3.13.54 \(\int \frac {(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^{3/2}} \, dx\) [1254]

Optimal. Leaf size=216 \[ \frac {(i a+b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}-\frac {(i a-b)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}}-\frac {2 b \left (a d (2 b c-a d)-b^2 \left (2 c^2+d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^2 \left (c^2+d^2\right ) f} \]

[Out]

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

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Rubi [A]
time = 0.35, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3646, 3711, 3620, 3618, 65, 214} \begin {gather*} -\frac {2 b \left (a d (2 b c-a d)-b^2 \left (2 c^2+d^2\right )\right ) \sqrt {c+d \tan (e+f x)}}{d^2 f \left (c^2+d^2\right )}-\frac {2 (b c-a d)^2 (a+b \tan (e+f x))}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}-\frac {(-b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}+\frac {(b+i a)^3 \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

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

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 3711

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

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.84, size = 287, normalized size = 1.33 \begin {gather*} \frac {-i b \left (3 a^2-b^2\right ) \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )+\frac {4 b^2 (b c-2 a d)}{d \sqrt {c+d \tan (e+f x)}}+\frac {\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \left ((-i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c-i d}\right )+(i c+d) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{\left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {2 b^2 (a+b \tan (e+f x))}{\sqrt {c+d \tan (e+f x)}}}{d f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])^3/(c + d*Tan[e + f*x])^(3/2),x]

[Out]

((-I)*b*(3*a^2 - b^2)*(ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/Sqrt[c - I*d] - ArcTanh[Sqrt[c + d*Tan[
e + f*x]]/Sqrt[c + I*d]]/Sqrt[c + I*d]) + (4*b^2*(b*c - 2*a*d))/(d*Sqrt[c + d*Tan[e + f*x]]) + ((3*a^2*b*c - b
^3*c - a^3*d + 3*a*b^2*d)*(((-I)*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (I*c
+ d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)]))/((c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]])
+ (2*b^2*(a + b*Tan[e + f*x]))/Sqrt[c + d*Tan[e + f*x]])/(d*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4571\) vs. \(2(194)=388\).
time = 0.53, size = 4572, normalized size = 21.17

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(4572\)
default	\(\text {Expression too large to display}\)	\(4572\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/f/d^2*(b^3*(c+d*tan(f*x+e))^(1/2)+d^2/(c^2+d^2)*(1/4/d^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*(-9*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*a*b^2*c^2*d^5+3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^4-2*(c^2+d^2)^(1/2)*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d^4-18*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a^2*b*c^5*d^2-12*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3*d^4+6*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a^2*b*c*d^6+9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^6*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^4*d^3+(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^6*d+(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*a^3*c^4*d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d^5+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^7+
6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*b^3*c*d^6-3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*a^3*c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a^3*c*d^5-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^6-(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*a^3*d^7+9*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^4-9*(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*a*b^2*c^5*d-6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3*d^3-(c^2+d^2)^(3/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4+6*(c^2+d^2)^(1/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^4*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^5)*ln(d*ta
n(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*(12*a^3*c^5*d^3+8*a^3*c^3*d
^5-4*a^3*c*d^7-18*a^2*b*c^6*d^2+6*a^2*b*c^4*d^4+18*a^2*b*c^2*d^6-6*a^2*b*d^8-36*a*b^2*c^5*d^3-24*a*b^2*c^3*d^5
+12*a*b^2*c*d^7+6*b^3*c^6*d^2-2*b^3*c^4*d^4-6*b^3*c^2*d^6+2*b^3*d^8-1/2*(-9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^
2*c^2*d^5+3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^4-2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*b^3*c^4*d^2-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d^4-18*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a^2*b*c^5*d^2-12*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3*d^4+6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d^6+9*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^6*d-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^4*d^3+(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b^3*c^6-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^6*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^4*
d^3+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d^5+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^7+6*(2*(c^2+d^2)^(1/2)
+2*c)^(1/2)*b^3*c^5*d^2+4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3*d^4-2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^6-
3*(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*
c^5*d+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a^3*c*d^5-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^6-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^7+9*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^4-9*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
b^2*c^5*d-6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^3*d^3-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b^3*c^4+(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*d^4+6*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*a^2*b*c^4*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c*d^5)*(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^2/(3*c^2-d^2)/(c^2+d^2)^(3/2)*(1/2*(9*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^2*d^5-3*
(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^4+2*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^
4*d^2+3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^2*d^4+18*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^5*d
^2+12*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^3*d^4-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c*d^6-9*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*a*b^2*c^6*d+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^4*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2
*c)^(1/2)*b^3*c^6+3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^6*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^4*d^3-3*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^2*d^5-3*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*d^7-6*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
b^3*c^5*d^2-4*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c^3*d^4+2*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^3*c*d^6+3*(c^2+d^2)^
(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*d^4-3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^5*d-2*(c^2
+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c^3*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*c*d^5+
3*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^6+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^3*d^7-9*(c^2+d^2)^(1
/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*b*c^2*d^4+9*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b^2*c^5*d+6*
(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a...

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Maxima [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((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

Timed out

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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((a+b*tan(f*x+e))^3/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))**3/(c+d*tan(f*x+e))**(3/2),x)

[Out]

Integral((a + b*tan(e + f*x))**3/(c + d*tan(e + f*x))**(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((a+b*tan(f*x+e))^3/(c+d*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.The choi
ce was done

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Mupad [B]
time = 13.18, size = 2500, normalized size = 11.57 \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))^3/(c + d*tan(e + f*x))^(3/2),x)

[Out]

(2*b^3*(c + d*tan(e + f*x))^(1/2))/(d^2*f) - atan((((-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48
*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3
*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4
*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 -
24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3
*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b
^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-((
(8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 1
20*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 3
60*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*
d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2
) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2
+ 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180
*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3
*c^4*d^2*f^4)))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5
+ 64*c^11*d^2*f^5) - 32*b^3*d^12*f^4 + 96*a^2*b*d^12*f^4 + 64*a^3*c*d^11*f^4 + 256*a^3*c^3*d^9*f^4 + 384*a^3*c
^5*d^7*f^4 + 256*a^3*c^7*d^5*f^4 + 64*a^3*c^9*d^3*f^4 - 96*b^3*c^2*d^10*f^4 - 64*b^3*c^4*d^8*f^4 + 64*b^3*c^6*
d^6*f^4 + 96*b^3*c^8*d^4*f^4 + 32*b^3*c^10*d^2*f^4 - 192*a*b^2*c*d^11*f^4 - 768*a*b^2*c^3*d^9*f^4 - 1152*a*b^2
*c^5*d^7*f^4 - 768*a*b^2*c^7*d^5*f^4 - 192*a*b^2*c^9*d^3*f^4 + 288*a^2*b*c^2*d^10*f^4 + 192*a^2*b*c^4*d^8*f^4
- 192*a^2*b*c^6*d^6*f^4 - 288*a^2*b*c^8*d^4*f^4 - 96*a^2*b*c^10*d^2*f^4) + (c + d*tan(e + f*x))^(1/2)*(16*b^6*
d^10*f^3 - 16*a^6*d^10*f^3 - 240*a^2*b^4*d^10*f^3 + 240*a^4*b^2*d^10*f^3 - 32*a^6*c^2*d^8*f^3 + 32*a^6*c^6*d^4
*f^3 + 16*a^6*c^8*d^2*f^3 + 32*b^6*c^2*d^8*f^3 - 32*b^6*c^6*d^4*f^3 - 16*b^6*c^8*d^2*f^3 - 480*a^2*b^4*c^2*d^8
*f^3 + 480*a^2*b^4*c^6*d^4*f^3 + 240*a^2*b^4*c^8*d^2*f^3 - 1920*a^3*b^3*c^3*d^7*f^3 - 1920*a^3*b^3*c^5*d^5*f^3
- 640*a^3*b^3*c^7*d^3*f^3 + 480*a^4*b^2*c^2*d^8*f^3 - 480*a^4*b^2*c^6*d^4*f^3 - 240*a^4*b^2*c^8*d^2*f^3 + 192
*a*b^5*c*d^9*f^3 + 192*a^5*b*c*d^9*f^3 + 576*a*b^5*c^3*d^7*f^3 + 576*a*b^5*c^5*d^5*f^3 + 192*a*b^5*c^7*d^3*f^3
- 640*a^3*b^3*c*d^9*f^3 + 576*a^5*b*c^3*d^7*f^3 + 576*a^5*b*c^5*d^5*f^3 + 192*a^5*b*c^7*d^3*f^3))*(-(((8*a^6*
c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*
b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*
b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4
+ 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))^(1/2) + 4*a
^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^
2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 + 72*a^5*b*c^2*d*f^2 - 180*a^2*b^
4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^
2*f^4)))^(1/2)*1i + ((-(((8*a^6*c^3*f^2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f
^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2
+ 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f
^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^
8*b^4 + 6*a^10*b^2))^(1/2) + 4*a^6*c^3*f^2 - 4*b^6*c^3*f^2 - 24*a*b^5*d^3*f^2 - 24*a^5*b*d^3*f^2 - 12*a^6*c*d^
2*f^2 + 12*b^6*c*d^2*f^2 + 60*a^2*b^4*c^3*f^2 - 60*a^4*b^2*c^3*f^2 + 80*a^3*b^3*d^3*f^2 + 72*a*b^5*c^2*d*f^2 +
72*a^5*b*c^2*d*f^2 - 180*a^2*b^4*c*d^2*f^2 - 240*a^3*b^3*c^2*d*f^2 + 180*a^4*b^2*c*d^2*f^2)/(16*(c^6*f^4 + d^
6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4)))^(1/2)*(32*b^3*d^12*f^4 + (c + d*tan(e + f*x))^(1/2)*(-(((8*a^6*c^3*f^
2 - 8*b^6*c^3*f^2 - 48*a*b^5*d^3*f^2 - 48*a^5*b*d^3*f^2 - 24*a^6*c*d^2*f^2 + 24*b^6*c*d^2*f^2 + 120*a^2*b^4*c^
3*f^2 - 120*a^4*b^2*c^3*f^2 + 160*a^3*b^3*d^3*f^2 + 144*a*b^5*c^2*d*f^2 + 144*a^5*b*c^2*d*f^2 - 360*a^2*b^4*c*
d^2*f^2 - 480*a^3*b^3*c^2*d*f^2 + 360*a^4*b^2*c*d^2*f^2)^2/4 - (16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*
c^4*d^2*f^4)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4...

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