3.13.45 \(\int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^2} \, dx\) [1245]

Optimal. Leaf size=243 \[ -\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {(b c-a d)^{3/2} \left (4 a b c+a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

-I*(c-I*d)^(5/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/f+I*(c+I*d)^(5/2)*arctanh((c+d*tan(f*
x+e))^(1/2)/(c+I*d)^(1/2))/(a+I*b)^2/f-(-a*d+b*c)^(3/2)*(a^2*d+4*a*b*c+5*b^2*d)*arctanh(b^(1/2)*(c+d*tan(f*x+e
))^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(a^2+b^2)^2/f-(-a*d+b*c)^2*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f*
x+e))

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Rubi [A]
time = 0.77, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {3646, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {(b c-a d)^{3/2} \left (a^2 d+4 a b c+5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} f \left (a^2+b^2\right )^2}-\frac {i (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(c - I*d)^(5/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(5/2)*Ar
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - ((b*c - a*d)^(3/2)*(4*a*b*c + a^2*d + 5*b^2*d
)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)^2*f) - ((b*c - a*d)^2*Sqrt
[c + d*Tan[e + f*x]])/(b*(a^2 + b^2)*f*(a + b*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 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]

Rule 3646

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

Rule 3715

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

Rule 3734

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

Rubi steps

\begin {align*} \int \frac {(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^2} \, dx &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\int \frac {\frac {1}{2} \left (5 b^2 c^2 d+a^2 d^3+2 a b c \left (c^2-2 d^2\right )\right )+b \left (a d \left (3 c^2-d^2\right )-b \left (c^3-3 c d^2\right )\right ) \tan (e+f x)+\frac {1}{2} d \left (\left (a^2+2 b^2\right ) d^2-b c (b c-2 a d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\int \frac {-b \left (b^2 c \left (c^2-3 d^2\right )-a^2 \left (c^3-3 c d^2\right )-a b \left (6 c^2 d-2 d^3\right )\right )-b \left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac {\left ((b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 b \left (a^2+b^2\right )^2}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \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}{2 (a-i b)^2}+\frac {(c+i d)^3 \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac {\left ((b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b \left (a^2+b^2\right )^2 f}\\ &=-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(i c-d)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (a+i b)^2 f}-\frac {(i c+d)^3 \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (a-i b)^2 f}+\frac {\left ((b c-a d)^2 \left (4 a b c+a^2 d+5 b^2 d\right )\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{b \left (a^2+b^2\right )^2 d f}\\ &=-\frac {(b c-a d)^{3/2} \left (4 a b c+a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(c-i d)^3 \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 )}{(a-i b)^2 d f}-\frac {(c+i d)^3 \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 )}{(a+i b)^2 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 )}{(a-i b)^2 f}+\frac {i (c+i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {(b c-a d)^{3/2} \left (4 a b c+a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 f}-\frac {(b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 5.12, size = 333, normalized size = 1.37 \begin {gather*} \frac {-\frac {i (a+i b)^2 b^{3/2} (c-i d)^{5/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+i b^{3/2} (i a+b)^2 (c+i d)^{5/2} (b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )+(b c-a d)^{5/2} \left (4 a b c+a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )}+\frac {d (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}{b}+d (b c-a d) (c+d \tan (e+f x))^{3/2}+b d (c+d \tan (e+f x))^{5/2}-\frac {b^2 (c+d \tan (e+f x))^{7/2}}{a+b \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((I*(a + I*b)^2*b^(3/2)*(c - I*d)^(5/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] + I*b^(3
/2)*(I*a + b)^2*(c + I*d)^(5/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (b*c - a*d)^(5/2
)*(4*a*b*c + a^2*d + 5*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2
))) + (d*(b*c - a*d)^2*Sqrt[c + d*Tan[e + f*x]])/b + d*(b*c - a*d)*(c + d*Tan[e + f*x])^(3/2) + b*d*(c + d*Tan
[e + f*x])^(5/2) - (b^2*(c + d*Tan[e + f*x])^(7/2))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1862\) vs. \(2(211)=422\).
time = 0.54, size = 1863, normalized size = 7.67

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1863\)
default	\(\text {Expression too large to display}\)	\(1863\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*d^3*(1/d^3/(a^2+b^2)^2*(1/4/d*(-1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+(c^2+d^2)^(1/2
)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2-4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d+(c^2+d^2)^(1/2
)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2+(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*a^2*c^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d-2*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*
c*d^2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))+2*(4*(c^2+d^2)^
(1/2)*a^2*c*d^2-4*(c^2+d^2)^(1/2)*a*b*c^2*d+4*(c^2+d^2)^(1/2)*a*b*d^3-4*(c^2+d^2)^(1/2)*b^2*c*d^2+1/2*(-(c^2+d
^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2-4*(c^2+d
^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2-(c^2+d^2
)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3-3*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*a^2*c*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^3-(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b^2*c^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^2)*(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^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1
/2)))+1/4/d*(1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*a^2*d^2-4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*
c)^(1/2)*b^2*c^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3-3
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^2+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d-2*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)*a*b*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^2)*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*(-4*(c^2+d^2)^(1/2)*a^2*c*d^2+4*(c^2
+d^2)^(1/2)*a*b*c^2*d-4*(c^2+d^2)^(1/2)*a*b*d^3+4*(c^2+d^2)^(1/2)*b^2*c*d^2-1/2*(-(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a^2*c^2+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d^2-4*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a*b*c*d+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*b^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^3-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c*d^2+6*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c^2*d-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d^3-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*
c^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c*d^2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*ar
ctan((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))))+(a*d-b*c)^2/d^3
/(a^2+b^2)^2*(-1/2*d*(a^2+b^2)/b*(c+d*tan(f*x+e))^(1/2)/((c+d*tan(f*x+e))*b+a*d-b*c)+1/2*(a^2*d+4*a*b*c+5*b^2*
d)/b/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2))))

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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((c+d*tan(f*x+e))^(5/2)/(a+b*tan(f*x+e))^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(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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+b*tan(f*x+e))^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

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Mupad [B]
time = 16.40, size = 2500, normalized size = 10.29 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(((((8*(4*b^12*c*d^17*f^2 - 16*a^11*b*d^18*f^2 - 8*a^12*c*d^17*f^2 - 4*a*b^11*d^18*f^2 + 304*a^3*b^9*d^1
8*f^2 + 120*a^5*b^7*d^18*f^2 - 320*a^7*b^5*d^18*f^2 - 148*a^9*b^3*d^18*f^2 - 8*a^12*c^3*d^15*f^2 - 400*b^12*c^
3*d^15*f^2 + 176*b^12*c^5*d^13*f^2 + 488*b^12*c^7*d^11*f^2 - 92*b^12*c^9*d^9*f^2 + 6576*a^2*b^10*c^3*d^15*f^2
+ 400*a^2*b^10*c^5*d^13*f^2 - 6144*a^2*b^10*c^7*d^11*f^2 + 1056*a^2*b^10*c^9*d^9*f^2 - 5408*a^3*b^9*c^2*d^16*f
^2 + 3888*a^3*b^9*c^4*d^14*f^2 + 6400*a^3*b^9*c^6*d^12*f^2 - 3072*a^3*b^9*c^8*d^10*f^2 + 128*a^3*b^9*c^10*d^8*
f^2 - 648*a^4*b^8*c^3*d^15*f^2 - 1952*a^4*b^8*c^5*d^13*f^2 + 208*a^4*b^8*c^7*d^11*f^2 + 200*a^4*b^8*c^9*d^9*f^
2 - 4288*a^5*b^7*c^2*d^16*f^2 + 4112*a^5*b^7*c^4*d^14*f^2 + 5120*a^5*b^7*c^6*d^12*f^2 - 3208*a^5*b^7*c^8*d^10*
f^2 + 192*a^5*b^7*c^10*d^8*f^2 - 6688*a^6*b^6*c^3*d^15*f^2 - 2464*a^6*b^6*c^5*d^13*f^2 + 5952*a^6*b^6*c^7*d^11
*f^2 - 960*a^6*b^6*c^9*d^9*f^2 + 2624*a^7*b^5*c^2*d^16*f^2 - 2016*a^7*b^5*c^4*d^14*f^2 - 3456*a^7*b^5*c^6*d^12
*f^2 + 1504*a^7*b^5*c^8*d^10*f^2 + 992*a^8*b^4*c^3*d^15*f^2 - 144*a^8*b^4*c^5*d^13*f^2 - 888*a^8*b^4*c^7*d^11*
f^2 - 12*a^8*b^4*c^9*d^9*f^2 + 352*a^9*b^3*c^2*d^16*f^2 + 520*a^9*b^3*c^4*d^14*f^2 + 32*a^9*b^3*c^6*d^12*f^2 +
12*a^9*b^3*c^8*d^10*f^2 + 48*a^10*b^2*c^3*d^15*f^2 + 144*a^10*b^2*c^5*d^13*f^2 + 1120*a*b^11*c^2*d^16*f^2 - 2
776*a*b^11*c^4*d^14*f^2 - 2208*a*b^11*c^6*d^12*f^2 + 1628*a*b^11*c^8*d^10*f^2 - 64*a*b^11*c^10*d^8*f^2 - 1024*
a^2*b^10*c*d^17*f^2 + 1312*a^4*b^8*c*d^17*f^2 + 2688*a^6*b^6*c*d^17*f^2 + 260*a^8*b^4*c*d^17*f^2 - 96*a^10*b^2
*c*d^17*f^2 - 32*a^11*b*c^2*d^16*f^2 - 16*a^11*b*c^4*d^14*f^2))/(b^9*f^5 + a^8*b*f^5 + 4*a^2*b^7*f^5 + 6*a^4*b
^5*f^5 + 4*a^6*b^3*f^5) - (((8*(96*a^2*b^14*d^13*f^4 + 480*a^4*b^12*d^13*f^4 + 960*a^6*b^10*d^13*f^4 + 960*a^8
*b^8*d^13*f^4 + 480*a^10*b^6*d^13*f^4 + 96*a^12*b^4*d^13*f^4 + 128*b^16*c^2*d^11*f^4 + 128*b^16*c^4*d^9*f^4 +
640*a^2*b^14*c^2*d^11*f^4 + 544*a^2*b^14*c^4*d^9*f^4 - 768*a^3*b^13*c^3*d^10*f^4 + 320*a^3*b^13*c^5*d^8*f^4 +
1280*a^4*b^12*c^2*d^11*f^4 + 800*a^4*b^12*c^4*d^9*f^4 - 1440*a^5*b^11*c^3*d^10*f^4 + 640*a^5*b^11*c^5*d^8*f^4
+ 1280*a^6*b^10*c^2*d^11*f^4 + 320*a^6*b^10*c^4*d^9*f^4 - 1280*a^7*b^9*c^3*d^10*f^4 + 640*a^7*b^9*c^5*d^8*f^4
+ 640*a^8*b^8*c^2*d^11*f^4 - 320*a^8*b^8*c^4*d^9*f^4 - 480*a^9*b^7*c^3*d^10*f^4 + 320*a^9*b^7*c^5*d^8*f^4 + 12
8*a^10*b^6*c^2*d^11*f^4 - 352*a^10*b^6*c^4*d^9*f^4 + 64*a^11*b^5*c^5*d^8*f^4 - 96*a^12*b^4*c^4*d^9*f^4 + 32*a^
13*b^3*c^3*d^10*f^4 - 224*a*b^15*c*d^12*f^4 - 160*a*b^15*c^3*d^10*f^4 + 64*a*b^15*c^5*d^8*f^4 - 1088*a^3*b^13*
c*d^12*f^4 - 2080*a^5*b^11*c*d^12*f^4 - 1920*a^7*b^9*c*d^12*f^4 - 800*a^9*b^7*c*d^12*f^4 - 64*a^11*b^5*c*d^12*
f^4 + 32*a^13*b^3*c*d^12*f^4))/(b^9*f^5 + a^8*b*f^5 + 4*a^2*b^7*f^5 + 6*a^4*b^5*f^5 + 4*a^6*b^3*f^5) - (16*(c
+ d*tan(e + f*x))^(1/2)*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 - 32*a*b^3*d^5*f^2 + 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4
*f^2 + 40*b^4*c*d^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f
^2 - 160*a*b^3*c^4*d*f^2 + 160*a^3*b*c^4*d*f^2 + 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 - 320*a^3*b*c^2
*d^3*f^2)^2/4 - (16*a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4*b^4*f^4 + 64*a^6*b^2*f^4)*(c^10 + d^10 + 5*
c^2*d^8 + 10*c^4*d^6 + 10*c^6*d^4 + 5*c^8*d^2))^(1/2) - 4*a^4*c^5*f^2 - 4*b^4*c^5*f^2 + 16*a*b^3*d^5*f^2 - 16*
a^3*b*d^5*f^2 - 20*a^4*c*d^4*f^2 - 20*b^4*c*d^4*f^2 + 24*a^2*b^2*c^5*f^2 + 40*a^4*c^3*d^2*f^2 + 40*b^4*c^3*d^2
*f^2 - 240*a^2*b^2*c^3*d^2*f^2 + 80*a*b^3*c^4*d*f^2 - 80*a^3*b*c^4*d*f^2 - 160*a*b^3*c^2*d^3*f^2 + 120*a^2*b^2
*c*d^4*f^2 + 160*a^3*b*c^2*d^3*f^2)/(16*(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))^
(1/2)*(32*b^18*d^10*f^4 + 160*a^2*b^16*d^10*f^4 + 288*a^4*b^14*d^10*f^4 + 160*a^6*b^12*d^10*f^4 - 160*a^8*b^10
*d^10*f^4 - 288*a^10*b^8*d^10*f^4 - 160*a^12*b^6*d^10*f^4 - 32*a^14*b^4*d^10*f^4 + 48*b^18*c^2*d^8*f^4 + 272*a
^2*b^16*c^2*d^8*f^4 + 624*a^4*b^14*c^2*d^8*f^4 + 720*a^6*b^12*c^2*d^8*f^4 + 400*a^8*b^10*c^2*d^8*f^4 + 48*a^10
*b^8*c^2*d^8*f^4 - 48*a^12*b^6*c^2*d^8*f^4 - 16*a^14*b^4*c^2*d^8*f^4 + 16*a*b^17*c*d^9*f^4 + 112*a^3*b^15*c*d^
9*f^4 + 336*a^5*b^13*c*d^9*f^4 + 560*a^7*b^11*c*d^9*f^4 + 560*a^9*b^9*c*d^9*f^4 + 336*a^11*b^7*c*d^9*f^4 + 112
*a^13*b^5*c*d^9*f^4 + 16*a^15*b^3*c*d^9*f^4))/(b^9*f^4 + a^8*b*f^4 + 4*a^2*b^7*f^4 + 6*a^4*b^5*f^4 + 4*a^6*b^3
*f^4))*((((8*a^4*c^5*f^2 + 8*b^4*c^5*f^2 - 32*a*b^3*d^5*f^2 + 32*a^3*b*d^5*f^2 + 40*a^4*c*d^4*f^2 + 40*b^4*c*d
^4*f^2 - 48*a^2*b^2*c^5*f^2 - 80*a^4*c^3*d^2*f^2 - 80*b^4*c^3*d^2*f^2 + 480*a^2*b^2*c^3*d^2*f^2 - 160*a*b^3*c^
4*d*f^2 + 160*a^3*b*c^4*d*f^2 + 320*a*b^3*c^2*d^3*f^2 - 240*a^2*b^2*c*d^4*f^2 - 320*a^3*b*c^2*d^3*f^2)^2/4 - (
16*a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4*b^4*f^4 + 64*a^6*b^2*f^4)*(c^10 + d^10 + 5*c^2*d^8 + 10*c^4*
d^6 + 10*c^6*d^4 + 5*c^8*d^2))^(1/2) - 4*a^4*c^5*f^2 - 4*b^4*c^5*f^2 + 16*a*b^3*d^5*f^2 - 16*a^3*b*d^5*f^2 - 2
0*a^4*c*d^4*f^2 - 20*b^4*c*d^4*f^2 + 24*a^2*b^2*c^5*f^2 + 40*a^4*c^3*d^2*f^2 + 40*b^4*c^3*d^2*f^2 - 240*a^2*b^
2*c^3*d^2*f^2 + 80*a*b^3*c^4*d*f^2 - 80*a^3*b*c...

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