Integrand size = 16, antiderivative size = 68 \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{2 d}+\frac {a \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{2 d} \]
1/2*a^(3/2)*arctanh(a^(1/2)*tan(d*x+c)/(a*sec(d*x+c)^2)^(1/2))/d+1/2*a*(a* sec(d*x+c)^2)^(1/2)*tan(d*x+c)/d
Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63 \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\frac {a \sqrt {a \sec ^2(c+d x)} (\text {arctanh}(\sin (c+d x)) \cos (c+d x)+\tan (c+d x))}{2 d} \]
Time = 0.25 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {3042, 4140, 3042, 4610, 211, 224, 219}
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.
\(\displaystyle \int \left (a \tan ^2(c+d x)+a\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \tan (c+d x)^2+a\right )^{3/2}dx\) |
\(\Big \downarrow \) 4140 |
\(\displaystyle \int \left (a \sec ^2(c+d x)\right )^{3/2}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (c+d x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle \frac {a \int \sqrt {a \tan ^2(c+d x)+a}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {a \left (\frac {1}{2} a \int \frac {1}{\sqrt {a \tan ^2(c+d x)+a}}d\tan (c+d x)+\frac {1}{2} \tan (c+d x) \sqrt {a \tan ^2(c+d x)+a}\right )}{d}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {a \left (\frac {1}{2} a \int \frac {1}{1-\frac {a \tan ^2(c+d x)}{a \tan ^2(c+d x)+a}}d\frac {\tan (c+d x)}{\sqrt {a \tan ^2(c+d x)+a}}+\frac {1}{2} \tan (c+d x) \sqrt {a \tan ^2(c+d x)+a}\right )}{d}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {a \left (\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \tan ^2(c+d x)+a}}\right )+\frac {1}{2} \tan (c+d x) \sqrt {a \tan ^2(c+d x)+a}\right )}{d}\) |
(a*((Sqrt[a]*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Tan[c + d*x]^2]])/2 + (Tan[c + d*x]*Sqrt[a + a*Tan[c + d*x]^2])/2))/d
3.3.71.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*sec[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a, b]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\tan \left (d x +c \right ) \sqrt {a +a \tan \left (d x +c \right )^{2}}}{2}+\frac {\sqrt {a}\, \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \tan \left (d x +c \right )^{2}}\right )}{2}\right )}{d}\) | \(60\) |
default | \(\frac {a \left (\frac {\tan \left (d x +c \right ) \sqrt {a +a \tan \left (d x +c \right )^{2}}}{2}+\frac {\sqrt {a}\, \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \tan \left (d x +c \right )^{2}}\right )}{2}\right )}{d}\) | \(60\) |
risch | \(-\frac {i a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) d}-\frac {\ln \left ({\mathrm e}^{i d x}-i {\mathrm e}^{-i c}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, a \cos \left (d x +c \right )}{d}+\frac {\ln \left ({\mathrm e}^{i d x}+i {\mathrm e}^{-i c}\right ) \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, a \cos \left (d x +c \right )}{d}\) | \(166\) |
1/d*a*(1/2*tan(d*x+c)*(a+a*tan(d*x+c)^2)^(1/2)+1/2*a^(1/2)*ln(a^(1/2)*tan( d*x+c)+(a+a*tan(d*x+c)^2)^(1/2)))
Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.06 \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\frac {a^{\frac {3}{2}} \log \left (2 \, a \tan \left (d x + c\right )^{2} + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} \sqrt {a} \tan \left (d x + c\right ) + a\right ) + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} a \tan \left (d x + c\right )}{4 \, d} \]
1/4*(a^(3/2)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a) *tan(d*x + c) + a) + 2*sqrt(a*tan(d*x + c)^2 + a)*a*tan(d*x + c))/d
\[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\int \left (a \tan ^{2}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 556 vs. \(2 (56) = 112\).
Time = 0.36 (sec) , antiderivative size = 556, normalized size of antiderivative = 8.18 \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=-\frac {{\left (8 \, a \cos \left (3 \, d x + 3 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - 8 \, a \cos \left (d x + c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 8 \, a \cos \left (2 \, d x + 2 \, c\right ) \sin \left (d x + c\right ) - 4 \, {\left (a \sin \left (3 \, d x + 3 \, c\right ) - a \sin \left (d x + c\right )\right )} \cos \left (4 \, d x + 4 \, c\right ) - {\left (a \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \cos \left (2 \, d x + 2 \, c\right )^{2} + a \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, a \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \cos \left (4 \, d x + 4 \, c\right ) + 4 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + {\left (a \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \cos \left (2 \, d x + 2 \, c\right )^{2} + a \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, a \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \cos \left (4 \, d x + 4 \, c\right ) + 4 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (a \cos \left (3 \, d x + 3 \, c\right ) - a \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) - 4 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \sin \left (3 \, d x + 3 \, c\right ) + 4 \, a \sin \left (d x + c\right )\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 4 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
-1/4*(8*a*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 8*a*cos(d*x + c)*sin(2*d*x + 2*c) + 8*a*cos(2*d*x + 2*c)*sin(d*x + c) - 4*(a*sin(3*d*x + 3*c) - a*sin( d*x + c))*cos(4*d*x + 4*c) - (a*cos(4*d*x + 4*c)^2 + 4*a*cos(2*d*x + 2*c)^ 2 + a*sin(4*d*x + 4*c)^2 + 4*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*sin (2*d*x + 2*c)^2 + 2*(2*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 4*a*cos( 2*d*x + 2*c) + a)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1 ) + (a*cos(4*d*x + 4*c)^2 + 4*a*cos(2*d*x + 2*c)^2 + a*sin(4*d*x + 4*c)^2 + 4*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*sin(2*d*x + 2*c)^2 + 2*(2*a* cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*log(cos (d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(a*cos(3*d*x + 3*c) - a*cos(d*x + c))*sin(4*d*x + 4*c) - 4*(2*a*cos(2*d*x + 2*c) + a)*sin(3*d *x + 3*c) + 4*a*sin(d*x + c))*sqrt(a)/((2*(2*cos(2*d*x + 2*c) + 1)*cos(4*d *x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d* x + 2*c) + 1)*d)
Leaf count of result is larger than twice the leaf count of optimal. 2125 vs. \(2 (56) = 112\).
Time = 1.77 (sec) , antiderivative size = 2125, normalized size of antiderivative = 31.25 \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\text {Too large to display} \]
1/2*((a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c )^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/ 2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) - a^(3/2 )*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/ 2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan (1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*tan(1/2* c) + tan(1/2*d*x) + tan(1/2*c) + 1))/(tan(1/2*c) - 1) - (a^(3/2)*sgn(tan(1 /2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) + a^(3/2)*sgn(tan(1/2*d*x)^4*ta n(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d* x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d *x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan (1/2*c) + 1))/(tan(1/2*c) + 1) - 2*(a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^ 4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan( 1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1 /2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^8 + 6*a^(3/2)*sgn(tan(1/2*d*x)^4*tan( 1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x) ^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x )*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^6 + 2*a^(3/2)*sgn(tan(1/2*d...
Timed out. \[ \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{3/2} \,d x \]