Integrand size = 41, antiderivative size = 122 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=-\frac {\left (a^2+b^2\right ) \log (\cos (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}{e (b+a \tan (d+e x))}+\frac {a^2 b \tan (d+e x) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}{e \left (a b+a^2 \tan (d+e x)\right )} \] Output:
-(a^2+b^2)*ln(cos(e*x+d))*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(1/2)/e/ (b+a*tan(e*x+d))+a^2*b*tan(e*x+d)*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^ (1/2)/e/(a*b+a^2*tan(e*x+d))
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.48 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\frac {\sqrt {(b+a \tan (d+e x))^2} \left (-\left (\left (a^2+b^2\right ) \log (\cos (d+e x))\right )+a b \tan (d+e x)\right )}{e (b+a \tan (d+e x))} \] Input:
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]
Output:
(Sqrt[(b + a*Tan[d + e*x])^2]*(-((a^2 + b^2)*Log[Cos[d + e*x]]) + a*b*Tan[ d + e*x]))/(e*(b + a*Tan[d + e*x]))
Time = 0.47 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.67, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4193, 27, 3042, 4008, 3042, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int (a+b \tan (d+e x)) \sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+b \tan (d+e x)) \sqrt {a^2 \tan (d+e x)^2+2 a b \tan (d+e x)+b^2}dx\) |
\(\Big \downarrow \) 4193 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \int 2 \left (\tan (d+e x) a^2+b a\right ) (a+b \tan (d+e x))dx}{2 \left (a^2 \tan (d+e x)+a b\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \int \left (\tan (d+e x) a^2+b a\right ) (a+b \tan (d+e x))dx}{a^2 \tan (d+e x)+a b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \int \left (\tan (d+e x) a^2+b a\right ) (a+b \tan (d+e x))dx}{a^2 \tan (d+e x)+a b}\) |
\(\Big \downarrow \) 4008 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \left (a \left (a^2+b^2\right ) \int \tan (d+e x)dx+\frac {a^2 b \tan (d+e x)}{e}\right )}{a^2 \tan (d+e x)+a b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \left (a \left (a^2+b^2\right ) \int \tan (d+e x)dx+\frac {a^2 b \tan (d+e x)}{e}\right )}{a^2 \tan (d+e x)+a b}\) |
\(\Big \downarrow \) 3956 |
\(\displaystyle \frac {\sqrt {a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2} \left (\frac {a^2 b \tan (d+e x)}{e}-\frac {a \left (a^2+b^2\right ) \log (\cos (d+e x))}{e}\right )}{a^2 \tan (d+e x)+a b}\) |
Input:
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]
Output:
((-((a*(a^2 + b^2)*Log[Cos[d + e*x]])/e) + (a^2*b*Tan[d + e*x])/e)*Sqrt[b^ 2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2])/(a*b + a^2*Tan[d + e*x])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[((A_) + (B_.)*tan[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*tan[(d_.) + (e_.)* (x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[(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*T an[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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.42 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.61
method | result | size |
derivativedivides | \(\frac {\operatorname {csgn}\left (b +a \tan \left (e x +d \right )\right ) \left (\ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right ) a^{2}+\ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right ) b^{2}+2 a b \tan \left (e x +d \right )+2 b^{2}\right )}{2 e}\) | \(75\) |
default | \(\frac {\operatorname {csgn}\left (b +a \tan \left (e x +d \right )\right ) \left (\ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right ) a^{2}+\ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right ) b^{2}+2 a b \tan \left (e x +d \right )+2 b^{2}\right )}{2 e}\) | \(75\) |
parts | \(\frac {a \,\operatorname {csgn}\left (b +a \tan \left (e x +d \right )\right ) \left (\ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right ) a +2 b \arctan \left (\tan \left (e x +d \right )\right )\right )}{2 e}+\frac {b \,\operatorname {csgn}\left (b +a \tan \left (e x +d \right )\right ) \left (b \ln \left (a^{2} \tan \left (e x +d \right )^{2}+a^{2}\right )-2 a \arctan \left (\tan \left (e x +d \right )\right )+2 a \tan \left (e x +d \right )+2 b \right )}{2 e}\) | \(108\) |
Input:
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,metho d=_RETURNVERBOSE)
Output:
1/2/e*csgn(b+a*tan(e*x+d))*(ln(a^2*tan(e*x+d)^2+a^2)*a^2+ln(a^2*tan(e*x+d) ^2+a^2)*b^2+2*a*b*tan(e*x+d)+2*b^2)
Time = 0.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.31 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\frac {2 \, a b \tan \left (e x + d\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\frac {1}{\tan \left (e x + d\right )^{2} + 1}\right )}{2 \, e} \] Input:
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")
Output:
1/2*(2*a*b*tan(e*x + d) - (a^2 + b^2)*log(1/(tan(e*x + d)^2 + 1)))/e
\[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\int \left (a + b \tan {\left (d + e x \right )}\right ) \sqrt {\left (a \tan {\left (d + e x \right )} + b\right )^{2}}\, dx \] Input:
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)
Output:
Integral((a + b*tan(d + e*x))*sqrt((a*tan(d + e*x) + b)**2), x)
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.53 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\frac {{\left (2 \, {\left (e x + d\right )} b + a \log \left (\tan \left (e x + d\right )^{2} + 1\right )\right )} a - {\left (2 \, {\left (e x + d\right )} a - b \log \left (\tan \left (e x + d\right )^{2} + 1\right ) - 2 \, a \tan \left (e x + d\right )\right )} b}{2 \, e} \] Input:
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")
Output:
1/2*((2*(e*x + d)*b + a*log(tan(e*x + d)^2 + 1))*a - (2*(e*x + d)*a - b*lo g(tan(e*x + d)^2 + 1) - 2*a*tan(e*x + d))*b)/e
Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.58 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\frac {a b \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) \tan \left (e x + d\right )}{e} + \frac {{\left (a^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right ) + b^{2} \mathrm {sgn}\left (a \tan \left (e x + d\right ) + b\right )\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{2 \, e} \] Input:
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")
Output:
a*b*sgn(a*tan(e*x + d) + b)*tan(e*x + d)/e + 1/2*(a^2*sgn(a*tan(e*x + d) + b) + b^2*sgn(a*tan(e*x + d) + b))*log(tan(e*x + d)^2 + 1)/e
Timed out. \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\int \left (a+b\,\mathrm {tan}\left (d+e\,x\right )\right )\,\sqrt {a^2\,{\mathrm {tan}\left (d+e\,x\right )}^2+2\,a\,b\,\mathrm {tan}\left (d+e\,x\right )+b^2} \,d x \] Input:
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)
Output:
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)
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.38 \[ \int (a+b \tan (d+e x)) \sqrt {b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx=\frac {\mathrm {log}\left (\tan \left (e x +d \right )^{2}+1\right ) a^{2}+\mathrm {log}\left (\tan \left (e x +d \right )^{2}+1\right ) b^{2}+2 \tan \left (e x +d \right ) a b}{2 e} \] Input:
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)
Output:
(log(tan(d + e*x)**2 + 1)*a**2 + log(tan(d + e*x)**2 + 1)*b**2 + 2*tan(d + e*x)*a*b)/(2*e)