3.1.32 \(\int \frac {\tan ^2(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{(a+b \tan (c+d x))^2} \, dx\) [32]

Optimal. Leaf size=208 \[ -\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 B-b^2 B+2 a b C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}-\frac {\left (a b B-2 a^2 C-b^2 C\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))} \]

[Out]

-(2*B*a*b-C*a^2+C*b^2)*x/(a^2+b^2)^2+(B*a^2-B*b^2+2*C*a*b)*ln(cos(d*x+c))/(a^2+b^2)^2/d+a^2*(B*a^2*b+3*B*b^3-2
*C*a^3-4*C*a*b^2)*ln(a+b*tan(d*x+c))/b^3/(a^2+b^2)^2/d-(B*a*b-2*C*a^2-C*b^2)*tan(d*x+c)/b^2/(a^2+b^2)/d+a*(B*b
-C*a)*tan(d*x+c)^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))

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Rubi [A]
time = 0.36, antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {3713, 3686, 3728, 3707, 3698, 31, 3556} \begin {gather*} \frac {a (b B-a C) \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (-2 a^2 C+a b B-b^2 C\right ) \tan (c+d x)}{b^2 d \left (a^2+b^2\right )}+\frac {\left (a^2 B+2 a b C-b^2 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {x \left (a^2 (-C)+2 a b B+b^2 C\right )}{\left (a^2+b^2\right )^2}+\frac {a^2 \left (-2 a^3 C+a^2 b B-4 a b^2 C+3 b^3 B\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((2*a*b*B - a^2*C + b^2*C)*x)/(a^2 + b^2)^2) + ((a^2*B - b^2*B + 2*a*b*C)*Log[Cos[c + d*x]])/((a^2 + b^2)^2*
d) + (a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[a + b*Tan[c + d*x]])/(b^3*(a^2 + b^2)^2*d) - ((a*b*B -
2*a^2*C - b^2*C)*Tan[c + d*x])/(b^2*(a^2 + b^2)*d) + (a*(b*B - a*C)*Tan[c + d*x]^2)/(b*(a^2 + b^2)*d*(a + b*T
an[c + d*x]))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3686

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

Rule 3698

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

Rule 3707

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

Rule 3713

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] :> Dist[1/b^2, Int[(a + b*Tan[e + f*x])
^(m + 1)*(c + d*Tan[e + f*x])^n*(b*B - a*C + b*C*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m,
n}, x] && NeQ[b*c - a*d, 0] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

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])))

Rubi steps

\begin {align*} \int \frac {\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx &=\int \frac {\tan ^3(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx\\ &=\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-2 a (b B-a C)+b (b B-a C) \tan (c+d x)-\left (a b B-2 a^2 C-b^2 C\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {\left (a b B-2 a^2 C-b^2 C\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\int \frac {a \left (a b B-2 a^2 C-b^2 C\right )-b^2 (a B+b C) \tan (c+d x)+\left (a^2+b^2\right ) (b B-2 a C) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=-\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}-\frac {\left (a b B-2 a^2 C-b^2 C\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac {\left (a^2 B-b^2 B+2 a b C\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 B-b^2 B+2 a b C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {\left (a b B-2 a^2 C-b^2 C\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right )\right ) \text {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^2 d}\\ &=-\frac {\left (2 a b B-a^2 C+b^2 C\right ) x}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 B-b^2 B+2 a b C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}-\frac {\left (a b B-2 a^2 C-b^2 C\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}+\frac {a (b B-a C) \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.82, size = 444, normalized size = 2.13 \begin {gather*} \frac {a \left (2 (a+i b)^2 \left (2 a b^2 (B+i C)+i a^2 b (B+4 i C)-2 i a^3 C+b^3 C\right ) (c+d x)+2 \left (a^2+b^2\right )^2 (-b B+2 a C) \log (\cos (c+d x))+a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right )+b \left (2 \left (a^3 b^2 C (3-4 i c-4 i d x)-b^5 C (c+d x)+i a^4 b B (i+c+d x)-2 i a^5 C (i+c+d x)+a b^4 (C-2 B (c+d x))+a^2 b^3 (C (c+d x)+i B (i+3 c+3 d x))\right )+2 \left (a^2+b^2\right )^2 (-b B+2 a C) \log (\cos (c+d x))+a^2 \left (a^2 b B+3 b^3 B-2 a^3 C-4 a b^2 C\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right ) \tan (c+d x)+2 b^2 \left (a^2+b^2\right )^2 C \tan ^2(c+d x)+2 i a^2 \left (-a^2 b B-3 b^3 B+2 a^3 C+4 a b^2 C\right ) \text {ArcTan}(\tan (c+d x)) (a+b \tan (c+d x))}{2 b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*(a + I*b)^2*(2*a*b^2*(B + I*C) + I*a^2*b*(B + (4*I)*C) - (2*I)*a^3*C + b^3*C)*(c + d*x) + 2*(a^2 + b^2)^
2*(-(b*B) + 2*a*C)*Log[Cos[c + d*x]] + a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[(a*Cos[c + d*x] + b*S
in[c + d*x])^2]) + b*(2*(a^3*b^2*C*(3 - (4*I)*c - (4*I)*d*x) - b^5*C*(c + d*x) + I*a^4*b*B*(I + c + d*x) - (2*
I)*a^5*C*(I + c + d*x) + a*b^4*(C - 2*B*(c + d*x)) + a^2*b^3*(C*(c + d*x) + I*B*(I + 3*c + 3*d*x))) + 2*(a^2 +
b^2)^2*(-(b*B) + 2*a*C)*Log[Cos[c + d*x]] + a^2*(a^2*b*B + 3*b^3*B - 2*a^3*C - 4*a*b^2*C)*Log[(a*Cos[c + d*x]
+ b*Sin[c + d*x])^2])*Tan[c + d*x] + 2*b^2*(a^2 + b^2)^2*C*Tan[c + d*x]^2 + (2*I)*a^2*(-(a^2*b*B) - 3*b^3*B +
2*a^3*C + 4*a*b^2*C)*ArcTan[Tan[c + d*x]]*(a + b*Tan[c + d*x]))/(2*b^3*(a^2 + b^2)^2*d*(a + b*Tan[c + d*x]))

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Maple [A]
time = 0.23, size = 172, normalized size = 0.83

method	result	size
derivativedivides	\(\frac {\frac {C \tan \left (d x +c \right )}{b^{2}}+\frac {a^{2} \left (B \,a^{2} b +3 B \,b^{3}-2 C \,a^{3}-4 C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{3} \left (B b -C a \right )}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-a^{2} B +b^{2} B -2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(172\)
default	\(\frac {\frac {C \tan \left (d x +c \right )}{b^{2}}+\frac {a^{2} \left (B \,a^{2} b +3 B \,b^{3}-2 C \,a^{3}-4 C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )^{2}}+\frac {a^{3} \left (B b -C a \right )}{b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )}+\frac {\frac {\left (-a^{2} B +b^{2} B -2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-2 B a b +C \,a^{2}-b^{2} C \right ) \arctan \left (\tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}}{d}\)	\(172\)
norman	\(\frac {\frac {C \left (\tan ^{2}\left (d x +c \right )\right )}{b d}+\frac {\left (B \,a^{2} b -2 C \,a^{3}-C a \,b^{2}\right ) a}{d \,b^{3} \left (a^{2}+b^{2}\right )}-\frac {a \left (2 B a b -C \,a^{2}+b^{2} C \right ) x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {b \left (2 B a b -C \,a^{2}+b^{2} C \right ) x \tan \left (d x +c \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}}{a +b \tan \left (d x +c \right )}+\frac {a^{2} \left (B \,a^{2} b +3 B \,b^{3}-2 C \,a^{3}-4 C a \,b^{2}\right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}-\frac {\left (a^{2} B -b^{2} B +2 C a b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(265\)
risch	\(\frac {2 i B x}{b^{2}}-\frac {x C}{2 i b a -a^{2}+b^{2}}-\frac {2 i a^{4} B c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d \,b^{2}}-\frac {6 i a^{2} B x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 i C a c}{b^{3} d}-\frac {6 i a^{2} B c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {4 i C a x}{b^{3}}+\frac {8 i a^{3} C x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b}+\frac {2 i B c}{d \,b^{2}}+\frac {4 i a^{5} C x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3}}-\frac {2 i a^{4} B x}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{2}}+\frac {8 i a^{3} C c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}+\frac {2 i \left (-B \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 C \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}-C \,b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 i C \,a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}-2 i C a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-B \,a^{3} b +2 C \,a^{4}+2 C \,a^{2} b^{2}+b^{4} C \right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \left (i b +a \right ) \left (-i b +a \right )^{2} \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+a \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +a \right ) b^{2} d}+\frac {4 i a^{5} C c}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}-\frac {i x B}{2 i b a -a^{2}+b^{2}}-\frac {B \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d \,b^{2}}+\frac {2 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C a}{b^{3} d}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d \,b^{2}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {2 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b^{3} d}-\frac {4 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) b d}\)	\(762\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(C/b^2*tan(d*x+c)+1/b^3*a^2*(B*a^2*b+3*B*b^3-2*C*a^3-4*C*a*b^2)/(a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/b^3*a^3*(
B*b-C*a)/(a^2+b^2)/(a+b*tan(d*x+c))+1/(a^2+b^2)^2*(1/2*(-B*a^2+B*b^2-2*C*a*b)*ln(1+tan(d*x+c)^2)+(-2*B*a*b+C*a
^2-C*b^2)*arctan(tan(d*x+c))))

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Maxima [A]
time = 0.51, size = 220, normalized size = 1.06 \begin {gather*} \frac {\frac {2 \, {\left (C a^{2} - 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (2 \, C a^{5} - B a^{4} b + 4 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac {{\left (B a^{2} + 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (C a^{4} - B a^{3} b\right )}}{a^{3} b^{3} + a b^{5} + {\left (a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )} + \frac {2 \, C \tan \left (d x + c\right )}{b^{2}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

1/2*(2*(C*a^2 - 2*B*a*b - C*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - 2*(2*C*a^5 - B*a^4*b + 4*C*a^3*b^2 - 3*B*
a^2*b^3)*log(b*tan(d*x + c) + a)/(a^4*b^3 + 2*a^2*b^5 + b^7) - (B*a^2 + 2*C*a*b - B*b^2)*log(tan(d*x + c)^2 +
1)/(a^4 + 2*a^2*b^2 + b^4) - 2*(C*a^4 - B*a^3*b)/(a^3*b^3 + a*b^5 + (a^2*b^4 + b^6)*tan(d*x + c)) + 2*C*tan(d*
x + c)/b^2)/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (209) = 418\).
time = 3.06, size = 434, normalized size = 2.09 \begin {gather*} -\frac {2 \, C a^{4} b^{2} - 2 \, B a^{3} b^{3} - 2 \, {\left (C a^{3} b^{3} - 2 \, B a^{2} b^{4} - C a b^{5}\right )} d x - 2 \, {\left (C a^{4} b^{2} + 2 \, C a^{2} b^{4} + C b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (2 \, C a^{6} - B a^{5} b + 4 \, C a^{4} b^{2} - 3 \, B a^{3} b^{3} + {\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 3 \, B a^{2} b^{4}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, C a^{6} - B a^{5} b + 4 \, C a^{4} b^{2} - 2 \, B a^{3} b^{3} + 2 \, C a^{2} b^{4} - B a b^{5} + {\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 2 \, B a^{2} b^{4} + 2 \, C a b^{5} - B b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (2 \, C a^{5} b - B a^{4} b^{2} + 2 \, C a^{3} b^{3} + C a b^{5} + {\left (C a^{2} b^{4} - 2 \, B a b^{5} - C b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right ) + {\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/2*(2*C*a^4*b^2 - 2*B*a^3*b^3 - 2*(C*a^3*b^3 - 2*B*a^2*b^4 - C*a*b^5)*d*x - 2*(C*a^4*b^2 + 2*C*a^2*b^4 + C*b
^6)*tan(d*x + c)^2 + (2*C*a^6 - B*a^5*b + 4*C*a^4*b^2 - 3*B*a^3*b^3 + (2*C*a^5*b - B*a^4*b^2 + 4*C*a^3*b^3 - 3
*B*a^2*b^4)*tan(d*x + c))*log((b^2*tan(d*x + c)^2 + 2*a*b*tan(d*x + c) + a^2)/(tan(d*x + c)^2 + 1)) - (2*C*a^6
- B*a^5*b + 4*C*a^4*b^2 - 2*B*a^3*b^3 + 2*C*a^2*b^4 - B*a*b^5 + (2*C*a^5*b - B*a^4*b^2 + 4*C*a^3*b^3 - 2*B*a^
2*b^4 + 2*C*a*b^5 - B*b^6)*tan(d*x + c))*log(1/(tan(d*x + c)^2 + 1)) - 2*(2*C*a^5*b - B*a^4*b^2 + 2*C*a^3*b^3
+ C*a*b^5 + (C*a^2*b^4 - 2*B*a*b^5 - C*b^6)*d*x)*tan(d*x + c))/((a^4*b^4 + 2*a^2*b^6 + b^8)*d*tan(d*x + c) + (
a^5*b^3 + 2*a^3*b^5 + a*b^7)*d)

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Sympy [C] Result contains complex when optimal does not.
time = 1.20, size = 4541, normalized size = 21.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*(B*tan(c) + C*tan(c)**2), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), ((-B*log(tan(c + d*x)**2 + 1)/(2*d
) + B*tan(c + d*x)**2/(2*d) + C*x + C*tan(c + d*x)**3/(3*d) - C*tan(c + d*x)/d)/a**2, Eq(b, 0)), (3*I*B*d*x*ta
n(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*B*d*x*tan(c + d*x)/(4*b**2*d
*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 3*I*B*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(
c + d*x) - 4*b**2*d) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan
(c + d*x) - 4*b**2*d) - 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan
(c + d*x) - 4*b**2*d) - 2*B*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b
**2*d) - 5*I*B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*B/(4*b**2*d*ta
n(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 9*C*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*
b**2*d*tan(c + d*x) - 4*b**2*d) + 18*I*C*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x)
- 4*b**2*d) + 9*C*d*x/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*C*log(tan(c + d*x)
**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*C*log(tan(c + d*x
)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*C*log(tan(c + d*x
)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*C*tan(c + d*x)**3/(4*b**2*d*tan(
c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 19*C*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*t
an(c + d*x) - 4*b**2*d) - 14*I*C/(4*b**2*d*tan(c + d*x)**2 - 8*I*b**2*d*tan(c + d*x) - 4*b**2*d), Eq(a, -I*b))
, (-3*I*B*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 6*B*d*x*tan(c
+ d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 3*I*B*d*x/(4*b**2*d*tan(c + d*x)**2 +
8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 2*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2
+ 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2
+ 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 2*B*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*ta
n(c + d*x) - 4*b**2*d) + 5*I*B*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) -
4*B/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 9*C*d*x*tan(c + d*x)**2/(4*b**2*d*tan(c
+ d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 18*I*C*d*x*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2
*d*tan(c + d*x) - 4*b**2*d) + 9*C*d*x/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) - 4*I*C*
log(tan(c + d*x)**2 + 1)*tan(c + d*x)**2/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 8*C
*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*I*C
*log(tan(c + d*x)**2 + 1)/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 4*C*tan(c + d*x)**
3/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 19*C*tan(c + d*x)/(4*b**2*d*tan(c + d*x)**
2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*d) + 14*I*C/(4*b**2*d*tan(c + d*x)**2 + 8*I*b**2*d*tan(c + d*x) - 4*b**2*
d), Eq(a, I*b)), (x*(B*tan(c) + C*tan(c)**2)*tan(c)**2/(a + b*tan(c))**2, Eq(d, 0)), (2*B*a**5*b*log(a/b + tan
(c + d*x))/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7
*d + 2*b**8*d*tan(c + d*x)) + 2*B*a**5*b/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*
b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(c + d*x)) + 2*B*a**4*b**2*log(a/b + tan(c + d*x))*tan(c + d*x)
/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**
8*d*tan(c + d*x)) + 6*B*a**3*b**3*log(a/b + tan(c + d*x))/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3
*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(c + d*x)) - B*a**3*b**3*log(tan(c + d*x)**2 +
1)/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*
b**8*d*tan(c + d*x)) + 2*B*a**3*b**3/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6
*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(c + d*x)) - 4*B*a**2*b**4*d*x/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c
+ d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(c + d*x)) + 6*B*a**2*b**4*log
(a/b + tan(c + d*x))*tan(c + d*x)/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*
tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(c + d*x)) - B*a**2*b**4*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*a**5
*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(
c + d*x)) - 4*B*a*b**5*d*x*tan(c + d*x)/(2*a**5*b**3*d + 2*a**4*b**4*d*tan(c + d*x) + 4*a**3*b**5*d + 4*a**2*b
**6*d*tan(c + d*x) + 2*a*b**7*d + 2*b**8*d*tan(...

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Giac [A]
time = 0.86, size = 290, normalized size = 1.39 \begin {gather*} \frac {\frac {2 \, {\left (C a^{2} - 2 \, B a b - C b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (B a^{2} + 2 \, C a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {2 \, {\left (2 \, C a^{5} - B a^{4} b + 4 \, C a^{3} b^{2} - 3 \, B a^{2} b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} + \frac {2 \, C \tan \left (d x + c\right )}{b^{2}} + \frac {2 \, {\left (2 \, C a^{5} b \tan \left (d x + c\right ) - B a^{4} b^{2} \tan \left (d x + c\right ) + 4 \, C a^{3} b^{3} \tan \left (d x + c\right ) - 3 \, B a^{2} b^{4} \tan \left (d x + c\right ) + C a^{6} + 3 \, C a^{4} b^{2} - 2 \, B a^{3} b^{3}\right )}}{{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(B*tan(d*x+c)+C*tan(d*x+c)^2)/(a+b*tan(d*x+c))^2,x, algorithm="giac")

[Out]

1/2*(2*(C*a^2 - 2*B*a*b - C*b^2)*(d*x + c)/(a^4 + 2*a^2*b^2 + b^4) - (B*a^2 + 2*C*a*b - B*b^2)*log(tan(d*x + c
)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) - 2*(2*C*a^5 - B*a^4*b + 4*C*a^3*b^2 - 3*B*a^2*b^3)*log(abs(b*tan(d*x + c) +
a))/(a^4*b^3 + 2*a^2*b^5 + b^7) + 2*C*tan(d*x + c)/b^2 + 2*(2*C*a^5*b*tan(d*x + c) - B*a^4*b^2*tan(d*x + c) +
4*C*a^3*b^3*tan(d*x + c) - 3*B*a^2*b^4*tan(d*x + c) + C*a^6 + 3*C*a^4*b^2 - 2*B*a^3*b^3)/((a^4*b^3 + 2*a^2*b^5
+ b^7)*(b*tan(d*x + c) + a)))/d

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Mupad [B]
time = 9.65, size = 210, normalized size = 1.01 \begin {gather*} \frac {C\,\mathrm {tan}\left (c+d\,x\right )}{b^2\,d}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (2\,C\,a^5-B\,a^4\,b+4\,C\,a^3\,b^2-3\,B\,a^2\,b^3\right )}{d\,\left (a^4\,b^3+2\,a^2\,b^5+b^7\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}-\frac {a^2\,\left (C\,a^2-B\,a\,b\right )}{b\,d\,\left (\mathrm {tan}\left (c+d\,x\right )\,b^3+a\,b^2\right )\,\left (a^2+b^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((tan(c + d*x)^2*(B*tan(c + d*x) + C*tan(c + d*x)^2))/(a + b*tan(c + d*x))^2,x)

[Out]

(C*tan(c + d*x))/(b^2*d) - (log(a + b*tan(c + d*x))*(2*C*a^5 - 3*B*a^2*b^3 + 4*C*a^3*b^2 - B*a^4*b))/(d*(b^7 +
2*a^2*b^5 + a^4*b^3)) - (log(tan(c + d*x) - 1i)*(B + C*1i))/(2*d*(a*b*2i + a^2 - b^2)) - (log(tan(c + d*x) +
1i)*(B*1i + C))/(2*d*(2*a*b + a^2*1i - b^2*1i)) - (a^2*(C*a^2 - B*a*b))/(b*d*(a*b^2 + b^3*tan(c + d*x))*(a^2 +
b^2))

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