Integrand size = 27, antiderivative size = 150 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {i (a-i b)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}+\frac {i (a+i b)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 (b c-a d)^2}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \] Output:
-I*(a-I*b)^2*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(c-I*d)^(3/2)/f +I*(a+I*b)^2*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(c+I*d)^(3/2)/f -2*(-a*d+b*c)^2/d/(c^2+d^2)/f/(c+d*tan(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {-\frac {2 b^2}{d}-\frac {(a-i b)^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )}{i c+d}+\frac {(a+i b)^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )}{i c-d}}{f \sqrt {c+d \tan (e+f x)}} \] Input:
Integrate[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]
Output:
((-2*b^2)/d - ((a - I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)])/(I*c + d) + ((a + I*b)^2*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c + I*d)])/(I*c - d))/(f*Sqrt[c + d*Tan[e + f*x]])
Time = 0.74 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.02, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4025, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
\(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 4025 |
\(\displaystyle \frac {\int \frac {c a^2+2 b d a-b^2 c+\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}-\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {c a^2+2 b d a-b^2 c+\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}-\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {1}{2} (a-i b)^2 (c+i d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c-i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {1}{2} (a-i b)^2 (c+i d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c-i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{c^2+d^2}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {i (a-i b)^2 (c+i d) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a+i b)^2 (c-i d) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}}{c^2+d^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {i (a+i b)^2 (c-i d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a-i b)^2 (c+i d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}}{c^2+d^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {(a-i b)^2 (c+i d) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b)^2 (c-i d) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{c^2+d^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 (b c-a d)^2}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}+\frac {\frac {(a-i b)^2 (c+i d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a+i b)^2 (c-i d) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{c^2+d^2}\) |
Input:
Int[(a + b*Tan[e + f*x])^2/(c + d*Tan[e + f*x])^(3/2),x]
Output:
(((a - I*b)^2*(c + I*d)*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d] *f) + ((a + I*b)^2*(c - I*d)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f))/(c^2 + d^2) - (2*(b*c - a*d)^2)/(d*(c^2 + d^2)*f*Sqrt[c + d*Tan[ e + f*x]])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(5618\) vs. \(2(128)=256\).
Time = 0.45 (sec) , antiderivative size = 5619, normalized size of antiderivative = 37.46
method | result | size |
parts | \(\text {Expression too large to display}\) | \(5619\) |
derivativedivides | \(\text {Expression too large to display}\) | \(11598\) |
default | \(\text {Expression too large to display}\) | \(11598\) |
Input:
int((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 6669 vs. \(2 (120) = 240\).
Time = 2.19 (sec) , antiderivative size = 6669, normalized size of antiderivative = 44.46 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{2}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*tan(f*x+e))**2/(c+d*tan(f*x+e))**(3/2),x)
Output:
Integral((a + b*tan(e + f*x))**2/(c + d*tan(e + f*x))**(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
Output:
Timed out
Exception generated. \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[3,9,3]%%%}+%%%{4,[3,7,3]%%%}+%%%{6,[3,5,3]%%%}+%%%{ 4,[3,3,3]
Time = 8.89 (sec) , antiderivative size = 7613, normalized size of antiderivative = 50.75 \[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \] Input:
int((a + b*tan(e + f*x))^2/(c + d*tan(e + f*x))^(3/2),x)
Output:
atan((((c + d*tan(e + f*x))^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96* a^2*b^2*d^10*f^3 + 32*a^4*c^2*d^8*f^3 - 32*a^4*c^6*d^4*f^3 - 16*a^4*c^8*d^ 2*f^3 + 32*b^4*c^2*d^8*f^3 - 32*b^4*c^6*d^4*f^3 - 16*b^4*c^8*d^2*f^3 - 192 *a^2*b^2*c^2*d^8*f^3 + 192*a^2*b^2*c^6*d^4*f^3 + 96*a^2*b^2*c^8*d^2*f^3 + 128*a*b^3*c*d^9*f^3 - 128*a^3*b*c*d^9*f^3 + 384*a*b^3*c^3*d^7*f^3 + 384*a* b^3*c^5*d^5*f^3 + 128*a*b^3*c^7*d^3*f^3 - 384*a^3*b*c^3*d^7*f^3 - 384*a^3* b*c^5*d^5*f^3 - 128*a^3*b*c^7*d^3*f^3) - (-(4*a*b^3 - 4*a^3*b + a^4*1i + b ^4*1i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2) ))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1 i - a^2*b^2*6i)/(4*(c^3*f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^( 1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 64*a^2*c*d^11*f^4 + 64*b^2*c*d^11* f^4 - 256*a^2*c^3*d^9*f^4 - 384*a^2*c^5*d^7*f^4 - 256*a^2*c^7*d^5*f^4 - 64 *a^2*c^9*d^3*f^4 + 256*b^2*c^3*d^9*f^4 + 384*b^2*c^5*d^7*f^4 + 256*b^2*c^7 *d^5*f^4 + 64*b^2*c^9*d^3*f^4 - 64*a*b*d^12*f^4 - 192*a*b*c^2*d^10*f^4 - 1 28*a*b*c^4*d^8*f^4 + 128*a*b*c^6*d^6*f^4 + 192*a*b*c^8*d^4*f^4 + 64*a*b*c^ 10*d^2*f^4))*(-(4*a*b^3 - 4*a^3*b + a^4*1i + b^4*1i - a^2*b^2*6i)/(4*(c^3* f^2*1i + d^3*f^2 - c*d^2*f^2*3i - 3*c^2*d*f^2)))^(1/2)*1i + ((c + d*tan(e + f*x))^(1/2)*(16*a^4*d^10*f^3 + 16*b^4*d^10*f^3 - 96*a^2*b^2*d^10*f^3 + 3 2*a^4*c^2*d^8*f^3 - 32*a^4*c^6*d^4*f^3 - 16*a^4*c^8*d^2*f^3 + 32*b^4*c^...
\[ \int \frac {(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {-2 \sqrt {d \tan \left (f x +e \right )+c}\, a^{2}-\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) \tan \left (f x +e \right ) a^{2} d^{2} f +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) \tan \left (f x +e \right ) b^{2} d^{2} f -\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) a^{2} c d f +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) b^{2} c d f +2 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) \tan \left (f x +e \right ) a b \,d^{2} f +2 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )}{\tan \left (f x +e \right )^{2} d^{2}+2 \tan \left (f x +e \right ) c d +c^{2}}d x \right ) a b c d f}{d f \left (d \tan \left (f x +e \right )+c \right )} \] Input:
int((a+b*tan(f*x+e))^2/(c+d*tan(f*x+e))^(3/2),x)
Output:
( - 2*sqrt(tan(e + f*x)*d + c)*a**2 - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f *x)*a**2*d**2*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*b**2*d**2*f - in t((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan (e + f*x)*c*d + c**2),x)*a**2*c*d*f + int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*b**2*c*d* f + 2*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c**2),x)*tan(e + f*x)*a*b*d**2*f + 2*int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e + f*x)**2*d**2 + 2*tan(e + f*x)*c*d + c **2),x)*a*b*c*d*f)/(d*f*(tan(e + f*x)*d + c))