3.1.26 \(\int \frac {\tan (c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{a+b \tan (c+d x)} \, dx\) [26]

Optimal. Leaf size=101 \[ -\frac {(a B+b C) x}{a^2+b^2}-\frac {(b B-a C) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac {a^2 (b B-a C) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac {C \tan (c+d x)}{b d} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3713, 3687, 3707, 3698, 31, 3556} \begin {gather*} \frac {a^2 (b B-a C) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )}-\frac {(b B-a C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {x (a B+b C)}{a^2+b^2}+\frac {C \tan (c+d x)}{b d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3687

Int[(((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)]))/((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)]), x_Symbol] :> Simp[b^2*B*(Tan[e + f*x]/(d*f)), x] + Dist[1/d, Int[(a^2*A*d - b^2*B*c + (2*a*
A*b + B*(a^2 - b^2))*d*Tan[e + f*x] + (A*b^2*d - b*B*(b*c - 2*a*d))*Tan[e + f*x]^2)/(c + d*Tan[e + f*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]

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.49, size = 118, normalized size = 1.17 \begin {gather*} \frac {\frac {i (B+i C) \log (i-\tan (c+d x))}{a+i b}-\frac {(i B+C) \log (i+\tan (c+d x))}{a-i b}+\frac {2 a^2 (b B-a C) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )}+\frac {2 C \tan (c+d x)}{b}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.18, size = 101, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {\frac {C \tan \left (d x +c \right )}{b}+\frac {a^{2} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (B b -C a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a B -C b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(101\)
default	\(\frac {\frac {C \tan \left (d x +c \right )}{b}+\frac {a^{2} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right )}+\frac {\frac {\left (B b -C a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-a B -C b \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}}{d}\)	\(101\)
norman	\(\frac {C \tan \left (d x +c \right )}{b d}-\frac {\left (a B +C b \right ) x}{a^{2}+b^{2}}+\frac {a^{2} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{2} \left (a^{2}+b^{2}\right ) d}+\frac {\left (B b -C a \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}\)	\(106\)
risch	\(\frac {x B}{i b -a}-\frac {i x C}{i b -a}-\frac {2 i a^{2} B x}{b \left (a^{2}+b^{2}\right )}-\frac {2 i a^{2} B c}{b d \left (a^{2}+b^{2}\right )}+\frac {2 i a^{3} C x}{b^{2} \left (a^{2}+b^{2}\right )}+\frac {2 i a^{3} C c}{b^{2} d \left (a^{2}+b^{2}\right )}+\frac {2 i B x}{b}+\frac {2 i B c}{b d}-\frac {2 i C a x}{b^{2}}-\frac {2 i C a c}{b^{2} d}+\frac {2 i C}{b d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{b d \left (a^{2}+b^{2}\right )}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{b^{2} d \left (a^{2}+b^{2}\right )}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) B}{b d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C a}{b^{2} d}\)	\(320\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.51, size = 109, normalized size = 1.08 \begin {gather*} -\frac {\frac {2 \, {\left (B a + C b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (C a^{3} - B a^{2} b\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{2} + b^{4}} + \frac {{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac {2 \, C \tan \left (d x + c\right )}{b}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 4.22, size = 149, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (B a b^{2} + C b^{3}\right )} d x + {\left (C a^{3} - B a^{2} b\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 (C a^{3} - B a^{2} b + C a b^{2} - B b^{3}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{2} b + C b^{3}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} + b^{4}\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.64, size = 1020, normalized size = 10.10 \begin {gather*} \begin {cases} \tilde {\infty } x \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {i B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {B d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {i B}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {3 C d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 i C d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {i C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 C \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 C}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\- \frac {i B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {B d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {i B}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 C d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 i C d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {i C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {2 C \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {3 C}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {- B x + \frac {B \tan {\left (c + d x \right )}}{d} - \frac {C \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C \tan ^{2}{\left (c + d x \right )}}{2 d}}{a} & \text {for}\: b = 0 \\\frac {x \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \tan {\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\\frac {2 B a^{2} b \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {2 B a b^{2} d x}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {B b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {2 C a^{3} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 C a^{2} b \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {C a b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} - \frac {2 C b^{3} d x}{2 a^{2} b^{2} d + 2 b^{4} d} + \frac {2 C b^{3} \tan {\left (c + d x \right )}}{2 a^{2} b^{2} d + 2 b^{4} d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x*(B*tan(c) + C*tan(c)**2), Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (I*B*d*x*tan(c + d*x)/(2*b*d*tan(c
+ d*x) - 2*I*b*d) + B*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) + B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan
(c + d*x) - 2*I*b*d) - I*B*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) - 2*I*b*d) - I*B/(2*b*d*tan(c + d*x) -
2*I*b*d) - 3*C*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 3*I*C*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) + I
*C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + C*log(tan(c + d*x)**2 + 1)/(2*b*d*ta
n(c + d*x) - 2*I*b*d) + 2*C*tan(c + d*x)**2/(2*b*d*tan(c + d*x) - 2*I*b*d) + 3*C/(2*b*d*tan(c + d*x) - 2*I*b*d
), Eq(a, -I*b)), (-I*B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + B*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d)
+ B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) + I*B*log(tan(c + d*x)**2 + 1)/(2*b*d
*tan(c + d*x) + 2*I*b*d) + I*B/(2*b*d*tan(c + d*x) + 2*I*b*d) - 3*C*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I
*b*d) - 3*I*C*d*x/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*
x) + 2*I*b*d) + C*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) + 2*C*tan(c + d*x)**2/(2*b*d*tan(c +
d*x) + 2*I*b*d) + 3*C/(2*b*d*tan(c + d*x) + 2*I*b*d), Eq(a, I*b)), ((-B*x + B*tan(c + d*x)/d - C*log(tan(c +
d*x)**2 + 1)/(2*d) + C*tan(c + d*x)**2/(2*d))/a, Eq(b, 0)), (x*(B*tan(c) + C*tan(c)**2)*tan(c)/(a + b*tan(c)),
Eq(d, 0)), (2*B*a**2*b*log(a/b + tan(c + d*x))/(2*a**2*b**2*d + 2*b**4*d) - 2*B*a*b**2*d*x/(2*a**2*b**2*d + 2
*b**4*d) + B*b**3*log(tan(c + d*x)**2 + 1)/(2*a**2*b**2*d + 2*b**4*d) - 2*C*a**3*log(a/b + tan(c + d*x))/(2*a*
*2*b**2*d + 2*b**4*d) + 2*C*a**2*b*tan(c + d*x)/(2*a**2*b**2*d + 2*b**4*d) - C*a*b**2*log(tan(c + d*x)**2 + 1)
/(2*a**2*b**2*d + 2*b**4*d) - 2*C*b**3*d*x/(2*a**2*b**2*d + 2*b**4*d) + 2*C*b**3*tan(c + d*x)/(2*a**2*b**2*d +
2*b**4*d), True))

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Giac [A]
time = 0.69, size = 110, normalized size = 1.09 \begin {gather*} -\frac {\frac {2 \, {\left (B a + C b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {{\left (C a - B b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (C a^{3} - B a^{2} b\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{2} + b^{4}} - \frac {2 \, C \tan \left (d x + c\right )}{b}}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 8.77, size = 117, normalized size = 1.16 \begin {gather*} \frac {C\,\mathrm {tan}\left (c+d\,x\right )}{b\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (b+a\,1{}\mathrm {i}\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (C\,a^3-B\,a^2\,b\right )}{d\,\left (a^2\,b^2+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (a+b\,1{}\mathrm {i}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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