\(\int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx\) [342]

Optimal result

Integrand size = 27, antiderivative size = 422 \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 b \sqrt {a+b \tan (c+d x)}}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3609, 12, 3566, 722, 1108, 648, 632, 212, 642} \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {\sqrt {a^2+b^2}+a}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} d \sqrt {a-\sqrt {a^2+b^2}}}+\frac {b \sqrt {a^2+b^2} \log \left (-\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}-\frac {b \sqrt {a^2+b^2} \log \left (\sqrt {2} \sqrt {\sqrt {a^2+b^2}+a} \sqrt {a+b \tan (c+d x)}+\sqrt {a^2+b^2}+a+b \tan (c+d x)\right )}{2 \sqrt {2} d \sqrt {\sqrt {a^2+b^2}+a}}+\frac {2 b \sqrt {a+b \tan (c+d x)}}{d} \]

[In]

Int[(-a + b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]],x]

[Out]

-((b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] - Sqrt[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2
+ b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d)) + (b*Sqrt[a^2 + b^2]*ArcTanh[(Sqrt[a + Sqrt[a^2 + b^2]] + Sqr
t[2]*Sqrt[a + b*Tan[c + d*x]])/Sqrt[a - Sqrt[a^2 + b^2]]])/(Sqrt[2]*Sqrt[a - Sqrt[a^2 + b^2]]*d) + (b*Sqrt[a^2
+ b^2]*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] - Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]]
)/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) - (b*Sqrt[a^2 + b^2]*Log[a + Sqrt[a^2 + b^2] + b*Tan[c + d*x] + Sqrt
[2]*Sqrt[a + Sqrt[a^2 + b^2]]*Sqrt[a + b*Tan[c + d*x]]])/(2*Sqrt[2]*Sqrt[a + Sqrt[a^2 + b^2]]*d) + (2*b*Sqrt[a
+ b*Tan[c + d*x]])/d

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

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

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*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] && GtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\int \frac {-a^2-b^2}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\left (-a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \tan (c+d x)}} \, dx \\ & = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}-\frac {\left (b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+x} \left (b^2+x^2\right )} \, dx,x,b \tan (c+d x)\right )}{d} \\ & = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}-\frac {\left (2 b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-2 a x^2+x^4} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{d} \\ & = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}-\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}-x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = \frac {2 b \sqrt {a+b \tan (c+d x)}}{d}-\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 d}-\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 d}+\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 x}{\sqrt {a^2+b^2}+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d} \\ & = \frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 b \sqrt {a+b \tan (c+d x)}}{d}+\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{d}+\frac {\left (b \sqrt {a^2+b^2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (a-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}}+2 \sqrt {a+b \tan (c+d x)}\right )}{d} \\ & = -\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\sqrt {a+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {a+b \tan (c+d x)}}{\sqrt {a-\sqrt {a^2+b^2}}}\right )}{\sqrt {2} \sqrt {a-\sqrt {a^2+b^2}} d}+\frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)-\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}-\frac {b \sqrt {a^2+b^2} \log \left (a+\sqrt {a^2+b^2}+b \tan (c+d x)+\sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} \sqrt {a+b \tan (c+d x)}\right )}{2 \sqrt {2} \sqrt {a+\sqrt {a^2+b^2}} d}+\frac {2 b \sqrt {a+b \tan (c+d x)}}{d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.37 \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=\frac {\cos (c+d x) (a-b \tan (c+d x)) \left (i \sqrt {a-i b} (a+i b) \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-i (a-i b) \sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )+2 b \sqrt {a+b \tan (c+d x)}\right )}{d (a \cos (c+d x)-b \sin (c+d x))} \]

[In]

Integrate[(-a + b*Tan[c + d*x])*Sqrt[a + b*Tan[c + d*x]],x]

[Out]

(Cos[c + d*x]*(a - b*Tan[c + d*x])*(I*Sqrt[a - I*b]*(a + I*b)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]]
- I*(a - I*b)*Sqrt[a + I*b]*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*b*Sqrt[a + b*Tan[c + d*x]]))/(
d*(a*Cos[c + d*x] - b*Sin[c + d*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(813\) vs. \(2(341)=682\).

Time = 0.07 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.93

method	result	size
parts	\(\frac {b \left (2 \sqrt {a +b \tan \left (d x +c \right )}-\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (a -\sqrt {a^{2}+b^{2}}\right ) \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right )}{4}+\frac {\left (\sqrt {a^{2}+b^{2}}-a \right ) \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right )}{d}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 b d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) \sqrt {a^{2}+b^{2}}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 b d}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a^{2}}{4 d b}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}\, a}{4 d b}+\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\)	\(814\)
derivativedivides	\(\text {Expression too large to display}\)	\(2285\)
default	\(\text {Expression too large to display}\)	\(2285\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (343) = 686\).

Time = 0.27 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.70 \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {d \sqrt {-\frac {a^{3} + a b^{2} + d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} + {\left (a d^{3} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}} + {\left (a^{2} b^{2} + b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} + a b^{2} + d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - d \sqrt {-\frac {a^{3} + a b^{2} + d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} - {\left (a d^{3} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}} + {\left (a^{2} b^{2} + b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} + a b^{2} + d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - d \sqrt {-\frac {a^{3} + a b^{2} - d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} + {\left (a d^{3} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}} - {\left (a^{2} b^{2} + b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} + a b^{2} - d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) + d \sqrt {-\frac {a^{3} + a b^{2} - d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}} \log \left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sqrt {b \tan \left (d x + c\right ) + a} - {\left (a d^{3} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}} - {\left (a^{2} b^{2} + b^{4}\right )} d\right )} \sqrt {-\frac {a^{3} + a b^{2} - d^{2} \sqrt {-\frac {a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}}{d^{4}}}}{d^{2}}}\right ) - 4 \, \sqrt {b \tan \left (d x + c\right ) + a} b}{2 \, d} \]

[In]

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

[Out]

-1/2*(d*sqrt(-(a^3 + a*b^2 + d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)*log((a^4*b + 2*a^2*b^3 + b^5)*sq
rt(b*tan(d*x + c) + a) + (a*d^3*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4) + (a^2*b^2 + b^4)*d)*sqrt(-(a^3 + a*b^2
+ d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt(-(a^3 + a*b^2 + d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 +
b^6)/d^4))/d^2)*log((a^4*b + 2*a^2*b^3 + b^5)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sqrt(-(a^4*b^2 + 2*a^2*b^4 +
b^6)/d^4) + (a^2*b^2 + b^4)*d)*sqrt(-(a^3 + a*b^2 + d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)) - d*sqrt
(-(a^3 + a*b^2 - d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)*log((a^4*b + 2*a^2*b^3 + b^5)*sqrt(b*tan(d*x
+ c) + a) + (a*d^3*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4) - (a^2*b^2 + b^4)*d)*sqrt(-(a^3 + a*b^2 - d^2*sqrt(
-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)) + d*sqrt(-(a^3 + a*b^2 - d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/
d^2)*log((a^4*b + 2*a^2*b^3 + b^5)*sqrt(b*tan(d*x + c) + a) - (a*d^3*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4) -
(a^2*b^2 + b^4)*d)*sqrt(-(a^3 + a*b^2 - d^2*sqrt(-(a^4*b^2 + 2*a^2*b^4 + b^6)/d^4))/d^2)) - 4*sqrt(b*tan(d*x +
c) + a)*b)/d

Sympy [F]

\[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=- \int a \sqrt {a + b \tan {\left (c + d x \right )}}\, dx - \int \left (- b \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}\right )\, dx \]

[In]

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

[Out]

-Integral(a*sqrt(a + b*tan(c + d*x)), x) - Integral(-b*sqrt(a + b*tan(c + d*x))*tan(c + d*x), x)

Maxima [F(-2)]

Exception generated. \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(b-a>0)', see `assume?` for mor
e details)Is

Giac [F(-1)]

Timed out. \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 10.21 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.38 \[ \int (-a+b \tan (c+d x)) \sqrt {a+b \tan (c+d x)} \, dx=\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (a^2\,b^4-a^4\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (a^3+1{}\mathrm {i}\,b\,a^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}\right )\,\sqrt {-\frac {a^3+1{}\mathrm {i}\,b\,a^2}{d^2}}}{16\,\left (a^5\,b^3+a^3\,b^5\right )}\right )\,\sqrt {-\frac {a^3+1{}\mathrm {i}\,b\,a^2}{d^2}}+\mathrm {atanh}\left (\frac {d^3\,\sqrt {\frac {-a^3+a^2\,b\,1{}\mathrm {i}}{d^2}}\,\left (\frac {16\,\left (a^2\,b^4-a^4\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}-\frac {16\,a\,b^2\,\left (-a^3+a^2\,b\,1{}\mathrm {i}\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}\right )}{16\,\left (a^5\,b^3+a^3\,b^5\right )}\right )\,\sqrt {\frac {-a^3+a^2\,b\,1{}\mathrm {i}}{d^2}}+\frac {2\,b\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d}-\mathrm {atan}\left (\frac {b^6\,\sqrt {\frac {a\,b^2}{4\,d^2}-\frac {b^3\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {b^8\,16{}\mathrm {i}}{d}+\frac {a^2\,b^6\,16{}\mathrm {i}}{d}}+\frac {32\,a\,b^5\,\sqrt {\frac {a\,b^2}{4\,d^2}-\frac {b^3\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {b^8\,16{}\mathrm {i}}{d}+\frac {a^2\,b^6\,16{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a\,b^2-b^3\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {b^6\,\sqrt {\frac {a\,b^2}{4\,d^2}+\frac {b^3\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,32{}\mathrm {i}}{\frac {b^8\,16{}\mathrm {i}}{d}+\frac {a^2\,b^6\,16{}\mathrm {i}}{d}}-\frac {32\,a\,b^5\,\sqrt {\frac {a\,b^2}{4\,d^2}+\frac {b^3\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {b^8\,16{}\mathrm {i}}{d}+\frac {a^2\,b^6\,16{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {b^3\,1{}\mathrm {i}+a\,b^2}{4\,d^2}}\,2{}\mathrm {i} \]

[In]

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

[Out]

atan((b^6*((b^3*1i)/(4*d^2) + (a*b^2)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((b^8*16i)/d + (a^2*b^6*1
6i)/d) - (32*a*b^5*((b^3*1i)/(4*d^2) + (a*b^2)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((b^8*16i)/d + (a^2*
b^6*16i)/d))*((a*b^2 + b^3*1i)/(4*d^2))^(1/2)*2i - atan((b^6*((a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2))^(1/2)*(a + b
*tan(c + d*x))^(1/2)*32i)/((b^8*16i)/d + (a^2*b^6*16i)/d) + (32*a*b^5*((a*b^2)/(4*d^2) - (b^3*1i)/(4*d^2))^(1/
2)*(a + b*tan(c + d*x))^(1/2))/((b^8*16i)/d + (a^2*b^6*16i)/d))*((a*b^2 - b^3*1i)/(4*d^2))^(1/2)*2i + atanh((d
^3*((16*(a^2*b^4 - a^4*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 + (16*a*b^2*(a^2*b*1i + a^3)*(a + b*tan(c + d*x))^
(1/2))/d^2)*(-(a^2*b*1i + a^3)/d^2)^(1/2))/(16*(a^3*b^5 + a^5*b^3)))*(-(a^2*b*1i + a^3)/d^2)^(1/2) + atanh((d^
3*((a^2*b*1i - a^3)/d^2)^(1/2)*((16*(a^2*b^4 - a^4*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 - (16*a*b^2*(a^2*b*1i
- a^3)*(a + b*tan(c + d*x))^(1/2))/d^2))/(16*(a^3*b^5 + a^5*b^3)))*((a^2*b*1i - a^3)/d^2)^(1/2) + (2*b*(a + b*
tan(c + d*x))^(1/2))/d