\(\int (a+b \tan (c+d x)) (B \tan (c+d x)+C \tan ^2(c+d x)) \, dx\) [2]

Optimal result

Integrand size = 30, antiderivative size = 66 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-((b B+a C) x)-\frac {(a B-b C) \log (\cos (c+d x))}{d}+\frac {b B \tan (c+d x)}{d}+\frac {C (a+b \tan (c+d x))^2}{2 b d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3711, 3606, 3556} \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {(a B-b C) \log (\cos (c+d x))}{d}-x (a C+b B)+\frac {C (a+b \tan (c+d x))^2}{2 b d}+\frac {b B \tan (c+d x)}{d} \]

[In]

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

[Out]

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

Rule 3556

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

Rule 3606

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

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*} \text {integral}& = \frac {C (a+b \tan (c+d x))^2}{2 b d}+\int (a+b \tan (c+d x)) (-C+B \tan (c+d x)) \, dx \\ & = -((b B+a C) x)+\frac {b B \tan (c+d x)}{d}+\frac {C (a+b \tan (c+d x))^2}{2 b d}+(a B-b C) \int \tan (c+d x) \, dx \\ & = -((b B+a C) x)-\frac {(a B-b C) \log (\cos (c+d x))}{d}+\frac {b B \tan (c+d x)}{d}+\frac {C (a+b \tan (c+d x))^2}{2 b d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {-2 (b B+a C) \arctan (\tan (c+d x))+2 (-a B+b C) \log (\cos (c+d x))+2 (b B+a C) \tan (c+d x)+b C \tan ^2(c+d x)}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.05

method	result	size
norman	\(\left (-B b -C a \right ) x +\frac {\left (B b +C a \right ) \tan \left (d x +c \right )}{d}+\frac {C b \tan \left (d x +c \right )^{2}}{2 d}+\frac {\left (B a -C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}\)	\(69\)
derivativedivides	\(\frac {\frac {C \tan \left (d x +c \right )^{2} b}{2}+B \tan \left (d x +c \right ) b +C \tan \left (d x +c \right ) a +\frac {\left (B a -C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B b -C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(74\)
default	\(\frac {\frac {C \tan \left (d x +c \right )^{2} b}{2}+B \tan \left (d x +c \right ) b +C \tan \left (d x +c \right ) a +\frac {\left (B a -C b \right ) \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\left (-B b -C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{d}\)	\(74\)
parts	\(\frac {\left (B b +C a \right ) \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}+\frac {B a \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2 d}+\frac {C b \left (\frac {\tan \left (d x +c \right )^{2}}{2}-\frac {\ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}\right )}{d}\)	\(77\)
parallelrisch	\(\frac {-2 B b d x -2 C a d x +C \tan \left (d x +c \right )^{2} b +B \ln \left (1+\tan \left (d x +c \right )^{2}\right ) a +2 B \tan \left (d x +c \right ) b -C \ln \left (1+\tan \left (d x +c \right )^{2}\right ) b +2 C \tan \left (d x +c \right ) a}{2 d}\)	\(79\)
risch	\(-B b x -C a x +i B a x -i C b x +\frac {2 i B a c}{d}-\frac {2 i C b c}{d}+\frac {2 i \left (-i C b \,{\mathrm e}^{2 i \left (d x +c \right )}+B b \,{\mathrm e}^{2 i \left (d x +c \right )}+C a \,{\mathrm e}^{2 i \left (d x +c \right )}+B b +C a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B a}{d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C b}{d}\)	\(143\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C b \tan \left (d x + c\right )^{2} - 2 \, {\left (C a + B b\right )} d x - {\left (B a - C b\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{2 \, d} \]

[In]

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

[Out]

1/2*(C*b*tan(d*x + c)^2 - 2*(C*a + B*b)*d*x - (B*a - C*b)*log(1/(tan(d*x + c)^2 + 1)) + 2*(C*a + B*b)*tan(d*x
+ c))/d

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.59 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\begin {cases} \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B b x + \frac {B b \tan {\left (c + d x \right )}}{d} - C a x + \frac {C a \tan {\left (c + d x \right )}}{d} - \frac {C b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\left (c \right )}\right ) \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((B*a*log(tan(c + d*x)**2 + 1)/(2*d) - B*b*x + B*b*tan(c + d*x)/d - C*a*x + C*a*tan(c + d*x)/d - C*b*
log(tan(c + d*x)**2 + 1)/(2*d) + C*b*tan(c + d*x)**2/(2*d), Ne(d, 0)), (x*(a + b*tan(c))*(B*tan(c) + C*tan(c)*
*2), True))

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {C b \tan \left (d x + c\right )^{2} - 2 \, {\left (C a + B b\right )} {\left (d x + c\right )} + {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{2 \, d} \]

[In]

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

[Out]

1/2*(C*b*tan(d*x + c)^2 - 2*(C*a + B*b)*(d*x + c) + (B*a - C*b)*log(tan(d*x + c)^2 + 1) + 2*(C*a + B*b)*tan(d*
x + c))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (64) = 128\).

Time = 0.65 (sec) , antiderivative size = 556, normalized size of antiderivative = 8.42 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=-\frac {2 \, C a d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + 2 \, B b d x \tan \left (d x\right )^{2} \tan \left (c\right )^{2} + B a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - C b \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 4 \, C a d x \tan \left (d x\right ) \tan \left (c\right ) - 4 \, B b d x \tan \left (d x\right ) \tan \left (c\right ) - C b \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, B a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, C b \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) \tan \left (d x\right ) \tan \left (c\right ) + 2 \, C a \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, B b \tan \left (d x\right )^{2} \tan \left (c\right ) + 2 \, C a \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, B b \tan \left (d x\right ) \tan \left (c\right )^{2} + 2 \, C a d x + 2 \, B b d x - C b \tan \left (d x\right )^{2} - C b \tan \left (c\right )^{2} + B a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) - C b \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, \tan \left (d x\right ) \tan \left (c\right ) + 1\right )}}{\tan \left (d x\right )^{2} \tan \left (c\right )^{2} + \tan \left (d x\right )^{2} + \tan \left (c\right )^{2} + 1}\right ) - 2 \, C a \tan \left (d x\right ) - 2 \, B b \tan \left (d x\right ) - 2 \, C a \tan \left (c\right ) - 2 \, B b \tan \left (c\right ) - C b}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \left (c\right )^{2} - 2 \, d \tan \left (d x\right ) \tan \left (c\right ) + d\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.97 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx=\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,b+C\,a\right )+\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {B\,a}{2}-\frac {C\,b}{2}\right )-d\,x\,\left (B\,b+C\,a\right )+\frac {C\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}}{d} \]

[In]

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

[Out]

(tan(c + d*x)*(B*b + C*a) + log(tan(c + d*x)^2 + 1)*((B*a)/2 - (C*b)/2) - d*x*(B*b + C*a) + (C*b*tan(c + d*x)^
2)/2)/d