3.2.1 \(\int \frac {(c+d \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{a+b \tan (e+f x)} \, dx\) [101]

Optimal. Leaf size=271 \[ -\frac {(i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b) f}-\frac {(A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{5/2} \left (a^2+b^2\right ) f}+\frac {2 (b c C+b B d-a C d) \sqrt {c+d \tan (e+f x)}}{b^2 f}+\frac {2 C (c+d \tan (e+f x))^{3/2}}{3 b f} \]

[Out]

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

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Rubi [A]
time = 1.21, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3728, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 (b c-a d)^{3/2} \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{5/2} f \left (a^2+b^2\right )}-\frac {(c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)}-\frac {(c+i d)^{3/2} (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)}+\frac {2 (-a C d+b B d+b c C) \sqrt {c+d \tan (e+f x)}}{b^2 f}+\frac {2 C (c+d \tan (e+f x))^{3/2}}{3 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)*f)) - ((A + I*B
- C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((I*a - b)*f) - (2*(A*b^2 - a*(b*B - a*
C))*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(5/2)*(a^2 + b^2)*f) + (
2*(b*c*C + b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/(b^2*f) + (2*C*(c + d*Tan[e + f*x])^(3/2))/(3*b*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 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 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, 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[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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

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Mathematica [A]
time = 1.65, size = 266, normalized size = 0.98 \begin {gather*} \frac {\frac {3 i b \left (-\left ((a+i b) (A-i B-C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )\right )+(a-i b) (A+i B-C) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )}{a^2+b^2}-\frac {6 \left (A b^2+a (-b B+a C)\right ) (b c-a d)^{3/2} \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 {6 (b c C+b B d-a C d) \sqrt {c+d \tan (e+f x)}}{b}+2 C (c+d \tan (e+f x))^{3/2}}{3 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((3*I)*b*(-((a + I*b)*(A - I*B - C)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]) + (a - I
*b)*(A + I*B - C)*(c + I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]))/(a^2 + b^2) - (6*(A*b^2 +
a*(-(b*B) + a*C))*(b*c - a*d)^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2
+ b^2)) + (6*(b*c*C + b*B*d - a*C*d)*Sqrt[c + d*Tan[e + f*x]])/b + 2*C*(c + d*Tan[e + f*x])^(3/2))/(3*b*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2290\) vs. \(2(234)=468\).
time = 0.64, size = 2291, normalized size = 8.45

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2291\)
default	\(\text {Expression too large to display}\)	\(2291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(2/b^2*(1/3*C*(c+d*tan(f*x+e))^(3/2)*b+B*b*d*(c+d*tan(f*x+e))^(1/2)-C*a*d*(c+d*tan(f*x+e))^(1/2)+C*b*c*(c+
d*tan(f*x+e))^(1/2))+2/(a^2+b^2)*(1/4/d*(1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-A*(c^2+d^2)
^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a*d^2+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d-B*(c^2+d^2)
^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)*b*c^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+C*(c^2+d^2)
^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)
*a*d^2-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*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*(-2*A*(c^2+d^2)^(1/2)*a*d^2+2*A*(c^2+d^2)^(1/2)*b*c*d-2*B*(c^2+d^2)^(1/2)*a*c*d-2*
B*(c^2+d^2)^(1/2)*b*d^2+2*C*(c^2+d^2)^(1/2)*a*d^2-2*C*(c^2+d^2)^(1/2)*b*c*d-1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*c-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
c^2-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c-2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a*c*d+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2+C*(c^2+d^2)^(1/2)*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*c+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
c^2+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*(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*(1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+A*(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2-2*A*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d+B*(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*c+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2+B*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c-C*(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^2+2*C*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*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*(-2*A*(c^2+d^2)^(1/2)*a*d^2+2*A*(c^2+d^2)^(1/2)*b*c*d-2*B*(c^2+d^2)^(1/2)*a*c*d-2*B*(c^2+d^2)^(1/
2)*b*d^2+2*C*(c^2+d^2)^(1/2)*a*d^2-2*C*(c^2+d^2)^(1/2)*b*c*d+1/2*(A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*c+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2+A*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*d^2-2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d-B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*d+B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d-B*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*c^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*a*c-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-C*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*a*d^2+2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d)*(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
))))+2/b^2*(A*a^2*b^2*d^2-2*A*a*b^3*c*d+A*b^4*c^2-B*a^3*b*d^2+2*B*a^2*b^2*c*d-B*a*b^3*c^2+C*a^4*d^2-2*C*a^3*b*
c*d+C*a^2*b^2*c^2)/(a^2+b^2)/((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))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),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))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),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 (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{a + b \tan {\left (e + f x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(a + b*tan(e + f*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e)),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 = 58.88, size = 2500, normalized size = 9.23 \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))^(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(a + b*tan(e + f*x)),x)

[Out]

atan(((((((32*(4*B*a*b^8*d^12*f^4 - 4*B*b^9*c*d^11*f^4 + 8*B*a^3*b^6*d^12*f^4 + 4*B*a^5*b^4*d^12*f^4 - 4*B*b^9
*c^3*d^9*f^4 + 8*B*a*b^8*c^2*d^10*f^4 + 4*B*a*b^8*c^4*d^8*f^4 - 12*B*a^2*b^7*c*d^11*f^4 - 12*B*a^4*b^5*c*d^11*
f^4 - 4*B*a^6*b^3*c*d^11*f^4 - 12*B*a^2*b^7*c^3*d^9*f^4 + 16*B*a^3*b^6*c^2*d^10*f^4 + 8*B*a^3*b^6*c^4*d^8*f^4
- 12*B*a^4*b^5*c^3*d^9*f^4 + 8*B*a^5*b^4*c^2*d^10*f^4 + 4*B*a^5*b^4*c^4*d^8*f^4 - 4*B*a^6*b^3*c^3*d^9*f^4))/(b
*f^5) - (32*(c + d*tan(e + f*x))^(1/2)*(-(((8*B^2*a^2*c^3*f^2 - 8*B^2*b^2*c^3*f^2 - 16*B^2*a*b*d^3*f^2 - 24*B^
2*a^2*c*d^2*f^2 + 24*B^2*b^2*c*d^2*f^2 + 48*B^2*a*b*c^2*d*f^2)^2/4 - (16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4
)*(B^4*c^6 + B^4*d^6 + 3*B^4*c^2*d^4 + 3*B^4*c^4*d^2))^(1/2) - 4*B^2*a^2*c^3*f^2 + 4*B^2*b^2*c^3*f^2 + 8*B^2*a
*b*d^3*f^2 + 12*B^2*a^2*c*d^2*f^2 - 12*B^2*b^2*c*d^2*f^2 - 24*B^2*a*b*c^2*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^
2*b^2*f^4)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^
10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 +
24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*(-(((8*B^2*a^2*c^3*f^2 - 8*B^2*b^
2*c^3*f^2 - 16*B^2*a*b*d^3*f^2 - 24*B^2*a^2*c*d^2*f^2 + 24*B^2*b^2*c*d^2*f^2 + 48*B^2*a*b*c^2*d*f^2)^2/4 - (16
*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4)*(B^4*c^6 + B^4*d^6 + 3*B^4*c^2*d^4 + 3*B^4*c^4*d^2))^(1/2) - 4*B^2*a^2
*c^3*f^2 + 4*B^2*b^2*c^3*f^2 + 8*B^2*a*b*d^3*f^2 + 12*B^2*a^2*c*d^2*f^2 - 12*B^2*b^2*c*d^2*f^2 - 24*B^2*a*b*c^
2*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(4*B^2*a^3*b^5*d^13*
f^2 + 2*B^2*a^5*b^3*d^13*f^2 + 28*B^2*b^8*c^3*d^10*f^2 - 10*B^2*b^8*c^5*d^8*f^2 - 14*B^2*a*b^7*d^13*f^2 + 16*B
^2*a^7*b*d^13*f^2 - 8*B^2*a^8*c*d^12*f^2 + 22*B^2*b^8*c*d^12*f^2 + 20*B^2*a*b^7*c^2*d^11*f^2 + 50*B^2*a*b^7*c^
4*d^9*f^2 - 28*B^2*a^2*b^6*c*d^12*f^2 - 2*B^2*a^4*b^4*c*d^12*f^2 - 56*B^2*a^6*b^2*c*d^12*f^2 + 32*B^2*a^7*b*c^
2*d^11*f^2 + 8*B^2*a^2*b^6*c^3*d^10*f^2 + 12*B^2*a^2*b^6*c^5*d^8*f^2 - 24*B^2*a^3*b^5*c^2*d^11*f^2 - 12*B^2*a^
3*b^5*c^4*d^9*f^2 - 4*B^2*a^4*b^4*c^3*d^10*f^2 - 10*B^2*a^4*b^4*c^5*d^8*f^2 + 52*B^2*a^5*b^3*c^2*d^11*f^2 + 34
*B^2*a^5*b^3*c^4*d^9*f^2 - 48*B^2*a^6*b^2*c^3*d^10*f^2))/(b*f^4))*(-(((8*B^2*a^2*c^3*f^2 - 8*B^2*b^2*c^3*f^2 -
16*B^2*a*b*d^3*f^2 - 24*B^2*a^2*c*d^2*f^2 + 24*B^2*b^2*c*d^2*f^2 + 48*B^2*a*b*c^2*d*f^2)^2/4 - (16*a^4*f^4 +
16*b^4*f^4 + 32*a^2*b^2*f^4)*(B^4*c^6 + B^4*d^6 + 3*B^4*c^2*d^4 + 3*B^4*c^4*d^2))^(1/2) - 4*B^2*a^2*c^3*f^2 +
4*B^2*b^2*c^3*f^2 + 8*B^2*a*b*d^3*f^2 + 12*B^2*a^2*c*d^2*f^2 - 12*B^2*b^2*c*d^2*f^2 - 24*B^2*a*b*c^2*d*f^2)/(1
6*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2) + (32*(15*B^3*a^4*b^3*d^15*f^2 - B^3*a^2*b^5*d^15*f^2 - 4*B^3*a^
7*c^3*d^12*f^2 + 2*B^3*b^7*c^2*d^13*f^2 + 4*B^3*b^7*c^4*d^11*f^2 + 2*B^3*b^7*c^6*d^9*f^2 - 12*B^3*a^6*b*d^15*f
^2 - 4*B^3*a^7*c*d^14*f^2 - B^3*a*b^6*c*d^14*f^2 - 27*B^3*a*b^6*c^3*d^12*f^2 - 19*B^3*a*b^6*c^5*d^10*f^2 + 7*B
^3*a*b^6*c^7*d^8*f^2 - 57*B^3*a^3*b^4*c*d^14*f^2 + 64*B^3*a^5*b^2*c*d^14*f^2 + 4*B^3*a^6*b*c^2*d^13*f^2 + 16*B
^3*a^6*b*c^4*d^11*f^2 + 65*B^3*a^2*b^5*c^2*d^13*f^2 + 9*B^3*a^2*b^5*c^4*d^11*f^2 - 57*B^3*a^2*b^5*c^6*d^9*f^2
+ 77*B^3*a^3*b^4*c^3*d^12*f^2 + 129*B^3*a^3*b^4*c^5*d^10*f^2 - 5*B^3*a^3*b^4*c^7*d^8*f^2 - 121*B^3*a^4*b^3*c^2
*d^13*f^2 - 119*B^3*a^4*b^3*c^4*d^11*f^2 + 17*B^3*a^4*b^3*c^6*d^9*f^2 + 40*B^3*a^5*b^2*c^3*d^12*f^2 - 24*B^3*a
^5*b^2*c^5*d^10*f^2))/(b*f^5))*(-(((8*B^2*a^2*c^3*f^2 - 8*B^2*b^2*c^3*f^2 - 16*B^2*a*b*d^3*f^2 - 24*B^2*a^2*c*
d^2*f^2 + 24*B^2*b^2*c*d^2*f^2 + 48*B^2*a*b*c^2*d*f^2)^2/4 - (16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4)*(B^4*c
^6 + B^4*d^6 + 3*B^4*c^2*d^4 + 3*B^4*c^4*d^2))^(1/2) - 4*B^2*a^2*c^3*f^2 + 4*B^2*b^2*c^3*f^2 + 8*B^2*a*b*d^3*f
^2 + 12*B^2*a^2*c*d^2*f^2 - 12*B^2*b^2*c*d^2*f^2 - 24*B^2*a*b*c^2*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^
4)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(B^4*b^6*d^16 - 2*B^4*a^6*d^16 + 12*B^4*a^6*c^2*d^14 - 2*B^4*a^6*c
^4*d^12 + 4*B^4*b^6*c^2*d^14 + 6*B^4*b^6*c^4*d^12 + 4*B^4*b^6*c^6*d^10 + B^4*b^6*c^8*d^8 - 2*B^4*a^2*b^4*c^4*d
^12 + 12*B^4*a^2*b^4*c^6*d^10 - 2*B^4*a^2*b^4*c^8*d^8 + 8*B^4*a^3*b^3*c^3*d^13 - 48*B^4*a^3*b^3*c^5*d^11 + 8*B
^4*a^3*b^3*c^7*d^9 - 12*B^4*a^4*b^2*c^2*d^14 + 72*B^4*a^4*b^2*c^4*d^12 - 12*B^4*a^4*b^2*c^6*d^10 + 8*B^4*a^5*b
*c*d^15 - 48*B^4*a^5*b*c^3*d^13 + 8*B^4*a^5*b*c^5*d^11))/(b*f^4))*(-(((8*B^2*a^2*c^3*f^2 - 8*B^2*b^2*c^3*f^2 -
16*B^2*a*b*d^3*f^2 - 24*B^2*a^2*c*d^2*f^2 + 24*B^2*b^2*c*d^2*f^2 + 48*B^2*a*b*c^2*d*f^2)^2/4 - (16*a^4*f^4 +
16*b^4*f^4 + 32*a^2*b^2*f^4)*(B^4*c^6 + B^4*d^6 + 3*B^4*c^2*d^4 + 3*B^4*c^4*d^2))^(1/2) - 4*B^2*a^2*c^3*f^2 +
4*B^2*b^2*c^3*f^2 + 8*B^2*a*b*d^3*f^2 + 12*B^2*a^2*c*d^2*f^2 - 12*B^2*b^2*c*d^2*f^2 - 24*B^2*a*b*c^2*d*f^2)/(1
6*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2)*1i - (((((32*(4*B*a*b^8*d^12*f^4 - 4*B*b^9*c*d^11*f^4 + 8*B*a^3*
b^6*d^12*f^4 + 4*B*a^5*b^4*d^12*f^4 - 4*B*b^9*c^3*d^9*f^4 + 8*B*a*b^8*c^2*d^10*f^4 + 4*B*a*b^8*c^4*d^8*f^4 - 1
2*B*a^2*b^7*c*d^11*f^4 - 12*B*a^4*b^5*c*d^11*f^...

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