3.13.62 \(\int \frac {a+b \tan (e+f x)}{(c+d \tan (e+f x))^{5/2}} \, dx\) [1262]

Optimal. Leaf size=186 \[ -\frac {(i a+b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{5/2} f}+\frac {(i a-b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{5/2} f}+\frac {2 (b c-a d)}{3 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 a c d-b \left (c^2-d^2\right )\right )}{\left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]

[Out]

-(I*a+b)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(5/2)/f+(I*a-b)*arctanh((c+d*tan(f*x+e))^(1/2)/
(c+I*d)^(1/2))/(c+I*d)^(5/2)/f-2*(2*a*c*d-b*(c^2-d^2))/(c^2+d^2)^2/f/(c+d*tan(f*x+e))^(1/2)+2/3*(-a*d+b*c)/(c^
2+d^2)/f/(c+d*tan(f*x+e))^(3/2)

________________________________________________________________________________________

Rubi [A]
time = 0.28, antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3610, 3620, 3618, 65, 214} \begin {gather*} \frac {2 (b c-a d)}{3 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^{3/2}}-\frac {2 \left (2 a c d-b \left (c^2-d^2\right )\right )}{f \left (c^2+d^2\right )^2 \sqrt {c+d \tan (e+f x)}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{5/2}}+\frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

-(((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((c - I*d)^(5/2)*f)) + ((I*a - b)*ArcTanh[Sqrt[c
+ d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(5/2)*f) + (2*(b*c - a*d))/(3*(c^2 + d^2)*f*(c + d*Tan[e + f*x])
^(3/2)) - (2*(2*a*c*d - b*(c^2 - d^2)))/((c^2 + d^2)^2*f*Sqrt[c + d*Tan[e + f*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 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 3610

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.18, size = 115, normalized size = 0.62 \begin {gather*} -\frac {i \left (-\frac {(a-i b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )}{c-i d}+\frac {(a+i b) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )}{c+i d}\right )}{3 f (c+d \tan (e+f x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])/(c + d*Tan[e + f*x])^(5/2),x]

[Out]

((-1/3*I)*(-(((a - I*b)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(c - I*d)) + ((a + I
*b)*Hypergeometric2F1[-3/2, 1, -1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(c + I*d)))/(f*(c + d*Tan[e + f*x])^(3/2
))

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3235\) vs. \(2(162)=324\).
time = 0.51, size = 3236, normalized size = 17.40

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(3236\)
default	\(\text {Expression too large to display}\)	\(3236\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/f*(-2/3*(a*d-b*c)/(c^2+d^2)/(c+d*tan(f*x+e))^(3/2)-2*(2*a*c*d-b*c^2+b*d^2)/(c^2+d^2)^2/(c+d*tan(f*x+e))^(1/2
)+2/(c^2+d^2)^2*(1/4/d/(5*c^4-10*c^2*d^2+d^4)/(c^2+d^2)^(3/2)*(1/2*(-3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a*c^6-5*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^2-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a*c^2*d^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)
^(1/2)*a*c^8+18*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d^2+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+
2*c)^(1/2)*a*c^4*d^4-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^6-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*b*c^7*d+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^3+18*(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d^5-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^7+5*(2*(c^2+d^2)^(1/2
)+2*c)^(1/2)*a*c^9-20*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^7*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d^4+28*(2*
(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^8+15*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
b*c^8*d-20*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6*d^3-22*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^5+12*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*b*c^2*d^7-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^9)*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(30*a*c^8*d^2-40*a*c^6*d^4-44*a*c^4*d^6+24*a*c^2*d^8-2*a*d^10-10*b
*c^9*d+40*b*c^7*d^3-12*b*c^5*d^5-56*b*c^3*d^7+6*b*c*d^9+1/2*(-3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*c^6-5*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^2-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*c^2*d^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^6-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*c^8+18*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d^2+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*c^4*d^4-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^6-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*b*c^7*d+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^3+18*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b*c^3*d^5-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^7+5*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a*c^9-20*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^7*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d^4+28*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*c^3*d^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^8+15*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^8*d
-20*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^6*d^3-22*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^5+12*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*b*c^2*d^7-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^9)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c
)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)-(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d/
(5*c^4-10*c^2*d^2+d^4)/(c^2+d^2)^(3/2)*(1/2*(3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6+5*(c^2+d^2)
^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^2+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^4-(c^2+d^
2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^6+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^8-18*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d^2-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^4+10*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^6+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^7
*d-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^3-18*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*b*c^3*d^5+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^7-5*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^9+20*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^7*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d^4-28*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*a*c^3*d^6+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^8-15*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^8*d+20*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)*b*c^6*d^3+22*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^4*d^5-12*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d
^7+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^9)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
+(c^2+d^2)^(1/2))+2*(30*a*c^8*d^2-40*a*c^6*d^4-44*a*c^4*d^6+24*a*c^2*d^8-2*a*d^10-10*b*c^9*d+40*b*c^7*d^3-12*b
*c^5*d^5-56*b*c^3*d^7+6*b*c*d^9-1/2*(3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6+5*(c^2+d^2)^(3/2)*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^2+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^4-(c^2+d^2)^(3/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^6+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^8-18*(c^2+d^2)^(1/2)*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^6*d^2-10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^4*d^4+10*(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^6+10*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^7*d-10*(c
^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^5*d^3-18*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d
^5+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^7-5*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^9+20*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*c^7*d^2-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^5*d^4-28*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3
*d^6+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^8-15...

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(d-c>0)', see `assume?` for mor
e details)Is

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 35738 vs. \(2 (161) = 322\).
time = 148.69, size = 35738, normalized size = 192.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/12*(12*sqrt(2)*((c^18 + c^16*d^2 - 20*c^14*d^4 - 84*c^12*d^6 - 154*c^10*d^8 - 154*c^8*d^10 - 84*c^6*d^12 -
20*c^4*d^14 + c^2*d^16 + d^18)*f^5*cos(f*x + e)^4 + 2*(3*c^16*d^2 + 20*c^14*d^4 + 56*c^12*d^6 + 84*c^10*d^8 +
70*c^8*d^10 + 28*c^6*d^12 - 4*c^2*d^16 - d^18)*f^5*cos(f*x + e)^2 + (c^14*d^4 + 7*c^12*d^6 + 21*c^10*d^8 + 35*
c^8*d^10 + 35*c^6*d^12 + 21*c^4*d^14 + 7*c^2*d^16 + d^18)*f^5 + 4*((c^17*d + 6*c^15*d^3 + 14*c^13*d^5 + 14*c^1
1*d^7 - 14*c^7*d^11 - 14*c^5*d^13 - 6*c^3*d^15 - c*d^17)*f^5*cos(f*x + e)^3 + (c^15*d^3 + 7*c^13*d^5 + 21*c^11
*d^7 + 35*c^9*d^9 + 35*c^7*d^11 + 21*c^5*d^13 + 7*c^3*d^15 + c*d^17)*f^5*cos(f*x + e))*sin(f*x + e))*sqrt(((a^
4 + 2*a^2*b^2 + b^4)*c^10 + 5*(a^4 + 2*a^2*b^2 + b^4)*c^8*d^2 + 10*(a^4 + 2*a^2*b^2 + b^4)*c^6*d^4 + 10*(a^4 +
2*a^2*b^2 + b^4)*c^4*d^6 + 5*(a^4 + 2*a^2*b^2 + b^4)*c^2*d^8 + (a^4 + 2*a^2*b^2 + b^4)*d^10 + (10*a*b*c^14*d
+ 30*a*b*c^12*d^3 + 2*a*b*c^10*d^5 - 90*a*b*c^8*d^7 - 130*a*b*c^6*d^9 - 70*a*b*c^4*d^11 - 10*a*b*c^2*d^13 + 2*
a*b*d^15 + (a^2 - b^2)*c^15 - 5*(a^2 - b^2)*c^13*d^2 - 35*(a^2 - b^2)*c^11*d^4 - 65*(a^2 - b^2)*c^9*d^6 - 45*(
a^2 - b^2)*c^7*d^8 + (a^2 - b^2)*c^5*d^10 + 15*(a^2 - b^2)*c^3*d^12 + 5*(a^2 - b^2)*c*d^14)*f^2*sqrt((a^4 + 2*
a^2*b^2 + b^4)/((c^10 + 5*c^8*d^2 + 10*c^6*d^4 + 10*c^4*d^6 + 5*c^2*d^8 + d^10)*f^4)))/(4*a^2*b^2*c^10 - 20*(a
^3*b - a*b^3)*c^9*d + 5*(5*a^4 - 26*a^2*b^2 + 5*b^4)*c^8*d^2 + 240*(a^3*b - a*b^3)*c^7*d^3 - 20*(5*a^4 - 32*a^
2*b^2 + 5*b^4)*c^6*d^4 - 504*(a^3*b - a*b^3)*c^5*d^5 + 10*(11*a^4 - 62*a^2*b^2 + 11*b^4)*c^4*d^6 + 240*(a^3*b
- a*b^3)*c^3*d^7 - 20*(a^4 - 7*a^2*b^2 + b^4)*c^2*d^8 - 20*(a^3*b - a*b^3)*c*d^9 + (a^4 - 2*a^2*b^2 + b^4)*d^1
0))*sqrt((4*a^2*b^2*c^10 - 20*(a^3*b - a*b^3)*c^9*d + 5*(5*a^4 - 26*a^2*b^2 + 5*b^4)*c^8*d^2 + 240*(a^3*b - a*
b^3)*c^7*d^3 - 20*(5*a^4 - 32*a^2*b^2 + 5*b^4)*c^6*d^4 - 504*(a^3*b - a*b^3)*c^5*d^5 + 10*(11*a^4 - 62*a^2*b^2
+ 11*b^4)*c^4*d^6 + 240*(a^3*b - a*b^3)*c^3*d^7 - 20*(a^4 - 7*a^2*b^2 + b^4)*c^2*d^8 - 20*(a^3*b - a*b^3)*c*d
^9 + (a^4 - 2*a^2*b^2 + b^4)*d^10)/((c^20 + 10*c^18*d^2 + 45*c^16*d^4 + 120*c^14*d^6 + 210*c^12*d^8 + 252*c^10
*d^10 + 210*c^8*d^12 + 120*c^6*d^14 + 45*c^4*d^16 + 10*c^2*d^18 + d^20)*f^4))*((a^4 + 2*a^2*b^2 + b^4)/((c^10
+ 5*c^8*d^2 + 10*c^6*d^4 + 10*c^4*d^6 + 5*c^2*d^8 + d^10)*f^4))^(3/4)*arctan(-((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b
^5 + a*b^7)*c^21 - 5*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^20*d - 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^
19*d^2 - 30*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^18*d^3 - 94*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^17*d^4
- 61*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^16*d^5 - 368*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^15*d^6 - 8*
(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^14*d^7 - 700*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^13*d^8 + 182*(a^8
+ 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^12*d^9 - 728*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^11*d^10 + 364*(a^8 +
2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^10*d^11 - 364*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^9*d^12 + 350*(a^8 + 2*a
^6*b^2 - 2*a^2*b^6 - b^8)*c^8*d^13 + 16*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^7*d^14 + 184*(a^8 + 2*a^6*b^
2 - 2*a^2*b^6 - b^8)*c^6*d^15 + 122*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^5*d^16 + 47*(a^8 + 2*a^6*b^2 - 2
*a^2*b^6 - b^8)*c^4*d^17 + 60*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3*d^18 + 2*(a^8 + 2*a^6*b^2 - 2*a^2*b^
6 - b^8)*c^2*d^19 + 10*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^20 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^
21)*f^4*sqrt((4*a^2*b^2*c^10 - 20*(a^3*b - a*b^3)*c^9*d + 5*(5*a^4 - 26*a^2*b^2 + 5*b^4)*c^8*d^2 + 240*(a^3*b
- a*b^3)*c^7*d^3 - 20*(5*a^4 - 32*a^2*b^2 + 5*b^4)*c^6*d^4 - 504*(a^3*b - a*b^3)*c^5*d^5 + 10*(11*a^4 - 62*a^2
*b^2 + 11*b^4)*c^4*d^6 + 240*(a^3*b - a*b^3)*c^3*d^7 - 20*(a^4 - 7*a^2*b^2 + b^4)*c^2*d^8 - 20*(a^3*b - a*b^3)
*c*d^9 + (a^4 - 2*a^2*b^2 + b^4)*d^10)/((c^20 + 10*c^18*d^2 + 45*c^16*d^4 + 120*c^14*d^6 + 210*c^12*d^8 + 252*
c^10*d^10 + 210*c^8*d^12 + 120*c^6*d^14 + 45*c^4*d^16 + 10*c^2*d^18 + d^20)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)
/((c^10 + 5*c^8*d^2 + 10*c^6*d^4 + 10*c^4*d^6 + 5*c^2*d^8 + d^10)*f^4)) + (2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 +
4*a^3*b^7 + a*b^9)*c^16 - 5*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^15*d - 10*(a^9*b +
4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^14*d^2 - 15*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^
8 - b^10)*c^13*d^3 - 70*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^12*d^4 - (a^10 + 3*a^8*b^2 + 2*a
^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^11*d^5 - 130*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^10
*d^6 + 45*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^9*d^7 - 90*(a^9*b + 4*a^7*b^3 + 6*a^
5*b^5 + 4*a^3*b^7 + a*b^9)*c^8*d^8 + 65*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - b^10)*c^7*d^9
+ 2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^6*d^10 + 35*(a^10 + 3*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^
6 - 3*a^2*b^8 - b^10)*c^5*d^11 + 30*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^4*d^12 + 5*(a^10 + 3
*a^8*b^2 + 2*a^6*b^4 - 2*a^4*b^6 - 3*a^2*b^8 - ...

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \tan {\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))**(5/2),x)

[Out]

Integral((a + b*tan(e + f*x))/(c + d*tan(e + f*x))**(5/2), x)

________________________________________________________________________________________

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((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(5/2),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.The choi
ce was done

________________________________________________________________________________________

Mupad [B]
time = 22.93, size = 2500, normalized size = 13.44 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))/(c + d*tan(e + f*x))^(5/2),x)

[Out]

(log(8*b^3*d^16*f^2 - ((((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4
- 400*b^4*c^8*d^2*f^4)^(1/2) + 4*b^2*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5
*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*
f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) + 4*b^2*c^5*f^2 + 20*b^2*c*d^4*
f^2 - 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f
^4))^(1/2)*(((((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*
c^8*d^2*f^4)^(1/2) + 4*b^2*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f
^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c
^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^
10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5))/4 + 736*b*c^3*d^18*f^4 +
2432*b*c^5*d^16*f^4 + 4480*b*c^7*d^14*f^4 + 4928*b*c^9*d^12*f^4 + 3136*b*c^11*d^10*f^4 + 896*b*c^13*d^8*f^4 -
128*b*c^15*d^6*f^4 - 160*b*c^17*d^4*f^4 - 32*b*c^19*d^2*f^4 + 96*b*c*d^20*f^4))/4 + (c + d*tan(e + f*x))^(1/2
)*(320*b^2*c^4*d^14*f^3 - 16*b^2*d^18*f^3 + 1024*b^2*c^6*d^12*f^3 + 1440*b^2*c^8*d^10*f^3 + 1024*b^2*c^10*d^8*
f^3 + 320*b^2*c^12*d^6*f^3 - 16*b^2*c^16*d^2*f^3)))/4 + 40*b^3*c^2*d^14*f^2 + 72*b^3*c^4*d^12*f^2 + 40*b^3*c^6
*d^10*f^2 - 40*b^3*c^8*d^8*f^2 - 72*b^3*c^10*d^6*f^2 - 40*b^3*c^12*d^4*f^2 - 8*b^3*c^14*d^2*f^2)*(((320*b^4*c^
2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) + 4*b^2
*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c
^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 + (log(8*b^3*d^16*f^2 - ((-((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 176
0*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) - 4*b^2*c^5*f^2 - 20*b^2*c*d^4*f^2 + 40*
b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2
)*(((-((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*
f^4)^(1/2) - 4*b^2*c^5*f^2 - 20*b^2*c*d^4*f^2 + 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*
c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(((-((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^
4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) - 4*b^2*c^5*f^2 - 20*b^2*c*d^4*f^2 + 40*b^2*c^3*
d^2*f^2)/(c^10*f^4 + d^10*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2)*(c + d
*tan(e + f*x))^(1/2)*(64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^1
4*f^5 + 16128*c^11*d^12*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 +
64*c^21*d^2*f^5))/4 + 736*b*c^3*d^18*f^4 + 2432*b*c^5*d^16*f^4 + 4480*b*c^7*d^14*f^4 + 4928*b*c^9*d^12*f^4 +
3136*b*c^11*d^10*f^4 + 896*b*c^13*d^8*f^4 - 128*b*c^15*d^6*f^4 - 160*b*c^17*d^4*f^4 - 32*b*c^19*d^2*f^4 + 96*b
*c*d^20*f^4))/4 + (c + d*tan(e + f*x))^(1/2)*(320*b^2*c^4*d^14*f^3 - 16*b^2*d^18*f^3 + 1024*b^2*c^6*d^12*f^3 +
1440*b^2*c^8*d^10*f^3 + 1024*b^2*c^10*d^8*f^3 + 320*b^2*c^12*d^6*f^3 - 16*b^2*c^16*d^2*f^3)))/4 + 40*b^3*c^2*
d^14*f^2 + 72*b^3*c^4*d^12*f^2 + 40*b^3*c^6*d^10*f^2 - 40*b^3*c^8*d^8*f^2 - 72*b^3*c^10*d^6*f^2 - 40*b^3*c^12*
d^4*f^2 - 8*b^3*c^14*d^2*f^2)*(-((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*
d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) - 4*b^2*c^5*f^2 - 20*b^2*c*d^4*f^2 + 40*b^2*c^3*d^2*f^2)/(c^10*f^4 + d^10
*f^4 + 5*c^2*d^8*f^4 + 10*c^4*d^6*f^4 + 10*c^6*d^4*f^4 + 5*c^8*d^2*f^4))^(1/2))/4 - log(8*b^3*d^16*f^2 - (((32
0*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2)
+ 4*b^2*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^2*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^
4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*((((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*
d^6*f^4 + 1600*b^4*c^6*d^4*f^4 - 400*b^4*c^8*d^2*f^4)^(1/2) + 4*b^2*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^
2*f^2)/(16*c^10*f^4 + 16*d^10*f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2
)*(736*b*c^3*d^18*f^4 - (((320*b^4*c^2*d^8*f^4 - 16*b^4*d^10*f^4 - 1760*b^4*c^4*d^6*f^4 + 1600*b^4*c^6*d^4*f^4
- 400*b^4*c^8*d^2*f^4)^(1/2) + 4*b^2*c^5*f^2 + 20*b^2*c*d^4*f^2 - 40*b^2*c^3*d^2*f^2)/(16*c^10*f^4 + 16*d^10*
f^4 + 80*c^2*d^8*f^4 + 160*c^4*d^6*f^4 + 160*c^6*d^4*f^4 + 80*c^8*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(
64*c*d^22*f^5 + 640*c^3*d^20*f^5 + 2880*c^5*d^18*f^5 + 7680*c^7*d^16*f^5 + 13440*c^9*d^14*f^5 + 16128*c^11*d^1
2*f^5 + 13440*c^13*d^10*f^5 + 7680*c^15*d^8*f^5 + 2880*c^17*d^6*f^5 + 640*c^19*d^4*f^5 + 64*c^21*d^2*f^5) + 24
32*b*c^5*d^16*f^4 + 4480*b*c^7*d^14*f^4 + 4928*...

________________________________________________________________________________________