\(\int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx\) [516]

Optimal result

Integrand size = 41, antiderivative size = 138 \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {\left (a^2-b^2\right ) \log (b \cos (d+e x)+a \sin (d+e x)) (b+a \tan (d+e x))}{\left (a^2+b^2\right ) e \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}+\frac {2 b x \left (a b+a^2 \tan (d+e x)\right )}{\left (a^2+b^2\right ) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \]

[Out]

(a^2-b^2)*ln(b*cos(e*x+d)+a*sin(e*x+d))*(b+a*tan(e*x+d))/(a^2+b^2)/e/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(
1/2)+2*b*x*(a*b+a^2*tan(e*x+d))/(a^2+b^2)/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2)

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {3791, 3612, 3611} \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {2 b x \left (a^2 \tan (d+e x)+a b\right )}{\left (a^2+b^2\right ) \sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}+\frac {\left (a^2-b^2\right ) (a \tan (d+e x)+b) \log (a \sin (d+e x)+b \cos (d+e x))}{e \left (a^2+b^2\right ) \sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}} \]

[In]

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

[Out]

((a^2 - b^2)*Log[b*Cos[d + e*x] + a*Sin[d + e*x]]*(b + a*Tan[d + e*x]))/((a^2 + b^2)*e*Sqrt[b^2 + 2*a*b*Tan[d
+ e*x] + a^2*Tan[d + e*x]^2]) + (2*b*x*(a*b + a^2*Tan[d + e*x]))/((a^2 + b^2)*Sqrt[b^2 + 2*a*b*Tan[d + e*x] +
a^2*Tan[d + e*x]^2])

Rule 3611

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

Rule 3612

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

Rule 3791

Int[((A_) + (B_.)*tan[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*tan[(d_.) + (e_.)*(x_)] + (c_.)*tan[(d_.) + (e_.)*(x_
)]^2)^(n_), x_Symbol] :> Dist[(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^n/(b + 2*c*Tan[d + e*x])^(2*n), Int[(A +
B*Tan[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0
] &&  !IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (2 a b+2 a^2 \tan (d+e x)\right ) \int \frac {a+b \tan (d+e x)}{2 a b+2 a^2 \tan (d+e x)} \, dx}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \\ & = \frac {2 b x \left (a b+a^2 \tan (d+e x)\right )}{\left (a^2+b^2\right ) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}+\frac {\left (\left (a^2-b^2\right ) \left (2 a b+2 a^2 \tan (d+e x)\right )\right ) \int \frac {2 a^2-2 a b \tan (d+e x)}{2 a b+2 a^2 \tan (d+e x)} \, dx}{2 a \left (a^2+b^2\right ) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \\ & = \frac {\left (a^2-b^2\right ) \log (b \cos (d+e x)+a \sin (d+e x)) (b+a \tan (d+e x))}{\left (a^2+b^2\right ) e \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}+\frac {2 b x \left (a b+a^2 \tan (d+e x)\right )}{\left (a^2+b^2\right ) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.84 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {\left (-(a+i b)^2 \log (i-\tan (d+e x))-(a-i b)^2 \log (i+\tan (d+e x))+2 \left (a^2-b^2\right ) \log (b+a \tan (d+e x))\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \sqrt {(b+a \tan (d+e x))^2}} \]

[In]

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

[Out]

((-((a + I*b)^2*Log[I - Tan[d + e*x]]) - (a - I*b)^2*Log[I + Tan[d + e*x]] + 2*(a^2 - b^2)*Log[b + a*Tan[d + e
*x]])*(b + a*Tan[d + e*x]))/(2*(a^2 + b^2)*e*Sqrt[(b + a*Tan[d + e*x])^2])

Maple [A] (verified)

Time = 0.84 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\left (b +a \tan \left (e x +d \right )\right ) \left (2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2}-2 \ln \left (b +a \tan \left (e x +d \right )\right ) b^{2}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2}+\ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{2}+4 a b \arctan \left (\tan \left (e x +d \right )\right )\right )}{2 e \sqrt {b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}}\, \left (a^{2}+b^{2}\right )}\)	\(114\)
default	\(\frac {\left (b +a \tan \left (e x +d \right )\right ) \left (2 \ln \left (b +a \tan \left (e x +d \right )\right ) a^{2}-2 \ln \left (b +a \tan \left (e x +d \right )\right ) b^{2}-\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2}+\ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{2}+4 a b \arctan \left (\tan \left (e x +d \right )\right )\right )}{2 e \sqrt {b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}}\, \left (a^{2}+b^{2}\right )}\)	\(114\)
parts	\(\frac {a \left (b +a \tan \left (e x +d \right )\right ) \left (2 a \ln \left (b +a \tan \left (e x +d \right )\right )-a \ln \left (1+\tan \left (e x +d \right )^{2}\right )+2 b \arctan \left (\tan \left (e x +d \right )\right )\right )}{2 e \sqrt {b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}}\, \left (a^{2}+b^{2}\right )}-\frac {b \left (b +a \tan \left (e x +d \right )\right ) \left (2 b \ln \left (b +a \tan \left (e x +d \right )\right )-b \ln \left (1+\tan \left (e x +d \right )^{2}\right )-2 a \arctan \left (\tan \left (e x +d \right )\right )\right )}{2 e \sqrt {b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}}\, \left (a^{2}+b^{2}\right )}\)	\(158\)

[In]

int((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/e*(b+a*tan(e*x+d))*(2*ln(b+a*tan(e*x+d))*a^2-2*ln(b+a*tan(e*x+d))*b^2-ln(1+tan(e*x+d)^2)*a^2+ln(1+tan(e*x+
d)^2)*b^2+4*a*b*arctan(tan(e*x+d)))/((b+a*tan(e*x+d))^2)^(1/2)/(a^2+b^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.51 \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {4 \, a b e x + {\left (a^{2} - b^{2}\right )} \log \left (\frac {a^{2} \tan \left (e x + d\right )^{2} + 2 \, a b \tan \left (e x + d\right ) + b^{2}}{\tan \left (e x + d\right )^{2} + 1}\right )}{2 \, {\left (a^{2} + b^{2}\right )} e} \]

[In]

integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\int \frac {a + b \tan {\left (d + e x \right )}}{\sqrt {\left (a \tan {\left (d + e x \right )} + b\right )^{2}}}\, dx \]

[In]

integrate((a+b*tan(e*x+d))/(b**2+2*a*b*tan(e*x+d)+a**2*tan(e*x+d)**2)**(1/2),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {a {\left (\frac {2 \, {\left (e x + d\right )} b}{a^{2} + b^{2}} + \frac {2 \, a \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{2} + b^{2}} - \frac {a \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} + {\left (\frac {2 \, {\left (e x + d\right )} a}{a^{2} + b^{2}} - \frac {2 \, b \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{2} + b^{2}} + \frac {b \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} b}{2 \, e} \]

[In]

integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*(a*(2*(e*x + d)*b/(a^2 + b^2) + 2*a*log(a*tan(e*x + d) + b)/(a^2 + b^2) - a*log(tan(e*x + d)^2 + 1)/(a^2 +
b^2)) + (2*(e*x + d)*a/(a^2 + b^2) - 2*b*log(a*tan(e*x + d) + b)/(a^2 + b^2) + b*log(tan(e*x + d)^2 + 1)/(a^2
+ b^2))*b)/e

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.17 \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\frac {\frac {4 \, {\left (e x + d\right )} a b}{a^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )} - \frac {{\left (a^{2} - b^{2}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )} + \frac {2 \, {\left (a^{3} - a b^{2}\right )} \log \left ({\left | a \tan \left (e x + d\right ) + b \right |}\right )}{a^{3} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + a b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )}}{2 \, e} \]

[In]

integrate((a+b*tan(e*x+d))/(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2),x, algorithm="giac")

[Out]

1/2*(4*(e*x + d)*a*b/(a^2*sgn(a*tan(e*x + d) + b) + b^2*sgn(a*tan(e*x + d) + b)) - (a^2 - b^2)*log(tan(e*x + d
)^2 + 1)/(a^2*sgn(a*tan(e*x + d) + b) + b^2*sgn(a*tan(e*x + d) + b)) + 2*(a^3 - a*b^2)*log(abs(a*tan(e*x + d)
+ b))/(a^3*sgn(a*tan(e*x + d) + b) + a*b^2*sgn(a*tan(e*x + d) + b)))/e

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \tan (d+e x)}{\sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (d+e\,x\right )}{\sqrt {a^2\,{\mathrm {tan}\left (d+e\,x\right )}^2+2\,a\,b\,\mathrm {tan}\left (d+e\,x\right )+b^2}} \,d x \]

[In]

int((a + b*tan(d + e*x))/(b^2 + a^2*tan(d + e*x)^2 + 2*a*b*tan(d + e*x))^(1/2),x)

[Out]

int((a + b*tan(d + e*x))/(b^2 + a^2*tan(d + e*x)^2 + 2*a*b*tan(d + e*x))^(1/2), x)