^2*(d^6 + c^2*d^4)))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*
d^5*160i + 32*c^7*d^4 - a^2*f^2*((((400*c^4*d^7 - 240*d^11 + 160*c^6*d^5)*1i)/(a^2*f^2) - (720*c*d^10 + 640*c^
3*d^8 - 48*c^5*d^6 + 32*c^7*d^4)/(a^2*f^2))^2 - 4*(256*d^6 + 256*c^2*d^4)*(((40*c*d^15 + 150*c^3*d^13 + 200*c^
5*d^11 + 100*c^7*d^9 - 10*c^11*d^5)*1i)/(a^4*f^4) - (35*c^4*d^12 - 31*c^2*d^14 - 16*d^16 + 130*c^6*d^10 + 110*
c^8*d^8 + 29*c^10*d^6 - c^12*d^4)/(a^4*f^4)))^(1/2))/(512*a^2*f^2*(d^6 + c^2*d^4)))^(1/2) - log((a*f*(12*d^11
- c*d^10*25i + 75*c^2*d^9 - c^3*d^8*35i + 113*c^4*d^7 + c^5*d^6*5i + 49*c^6*d^5 + c^7*d^4*15i - c^8*d^3))/2 -
(((a*f*(a^2*d^6*f^2*80i + 16*a^2*c*d^5*f^2 + a^2*c^2*d^4*f^2*80i + 16*a^2*c^3*d^3*f^2))/2 - 64*a^4*c*d^2*f^4*(
-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((c*d
^21*304640i - 73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c
^6*d^16 - c^7*d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*
d^6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^
4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((c*d^21*304640i - 73984*d^22 + 470272*c^2*d^20
+ c^3*d^19*112640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c^6*d^16 - c^7*d^15*1034240i + 70400*c^8*d^1
4 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2) - 2*a
^2*f^2*(c + d*tan(e + f*x))^(1/2)*(c*d^7*50i - 17*d^8 + 80*c^2*d^6 - 5*c^4*d^4 - c^5*d^3*10i + 2*c^6*d^2))*(-(
720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 + a^2*f^2*((c*d^2
1*304640i - 73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c^6
*d^16 - c^7*d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^
6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6
*d^5*160i + 32*c^7*d^4 + a^2*f^2*((c*d^21*304640i - 73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*
c^4*d^18 - c^5*d^17*957440i + 586240*c^6*d^16 - c^7*d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*
c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2) - log((a*f*(12*d^11 - c*d^10*25i +
75*c^2*d^9 - c^3*d^8*35i + 113*c^4*d^7 + c^5*d^6*5i + 49*c^6*d^5 + c^7*d^4*15i - c^8*d^3))/2 - (((a*f*(a^2*d^
6*f^2*80i + 16*a^2*c*d^5*f^2 + a^2*c^2*d^4*f^2*80i + 16*a^2*c^3*d^3*f^2))/2 - 64*a^4*c*d^2*f^4*(-(720*c*d^10 +
d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 - a^2*f^2*((c*d^21*304640i -
73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c^6*d^16 - c^7*
d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*
a^2*c^2*d^4*f^2))^(1/2)*(c + d*tan(e + f*x))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 4
8*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 - a^2*f^2*((c*d^21*304640i - 73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112
640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c^6*d^16 - c^7*d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*2
68800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2) - 2*a^2*f^2*(c + d*
tan(e + f*x))^(1/2)*(c*d^7*50i - 17*d^8 + 80*c^2*d^6 - 5*c^4*d^4 - c^5*d^3*10i + 2*c^6*d^2))*(-(720*c*d^10 + d
^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32*c^7*d^4 - a^2*f^2*((c*d^21*304640i - 73
984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*c^4*d^18 - c^5*d^17*957440i + 586240*c^6*d^16 - c^7*d^
15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*a^
2*c^2*d^4*f^2))^(1/2))*(-(720*c*d^10 + d^11*240i + 640*c^3*d^8 - c^4*d^7*400i - 48*c^5*d^6 - c^6*d^5*160i + 32
*c^7*d^4 - a^2*f^2*((c*d^21*304640i - 73984*d^22 + 470272*c^2*d^20 + c^3*d^19*112640i + 1117696*c^4*d^18 - c^5
*d^17*957440i + 586240*c^6*d^16 - c^7*d^15*1034240i + 70400*c^8*d^14 - c^9*d^13*268800i + 57600*c^10*d^12)/(a^
4*f^4))^(1/2))/(512*a^2*d^6*f^2 + 512*a^2*c^2*d^4*f^2))^(1/2) - (d^2*(c + d*tan(e + f*x))^(1/2)*2i)/(a*f) - ((
c + d*tan(e + f*x))^(1/2)*(c*d^2*2i + c^2*d - d^3))/(2*a*f*(d*1i - d*tan(e + f*x)))
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \left (\int \frac {c^{2} \sqrt {c + d \tan {\left (e + f x \right )}}}{\tan {\left (e + f x \right )} - i}\, dx + \int \frac {d^{2} \sqrt {c + d \tan {\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx + \int \frac {2 c d \sqrt {c + d \tan {\left (e + f x \right )}} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\, dx\right )}{a} \]
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)),x)
[Out]
-I*(Integral(c**2*sqrt(c + d*tan(e + f*x))/(tan(e + f*x) - I), x) + Integral(d**2*sqrt(c + d*tan(e + f*x))*tan
(e + f*x)**2/(tan(e + f*x) - I), x) + Integral(2*c*d*sqrt(c + d*tan(e + f*x))*tan(e + f*x)/(tan(e + f*x) - I),
x))/a
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