3.1.61 \(\int \frac {c+d x}{(a+b \tan (e+f x))^2} \, dx\) [61]

Optimal. Leaf size=214 \[ -\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac {(b d+2 a c f+2 a d f x)^2}{4 a (a+i b) \left (a^2+b^2\right ) d f^2}+\frac {b (b d+2 a c f+2 a d f x) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {i a b d \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

-1/2*(d*x+c)^2/(a^2+b^2)/d+1/4*(2*a*d*f*x+2*a*c*f+b*d)^2/a/(a+I*b)/(a^2+b^2)/d/f^2+b*(2*a*d*f*x+2*a*c*f+b*d)*l
n(1+(a^2+b^2)*exp(2*I*(f*x+e))/(a+I*b)^2)/(a^2+b^2)^2/f^2-I*a*b*d*polylog(2,-(a^2+b^2)*exp(2*I*(f*x+e))/(a+I*b
)^2)/(a^2+b^2)^2/f^2-b*(d*x+c)/(a^2+b^2)/f/(a+b*tan(f*x+e))

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Rubi [A]
time = 0.19, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3814, 3813, 2221, 2317, 2438} \begin {gather*} \frac {b (2 a c f+2 a d f x+b d) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f^2 \left (a^2+b^2\right )^2}-\frac {b (c+d x)}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {(2 a c f+2 a d f x+b d)^2}{4 a d f^2 (a+i b) \left (a^2+b^2\right )}-\frac {(c+d x)^2}{2 d \left (a^2+b^2\right )}-\frac {i a b d \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f^2 \left (a^2+b^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(c + d*x)^2/((a^2 + b^2)*d) + (b*d + 2*a*c*f + 2*a*d*f*x)^2/(4*a*(a + I*b)*(a^2 + b^2)*d*f^2) + (b*(b*d +
2*a*c*f + 2*a*d*f*x)*Log[1 + ((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2])/((a^2 + b^2)^2*f^2) - (I*a*b*d*P
olyLog[2, -(((a^2 + b^2)*E^((2*I)*(e + f*x)))/(a + I*b)^2)])/((a^2 + b^2)^2*f^2) - (b*(c + d*x))/((a^2 + b^2)*
f*(a + b*Tan[e + f*x]))

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3813

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + d*x)^(m + 1)/(d*
(m + 1)*(a + I*b)), x] + Dist[2*I*b, Int[(c + d*x)^m*(E^Simp[2*I*(e + f*x), x]/((a + I*b)^2 + (a^2 + b^2)*E^Si
mp[2*I*(e + f*x), x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0]

Rule 3814

Int[((c_.) + (d_.)*(x_))/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[-(c + d*x)^2/(2*d*(a^2 +
b^2)), x] + (Dist[1/(f*(a^2 + b^2)), Int[(b*d + 2*a*c*f + 2*a*d*f*x)/(a + b*Tan[e + f*x]), x], x] - Simp[b*((c
+ d*x)/(f*(a^2 + b^2)*(a + b*Tan[e + f*x]))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 + b^2, 0]

Rubi steps

\begin {align*} \int \frac {c+d x}{(a+b \tan (e+f x))^2} \, dx &=-\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\int \frac {b d+2 a c f+2 a d f x}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac {(b d+2 a c f+2 a d f x)^2}{4 a (a+i b) \left (a^2+b^2\right ) d f^2}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(2 i b) \int \frac {e^{2 i (e+f x)} (b d+2 a c f+2 a d f x)}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (e+f x)}} \, dx}{\left (a^2+b^2\right ) f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac {(b d+2 a c f+2 a d f x)^2}{4 a (a+i b) \left (a^2+b^2\right ) d f^2}+\frac {b (b d+2 a c f+2 a d f x) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(2 a b d) \int \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right )^2 f}\\ &=-\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac {(b d+2 a c f+2 a d f x)^2}{4 a (a+i b) \left (a^2+b^2\right ) d f^2}+\frac {b (b d+2 a c f+2 a d f x) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {(i a b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{\left (a^2+b^2\right )^2 f^2}\\ &=-\frac {(c+d x)^2}{2 \left (a^2+b^2\right ) d}+\frac {(b d+2 a c f+2 a d f x)^2}{4 a (a+i b) \left (a^2+b^2\right ) d f^2}+\frac {b (b d+2 a c f+2 a d f x) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {i a b d \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right )^2 f^2}-\frac {b (c+d x)}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(745\) vs. \(2(214)=428\).
time = 6.87, size = 745, normalized size = 3.48 \begin {gather*} \frac {(e+f x) (-2 d e+2 c f+d (e+f x)) \sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2}{2 (a-i b) (a+i b) f^2 (a+b \tan (e+f x))^2}+\frac {b^2 d (-b (e+f x)+a \log (a \cos (e+f x)+b \sin (e+f x))) \sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2}{a (a-i b) (a+i b) \left (a^2+b^2\right ) f^2 (a+b \tan (e+f x))^2}-\frac {2 b d e (-b (e+f x)+a \log (a \cos (e+f x)+b \sin (e+f x))) \sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2}{(a-i b) (a+i b) \left (a^2+b^2\right ) f^2 (a+b \tan (e+f x))^2}+\frac {2 b c (-b (e+f x)+a \log (a \cos (e+f x)+b \sin (e+f x))) \sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2}{(a-i b) (a+i b) \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {d \left (e^{i \text {ArcTan}\left (\frac {a}{b}\right )} (e+f x)^2+\frac {a \left (i (e+f x) \left (-\pi +2 \text {ArcTan}\left (\frac {a}{b}\right )\right )-\pi \log \left (1+e^{-2 i (e+f x)}\right )-2 \left (e+f x+\text {ArcTan}\left (\frac {a}{b}\right )\right ) \log \left (1-e^{2 i \left (e+f x+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )+\pi \log (\cos (e+f x))+2 \text {ArcTan}\left (\frac {a}{b}\right ) \log \left (\sin \left (e+f x+\text {ArcTan}\left (\frac {a}{b}\right )\right )\right )+i \text {PolyLog}\left (2,e^{2 i \left (e+f x+\text {ArcTan}\left (\frac {a}{b}\right )\right )}\right )\right )}{\sqrt {1+\frac {a^2}{b^2}} b}\right ) \sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x))^2}{(a-i b) (a+i b) \sqrt {\frac {a^2+b^2}{b^2}} f^2 (a+b \tan (e+f x))^2}+\frac {\sec ^2(e+f x) (a \cos (e+f x)+b \sin (e+f x)) \left (-b^2 d e \sin (e+f x)+b^2 c f \sin (e+f x)+b^2 d (e+f x) \sin (e+f x)\right )}{a (a-i b) (a+i b) f^2 (a+b \tan (e+f x))^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((e + f*x)*(-2*d*e + 2*c*f + d*(e + f*x))*Sec[e + f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)/(2*(a - I*b)*(a
+ I*b)*f^2*(a + b*Tan[e + f*x])^2) + (b^2*d*(-(b*(e + f*x)) + a*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])*Sec[e +
f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)/(a*(a - I*b)*(a + I*b)*(a^2 + b^2)*f^2*(a + b*Tan[e + f*x])^2) - (
2*b*d*e*(-(b*(e + f*x)) + a*Log[a*Cos[e + f*x] + b*Sin[e + f*x]])*Sec[e + f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f
*x])^2)/((a - I*b)*(a + I*b)*(a^2 + b^2)*f^2*(a + b*Tan[e + f*x])^2) + (2*b*c*(-(b*(e + f*x)) + a*Log[a*Cos[e
+ f*x] + b*Sin[e + f*x]])*Sec[e + f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)/((a - I*b)*(a + I*b)*(a^2 + b^2)
*f*(a + b*Tan[e + f*x])^2) - (d*(E^(I*ArcTan[a/b])*(e + f*x)^2 + (a*(I*(e + f*x)*(-Pi + 2*ArcTan[a/b]) - Pi*Lo
g[1 + E^((-2*I)*(e + f*x))] - 2*(e + f*x + ArcTan[a/b])*Log[1 - E^((2*I)*(e + f*x + ArcTan[a/b]))] + Pi*Log[Co
s[e + f*x]] + 2*ArcTan[a/b]*Log[Sin[e + f*x + ArcTan[a/b]]] + I*PolyLog[2, E^((2*I)*(e + f*x + ArcTan[a/b]))])
)/(Sqrt[1 + a^2/b^2]*b))*Sec[e + f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f*x])^2)/((a - I*b)*(a + I*b)*Sqrt[(a^2 +
b^2)/b^2]*f^2*(a + b*Tan[e + f*x])^2) + (Sec[e + f*x]^2*(a*Cos[e + f*x] + b*Sin[e + f*x])*(-(b^2*d*e*Sin[e + f
*x]) + b^2*c*f*Sin[e + f*x] + b^2*d*(e + f*x)*Sin[e + f*x]))/(a*(a - I*b)*(a + I*b)*f^2*(a + b*Tan[e + f*x])^2
)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 998 vs. \(2 (202 ) = 404\).
time = 0.52, size = 999, normalized size = 4.67

method	result	size
risch	\(-\frac {d \,x^{2}}{2 \left (2 i a b -a^{2}+b^{2}\right )}-\frac {c x}{2 i a b -a^{2}+b^{2}}-\frac {2 i b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right )}+\frac {4 i b a d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right )}+\frac {2 i b \,a^{2} d e \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}-\frac {b^{3} d \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}-\frac {2 i b \,a^{2} c \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{\left (i a +b \right )^{2} f \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}-\frac {4 i b a c \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{\left (i a +b \right )^{2} f \left (-i a +b \right ) \left (i b +a \right )}-\frac {2 b^{2} a c \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{\left (i a +b \right )^{2} f \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}+\frac {2 i b a d \ln \left (1-\frac {\left (i b -a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{i b +a}\right ) x}{\left (i a +b \right )^{2} f \left (-i a +b \right ) \left (i b +a \right )}-\frac {2 i b^{2} \left (d x +c \right )}{\left (-i a +b \right ) f \left (i a +b \right )^{2} \left (b \,{\mathrm e}^{2 i \left (f x +e \right )}+i a \,{\mathrm e}^{2 i \left (f x +e \right )}-b +i a \right )}+\frac {2 b^{2} a d e \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}-\frac {i b^{2} d \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right ) a}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right ) \left (i b -a \right )}+\frac {2 i b a d \ln \left (1-\frac {\left (i b -a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{i b +a}\right ) e}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right )}+\frac {2 b a d \,x^{2}}{\left (i a +b \right )^{2} \left (-i a +b \right ) \left (i b +a \right )}+\frac {4 b a d e x}{\left (i a +b \right )^{2} f \left (-i a +b \right ) \left (i b +a \right )}+\frac {2 b a d \,e^{2}}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right )}+\frac {b a d \polylog \left (2, \frac {\left (i b -a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{i b +a}\right )}{\left (i a +b \right )^{2} f^{2} \left (-i a +b \right ) \left (i b +a \right )}\)	\(999\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(2*I*a*b-a^2+b^2)*d*x^2-1/(2*I*a*b-a^2+b^2)*c*x-2*I/(I*a+b)^2/f^2/(b-I*a)*b^2*d/(a+I*b)*ln(exp(I*(f*x+e))
)+4*I/(I*a+b)^2/f^2/(b-I*a)*b*a*d*e/(a+I*b)*ln(exp(I*(f*x+e)))+2*I/(I*a+b)^2/f^2/(b-I*a)*b*a^2*d*e/(a+I*b)/(I*
b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)-1/(I*a+b)^2/f^2/(b-I*a)*b^3*d/(a+I*b)/(I*b-a)*ln(I*exp(
2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)-2*I/(I*a+b)^2/f/(b-I*a)*b*a^2*c/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))
*b-a*exp(2*I*(f*x+e))-I*b-a)-4*I/(I*a+b)^2/f/(b-I*a)*b*a*c/(a+I*b)*ln(exp(I*(f*x+e)))-2/(I*a+b)^2/f/(b-I*a)*b^
2*a*c/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)+2*I/(I*a+b)^2/f/(b-I*a)*b*a*d/(a+I*b)*
ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*x-2*I*b^2*(d*x+c)/(b-I*a)/f/(I*a+b)^2/(b*exp(2*I*(f*x+e))+I*a*exp(2*I*(
f*x+e))-b+I*a)+2/(I*a+b)^2/f^2/(b-I*a)*b^2*a*d*e/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*
b-a)-I/(I*a+b)^2/f^2/(b-I*a)*b^2*d/(a+I*b)/(I*b-a)*ln(I*exp(2*I*(f*x+e))*b-a*exp(2*I*(f*x+e))-I*b-a)*a+2*I/(I*
a+b)^2/f^2/(b-I*a)*b*a*d/(a+I*b)*ln(1-(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))*e+2/(I*a+b)^2/(b-I*a)*b*a*d/(a+I*b)*x^
2+4/(I*a+b)^2/f/(b-I*a)*b*a*d/(a+I*b)*e*x+2/(I*a+b)^2/f^2/(b-I*a)*b*a*d/(a+I*b)*e^2+1/(I*a+b)^2/f^2/(b-I*a)*b*
a*d/(a+I*b)*polylog(2,(I*b-a)*exp(2*I*(f*x+e))/(a+I*b))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1193 vs. \(2 (200) = 400\).
time = 0.88, size = 1193, normalized size = 5.57 \begin {gather*} \frac {{\left (a^{3} - i \, a^{2} b + a b^{2} - i \, b^{3}\right )} d f^{2} x^{2} + 2 \, {\left (a^{3} - i \, a^{2} b + a b^{2} - i \, b^{3}\right )} c f^{2} x - 4 \, {\left (-i \, a b^{2} + b^{3}\right )} c f - 2 \, {\left (2 \, {\left (-i \, a^{2} b + a b^{2}\right )} c f + {\left (-i \, a b^{2} + b^{3}\right )} d + {\left (2 \, {\left (-i \, a^{2} b - a b^{2}\right )} c f + {\left (-i \, a b^{2} - b^{3}\right )} d\right )} \cos \left (2 \, f x + 2 \, e\right ) + {\left (2 \, {\left (a^{2} b - i \, a b^{2}\right )} c f + {\left (a b^{2} - i \, b^{3}\right )} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \arctan \left (-b \cos \left (2 \, f x + 2 \, e\right ) + a \sin \left (2 \, f x + 2 \, e\right ) + b, a \cos \left (2 \, f x + 2 \, e\right ) + b \sin \left (2 \, f x + 2 \, e\right ) + a\right ) - 4 \, {\left ({\left (i \, a^{2} b + a b^{2}\right )} d f x \cos \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} b - i \, a b^{2}\right )} d f x \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, a^{2} b - a b^{2}\right )} d f x\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, f x + 2 \, e\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (a^{3} - 3 i \, a^{2} b - 3 \, a b^{2} + i \, b^{3}\right )} d f^{2} x^{2} + 2 \, {\left ({\left (a^{3} - 3 i \, a^{2} b - 3 \, a b^{2} + i \, b^{3}\right )} c f^{2} - 2 \, {\left (i \, a b^{2} + b^{3}\right )} d f\right )} x\right )} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, {\left ({\left (i \, a^{2} b + a b^{2}\right )} d \cos \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} b - i \, a b^{2}\right )} d \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, a^{2} b - a b^{2}\right )} d\right )} {\rm Li}_2\left (\frac {{\left (i \, a e^{\left (2 i \, e\right )} + b e^{\left (2 i \, e\right )}\right )} e^{\left (2 i \, f x\right )}}{-i \, a + b}\right ) + {\left (2 \, {\left (a^{2} b + i \, a b^{2}\right )} c f + {\left (a b^{2} + i \, b^{3}\right )} d + {\left (2 \, {\left (a^{2} b - i \, a b^{2}\right )} c f + {\left (a b^{2} - i \, b^{3}\right )} d\right )} \cos \left (2 \, f x + 2 \, e\right ) - {\left (2 \, {\left (-i \, a^{2} b - a b^{2}\right )} c f - {\left (i \, a b^{2} + b^{3}\right )} d\right )} \sin \left (2 \, f x + 2 \, e\right )\right )} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )\right ) + 2 \, {\left ({\left (a^{2} b - i \, a b^{2}\right )} d f x \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, a^{2} b - a b^{2}\right )} d f x \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} b + i \, a b^{2}\right )} d f x\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) + {\left ({\left (i \, a^{3} + 3 \, a^{2} b - 3 i \, a b^{2} - b^{3}\right )} d f^{2} x^{2} - 2 \, {\left ({\left (-i \, a^{3} - 3 \, a^{2} b + 3 i \, a b^{2} + b^{3}\right )} c f^{2} - 2 \, {\left (a b^{2} - i \, b^{3}\right )} d f\right )} x\right )} \sin \left (2 \, f x + 2 \, e\right )}{2 \, {\left ({\left (a^{5} - i \, a^{4} b + 2 \, a^{3} b^{2} - 2 i \, a^{2} b^{3} + a b^{4} - i \, b^{5}\right )} f^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, a^{5} - a^{4} b - 2 i \, a^{3} b^{2} - 2 \, a^{2} b^{3} - i \, a b^{4} - b^{5}\right )} f^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{5} + i \, a^{4} b + 2 \, a^{3} b^{2} + 2 i \, a^{2} b^{3} + a b^{4} + i \, b^{5}\right )} f^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^3 - I*a^2*b + a*b^2 - I*b^3)*d*f^2*x^2 + 2*(a^3 - I*a^2*b + a*b^2 - I*b^3)*c*f^2*x - 4*(-I*a*b^2 + b^3
)*c*f - 2*(2*(-I*a^2*b + a*b^2)*c*f + (-I*a*b^2 + b^3)*d + (2*(-I*a^2*b - a*b^2)*c*f + (-I*a*b^2 - b^3)*d)*cos
(2*f*x + 2*e) + (2*(a^2*b - I*a*b^2)*c*f + (a*b^2 - I*b^3)*d)*sin(2*f*x + 2*e))*arctan2(-b*cos(2*f*x + 2*e) +
a*sin(2*f*x + 2*e) + b, a*cos(2*f*x + 2*e) + b*sin(2*f*x + 2*e) + a) - 4*((I*a^2*b + a*b^2)*d*f*x*cos(2*f*x +
2*e) - (a^2*b - I*a*b^2)*d*f*x*sin(2*f*x + 2*e) + (I*a^2*b - a*b^2)*d*f*x)*arctan2((2*a*b*cos(2*f*x + 2*e) - (
a^2 - b^2)*sin(2*f*x + 2*e))/(a^2 + b^2), (2*a*b*sin(2*f*x + 2*e) + a^2 + b^2 + (a^2 - b^2)*cos(2*f*x + 2*e))/
(a^2 + b^2)) + ((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*d*f^2*x^2 + 2*((a^3 - 3*I*a^2*b - 3*a*b^2 + I*b^3)*c*f^2 -
2*(I*a*b^2 + b^3)*d*f)*x)*cos(2*f*x + 2*e) - 2*((I*a^2*b + a*b^2)*d*cos(2*f*x + 2*e) - (a^2*b - I*a*b^2)*d*si
n(2*f*x + 2*e) + (I*a^2*b - a*b^2)*d)*dilog((I*a*e^(2*I*e) + b*e^(2*I*e))*e^(2*I*f*x)/(-I*a + b)) + (2*(a^2*b
+ I*a*b^2)*c*f + (a*b^2 + I*b^3)*d + (2*(a^2*b - I*a*b^2)*c*f + (a*b^2 - I*b^3)*d)*cos(2*f*x + 2*e) - (2*(-I*a
^2*b - a*b^2)*c*f - (I*a*b^2 + b^3)*d)*sin(2*f*x + 2*e))*log((a^2 + b^2)*cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x
+ 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2*f*x + 2*e)) + 2*((a^2*b - I*a*b^2)*d
*f*x*cos(2*f*x + 2*e) - (-I*a^2*b - a*b^2)*d*f*x*sin(2*f*x + 2*e) + (a^2*b + I*a*b^2)*d*f*x)*log(((a^2 + b^2)*
cos(2*f*x + 2*e)^2 + 4*a*b*sin(2*f*x + 2*e) + (a^2 + b^2)*sin(2*f*x + 2*e)^2 + a^2 + b^2 + 2*(a^2 - b^2)*cos(2
*f*x + 2*e))/(a^2 + b^2)) + ((I*a^3 + 3*a^2*b - 3*I*a*b^2 - b^3)*d*f^2*x^2 - 2*((-I*a^3 - 3*a^2*b + 3*I*a*b^2
+ b^3)*c*f^2 - 2*(a*b^2 - I*b^3)*d*f)*x)*sin(2*f*x + 2*e))/((a^5 - I*a^4*b + 2*a^3*b^2 - 2*I*a^2*b^3 + a*b^4 -
I*b^5)*f^2*cos(2*f*x + 2*e) - (-I*a^5 - a^4*b - 2*I*a^3*b^2 - 2*a^2*b^3 - I*a*b^4 - b^5)*f^2*sin(2*f*x + 2*e)
+ (a^5 + I*a^4*b + 2*a^3*b^2 + 2*I*a^2*b^3 + a*b^4 + I*b^5)*f^2)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 871 vs. \(2 (200) = 400\).
time = 0.44, size = 871, normalized size = 4.07 \begin {gather*} \frac {{\left (a^{3} - a b^{2}\right )} d f^{2} x^{2} - 2 \, b^{3} c f - 2 \, {\left (b^{3} d f - {\left (a^{3} - a b^{2}\right )} c f^{2}\right )} x + {\left (i \, a b^{2} d \tan \left (f x + e\right ) + i \, a^{2} b d\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) + {\left (-i \, a b^{2} d \tan \left (f x + e\right ) - i \, a^{2} b d\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) + 2 \, {\left (a^{2} b d f x + a^{2} b d e + {\left (a b^{2} d f x + a b^{2} d e\right )} \tan \left (f x + e\right )\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 2 \, {\left (a^{2} b d f x + a^{2} b d e + {\left (a b^{2} d f x + a b^{2} d e\right )} \tan \left (f x + e\right )\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + {\left (2 \, a^{2} b c f - 2 \, a^{2} b d e + a b^{2} d + {\left (2 \, a b^{2} c f - 2 \, a b^{2} d e + b^{3} d\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left (2 \, a^{2} b c f - 2 \, a^{2} b d e + a b^{2} d + {\left (2 \, a b^{2} c f - 2 \, a b^{2} d e + b^{3} d\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + {\left ({\left (a^{2} b - b^{3}\right )} d f^{2} x^{2} + 2 \, a b^{2} c f + 2 \, {\left (a b^{2} d f + {\left (a^{2} b - b^{3}\right )} c f^{2}\right )} x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} f^{2} \tan \left (f x + e\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} f^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((a^3 - a*b^2)*d*f^2*x^2 - 2*b^3*c*f - 2*(b^3*d*f - (a^3 - a*b^2)*c*f^2)*x + (I*a*b^2*d*tan(f*x + e) + I*a
^2*b*d)*dilog(2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2 - I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^
2)*tan(f*x + e)^2 + a^2 + b^2) + 1) + (-I*a*b^2*d*tan(f*x + e) - I*a^2*b*d)*dilog(2*((-I*a*b - b^2)*tan(f*x +
e)^2 - a^2 + I*a*b + (-I*a^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2) + 1) + 2*
(a^2*b*d*f*x + a^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*tan(f*x + e))*log(-2*((I*a*b - b^2)*tan(f*x + e)^2 - a^2
- I*a*b + (I*a^2 - 2*a*b - I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + 2*(a^2*b*d*f*x + a
^2*b*d*e + (a*b^2*d*f*x + a*b^2*d*e)*tan(f*x + e))*log(-2*((-I*a*b - b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (-I*a
^2 - 2*a*b + I*b^2)*tan(f*x + e))/((a^2 + b^2)*tan(f*x + e)^2 + a^2 + b^2)) + (2*a^2*b*c*f - 2*a^2*b*d*e + a*b
^2*d + (2*a*b^2*c*f - 2*a*b^2*d*e + b^3*d)*tan(f*x + e))*log(((I*a*b + b^2)*tan(f*x + e)^2 - a^2 + I*a*b + (I*
a^2 + I*b^2)*tan(f*x + e))/(tan(f*x + e)^2 + 1)) + (2*a^2*b*c*f - 2*a^2*b*d*e + a*b^2*d + (2*a*b^2*c*f - 2*a*b
^2*d*e + b^3*d)*tan(f*x + e))*log(((I*a*b - b^2)*tan(f*x + e)^2 + a^2 + I*a*b + (I*a^2 + I*b^2)*tan(f*x + e))/
(tan(f*x + e)^2 + 1)) + ((a^2*b - b^3)*d*f^2*x^2 + 2*a*b^2*c*f + 2*(a*b^2*d*f + (a^2*b - b^3)*c*f^2)*x)*tan(f*
x + e))/((a^4*b + 2*a^2*b^3 + b^5)*f^2*tan(f*x + e) + (a^5 + 2*a^3*b^2 + a*b^4)*f^2)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {c + d x}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)/(b*tan(f*x + e) + a)^2, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {c+d\,x}{{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)/(a + b*tan(e + f*x))^2,x)

[Out]

int((c + d*x)/(a + b*tan(e + f*x))^2, x)

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