3.1.25 \(\int \frac {\tan ^2(c+d x) (B \tan (c+d x)+C \tan ^2(c+d x))}{a+b \tan (c+d x)} \, dx\) [25]

Optimal. Leaf size=127 \[ -\frac {(b B-a C) x}{a^2+b^2}+\frac {(a B+b C) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac {a^3 (b B-a C) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right ) d}+\frac {(b B-a C) \tan (c+d x)}{b^2 d}+\frac {C \tan ^2(c+d x)}{2 b d} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 127, 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, 3688, 3728, 3707, 3698, 31, 3556} \begin {gather*} \frac {(a B+b C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac {x (b B-a C)}{a^2+b^2}-\frac {a^3 (b B-a C) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )}+\frac {(b B-a C) \tan (c+d x)}{b^2 d}+\frac {C \tan ^2(c+d x)}{2 b d} \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]),x]

[Out]

-(((b*B - a*C)*x)/(a^2 + b^2)) + ((a*B + b*C)*Log[Cos[c + d*x]])/((a^2 + b^2)*d) - (a^3*(b*B - a*C)*Log[a + b*
Tan[c + d*x]])/(b^3*(a^2 + b^2)*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Tan[c + d*x]^2)/(2*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 3688

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*B*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^(n + 1)/(d*f
*(m + n))), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 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]

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.99, size = 138, normalized size = 1.09 \begin {gather*} \frac {-\frac {b (B+i C) \log (i-\tan (c+d x))}{a+i b}-\frac {b (B-i C) \log (i+\tan (c+d x))}{a-i b}+\frac {2 a^3 (-b B+a C) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )}+\frac {2 (b B-a C) \tan (c+d x)}{b}+C \tan ^2(c+d x)}{2 b d} \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]),x]

[Out]

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

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Maple [A]
time = 0.19, size = 127, normalized size = 1.00

method	result	size
derivativedivides	\(\frac {\frac {\frac {C b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+b B \tan \left (d x +c \right )-C \tan \left (d x +c \right ) a}{b^{2}}+\frac {\frac {\left (-a B -C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-B b +C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )}}{d}\)	\(127\)
default	\(\frac {\frac {\frac {C b \left (\tan ^{2}\left (d x +c \right )\right )}{2}+b B \tan \left (d x +c \right )-C \tan \left (d x +c \right ) a}{b^{2}}+\frac {\frac {\left (-a B -C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2}+\left (-B b +C a \right ) \arctan \left (\tan \left (d x +c \right )\right )}{a^{2}+b^{2}}-\frac {a^{3} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right )}}{d}\)	\(127\)
norman	\(\frac {\left (B b -C a \right ) \tan \left (d x +c \right )}{b^{2} d}-\frac {\left (B b -C a \right ) x}{a^{2}+b^{2}}+\frac {C \left (\tan ^{2}\left (d x +c \right )\right )}{2 b d}-\frac {\left (a B +C b \right ) \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )}-\frac {a^{3} \left (B b -C a \right ) \ln \left (a +b \tan \left (d x +c \right )\right )}{b^{3} \left (a^{2}+b^{2}\right ) d}\)	\(131\)
risch	\(-\frac {2 i a B c}{b^{2} d}-\frac {x C}{i b -a}+\frac {2 i C \,a^{2} x}{b^{3}}-\frac {2 i a^{4} C c}{\left (a^{2}+b^{2}\right ) b^{3} d}+\frac {2 i a^{3} B x}{b^{2} \left (a^{2}+b^{2}\right )}-\frac {2 i C c}{b d}+\frac {2 i \left (-i C b \,{\mathrm e}^{2 i \left (d x +c \right )}+B b \,{\mathrm e}^{2 i \left (d x +c \right )}-C a \,{\mathrm e}^{2 i \left (d x +c \right )}+B b -C a \right )}{b^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {i x B}{i b -a}+\frac {2 i a^{3} B c}{d \left (a^{2}+b^{2}\right ) b^{2}}-\frac {2 i a^{4} C x}{\left (a^{2}+b^{2}\right ) b^{3}}+\frac {2 i C \,a^{2} c}{b^{3} d}-\frac {2 i a B x}{b^{2}}-\frac {2 i C x}{b}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) a B}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C \,a^{2}}{b^{3} d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) C}{b d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) B}{d \left (a^{2}+b^{2}\right ) b^{2}}+\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right ) C}{\left (a^{2}+b^{2}\right ) b^{3} d}\)	\(415\)

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)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.51, size = 130, normalized size = 1.02 \begin {gather*} \frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} + \frac {2 \, {\left (C a^{4} - B a^{3} b\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{3} + b^{5}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {C b \tan \left (d x + c\right )^{2} - 2 \, {\left (C a - B b\right )} \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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 4.86, size = 190, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (C a b^{3} - B b^{4}\right )} d x + {\left (C a^{2} b^{2} + C b^{4}\right )} \tan \left (d x + c\right )^{2} + {\left (C a^{4} - B a^{3} 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^{4} - B a^{3} b - B a b^{3} - C b^{4}\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (C a^{3} b - B a^{2} b^{2} + C a b^{3} - B b^{4}\right )} \tan \left (d x + c\right )}{2 \, {\left (a^{2} b^{3} + b^{5}\right )} 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)),x, algorithm="fricas")

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.82, size = 1306, normalized size = 10.28 \begin {gather*} \begin {cases} \tilde {\infty } x \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \tan {\left (c \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {- \frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B \tan ^{2}{\left (c + d x \right )}}{2 d} + C x + \frac {C \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {C \tan {\left (c + d x \right )}}{d}}{a} & \text {for}\: b = 0 \\- \frac {3 B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 i B d x}{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 )} \tan {\left (c + d x \right )}}{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 )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {2 B \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 B}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {3 i C d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {3 C d x}{2 b d \tan {\left (c + d x \right )} - 2 i b d} - \frac {2 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 {2 i 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 {C \tan ^{3}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {i C \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} - 2 i b d} + \frac {3 i C}{2 b d \tan {\left (c + d x \right )} - 2 i b d} & \text {for}\: a = - i b \\- \frac {3 B d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 i B d x}{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 )} \tan {\left (c + d x \right )}}{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 )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {2 B \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {3 B}{2 b d \tan {\left (c + d x \right )} + 2 i b d} + \frac {3 i C d x \tan {\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 C d x}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {2 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 {2 i 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 {C \tan ^{3}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {i C \tan ^{2}{\left (c + d x \right )}}{2 b d \tan {\left (c + d x \right )} + 2 i b d} - \frac {3 i C}{2 b d \tan {\left (c + d x \right )} + 2 i b d} & \text {for}\: a = i b \\\frac {x \left (B \tan {\left (c \right )} + C \tan ^{2}{\left (c \right )}\right ) \tan ^{2}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {for}\: d = 0 \\- \frac {2 B a^{3} b \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {2 B a^{2} b^{2} \tan {\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} - \frac {B a b^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} - \frac {2 B b^{4} d x}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {2 B b^{4} \tan {\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {2 C a^{4} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} - \frac {2 C a^{3} b \tan {\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {C a^{2} b^{2} \tan ^{2}{\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {2 C a b^{3} d x}{2 a^{2} b^{3} d + 2 b^{5} d} - \frac {2 C a b^{3} \tan {\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} - \frac {C b^{4} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} + \frac {C b^{4} \tan ^{2}{\left (c + d x \right )}}{2 a^{2} b^{3} d + 2 b^{5} d} & \text {otherwise} \end {cases} \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)),x)

[Out]

Piecewise((zoo*x*(B*tan(c) + C*tan(c)**2)*tan(c), 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, Eq(b, 0)), (-3*B*d*x
*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 3*I*B*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) + I*B*log(tan(c + d*x)
**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + B*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) - 2*I*b*
d) + 2*B*tan(c + d*x)**2/(2*b*d*tan(c + d*x) - 2*I*b*d) + 3*B/(2*b*d*tan(c + d*x) - 2*I*b*d) - 3*I*C*d*x*tan(c
+ d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) - 3*C*d*x/(2*b*d*tan(c + d*x) - 2*I*b*d) - 2*C*log(tan(c + d*x)**2 + 1)
*tan(c + d*x)/(2*b*d*tan(c + d*x) - 2*I*b*d) + 2*I*C*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) - 2*I*b*d) +
C*tan(c + d*x)**3/(2*b*d*tan(c + d*x) - 2*I*b*d) + I*C*tan(c + d*x)**2/(2*b*d*tan(c + d*x) - 2*I*b*d) + 3*I*C
/(2*b*d*tan(c + d*x) - 2*I*b*d), Eq(a, -I*b)), (-3*B*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 3*I*B*d
*x/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*B*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) +
B*log(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) + 2*B*tan(c + d*x)**2/(2*b*d*tan(c + d*x) + 2*I*b*d
) + 3*B/(2*b*d*tan(c + d*x) + 2*I*b*d) + 3*I*C*d*x*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 3*C*d*x/(2*b*
d*tan(c + d*x) + 2*I*b*d) - 2*C*log(tan(c + d*x)**2 + 1)*tan(c + d*x)/(2*b*d*tan(c + d*x) + 2*I*b*d) - 2*I*C*l
og(tan(c + d*x)**2 + 1)/(2*b*d*tan(c + d*x) + 2*I*b*d) + C*tan(c + d*x)**3/(2*b*d*tan(c + d*x) + 2*I*b*d) - I*
C*tan(c + d*x)**2/(2*b*d*tan(c + d*x) + 2*I*b*d) - 3*I*C/(2*b*d*tan(c + d*x) + 2*I*b*d), Eq(a, I*b)), (x*(B*ta
n(c) + C*tan(c)**2)*tan(c)**2/(a + b*tan(c)), Eq(d, 0)), (-2*B*a**3*b*log(a/b + tan(c + d*x))/(2*a**2*b**3*d +
2*b**5*d) + 2*B*a**2*b**2*tan(c + d*x)/(2*a**2*b**3*d + 2*b**5*d) - B*a*b**3*log(tan(c + d*x)**2 + 1)/(2*a**2
*b**3*d + 2*b**5*d) - 2*B*b**4*d*x/(2*a**2*b**3*d + 2*b**5*d) + 2*B*b**4*tan(c + d*x)/(2*a**2*b**3*d + 2*b**5*
d) + 2*C*a**4*log(a/b + tan(c + d*x))/(2*a**2*b**3*d + 2*b**5*d) - 2*C*a**3*b*tan(c + d*x)/(2*a**2*b**3*d + 2*
b**5*d) + C*a**2*b**2*tan(c + d*x)**2/(2*a**2*b**3*d + 2*b**5*d) + 2*C*a*b**3*d*x/(2*a**2*b**3*d + 2*b**5*d) -
2*C*a*b**3*tan(c + d*x)/(2*a**2*b**3*d + 2*b**5*d) - C*b**4*log(tan(c + d*x)**2 + 1)/(2*a**2*b**3*d + 2*b**5*
d) + C*b**4*tan(c + d*x)**2/(2*a**2*b**3*d + 2*b**5*d), True))

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Giac [A]
time = 0.79, size = 135, normalized size = 1.06 \begin {gather*} \frac {\frac {2 \, {\left (C a - B b\right )} {\left (d x + c\right )}}{a^{2} + b^{2}} - \frac {{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac {2 \, {\left (C a^{4} - B a^{3} b\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{3} + b^{5}} + \frac {C b \tan \left (d x + c\right )^{2} - 2 \, C a \tan \left (d x + c\right ) + 2 \, B b \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)),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 9.07, size = 144, normalized size = 1.13 \begin {gather*} \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (\frac {B}{b}-\frac {C\,a}{b^2}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,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^4-B\,a^3\,b\right )}{d\,\left (a^2\,b^3+b^5\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (a-b\,1{}\mathrm {i}\right )}+\frac {C\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,b\,d} \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)),x)

[Out]

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

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