Integrand size = 32, antiderivative size = 195 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {2 (-1)^{3/4} d^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a} f}-\frac {i (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{\sqrt {2} \sqrt {a} f}+\frac {(i c-d) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \]
2*(-1)^(3/4)*d^(3/2)*arctanh((-1)^(3/4)*d^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a ^(1/2)/(c+d*tan(f*x+e))^(1/2))/f/a^(1/2)-1/2*I*(c-I*d)^(3/2)*arctanh(2^(1/ 2)*a^(1/2)*(c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2)/(a+I*a*tan(f*x+e))^(1/2))/ f*2^(1/2)/a^(1/2)+(I*c-d)*(c+d*tan(f*x+e))^(1/2)/f/(a+I*a*tan(f*x+e))^(1/2 )
Time = 5.70 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\frac {-\frac {i \sqrt {2} (c-i d)^2 \arctan \left (\frac {\sqrt {-a (c-i d)} \sqrt {a+i a \tan (e+f x)}}{\sqrt {2} a \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a (c-i d)}}+\frac {2 i (c+i d) \sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}+\frac {4 (-1)^{3/4} (i a)^{3/2} \sqrt {i a (c+i d)} d^{3/2} \text {arcsinh}\left (\frac {\sqrt [4]{-1} \sqrt {a} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {i a} \sqrt {i a (c+i d)}}\right ) \sqrt {\frac {c+d \tan (e+f x)}{c+i d}}}{a^{5/2} \sqrt {c+d \tan (e+f x)}}}{2 f} \]
(((-I)*Sqrt[2]*(c - I*d)^2*ArcTan[(Sqrt[-(a*(c - I*d))]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[2]*a*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[-(a*(c - I*d))] + (( 2*I)*(c + I*d)*Sqrt[c + d*Tan[e + f*x]])/Sqrt[a + I*a*Tan[e + f*x]] + (4*( -1)^(3/4)*(I*a)^(3/2)*Sqrt[I*a*(c + I*d)]*d^(3/2)*ArcSinh[((-1)^(1/4)*Sqrt [a]*Sqrt[d]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[I*a]*Sqrt[I*a*(c + I*d)])]*S qrt[(c + d*Tan[e + f*x])/(c + I*d)])/(a^(5/2)*Sqrt[c + d*Tan[e + f*x]]))/( 2*f)
Time = 1.19 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {3042, 4041, 27, 3042, 4084, 3042, 4027, 221, 4082, 66, 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.
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}}dx\) |
| \(\Big \downarrow \) 4041 |
| \(\displaystyle \frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}-\frac {\int -\frac {\sqrt {i \tan (e+f x) a+a} \left (a \left (c^2-2 i d c+d^2\right )-2 i a d^2 \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{a^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (a \left (c^2-2 i d c+d^2\right )-2 i a d^2 \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sqrt {i \tan (e+f x) a+a} \left (a \left (c^2-2 i d c+d^2\right )-2 i a d^2 \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4084 |
| \(\displaystyle \frac {2 d^2 \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a (c-i d)^2 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 d^2 \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx+a (c-i d)^2 \int \frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle \frac {2 d^2 \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {2 i a^3 (c-i d)^2 \int \frac {1}{a (c-i d)-\frac {2 a^2 (c+d \tan (e+f x))}{i \tan (e+f x) a+a}}d\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {i \tan (e+f x) a+a}}}{f}}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 d^2 \int \frac {(a-i a \tan (e+f x)) \sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}dx-\frac {i \sqrt {2} a^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 4082 |
| \(\displaystyle \frac {\frac {2 a^2 d^2 \int \frac {1}{\sqrt {i \tan (e+f x) a+a} \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f}-\frac {i \sqrt {2} a^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {\frac {4 a^2 d^2 \int \frac {1}{i a-\frac {d (i \tan (e+f x) a+a)}{c+d \tan (e+f x)}}d\frac {\sqrt {i \tan (e+f x) a+a}}{\sqrt {c+d \tan (e+f x)}}}{f}-\frac {i \sqrt {2} a^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {4 (-1)^{3/4} a^{3/2} d^{3/2} \text {arctanh}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {i \sqrt {2} a^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}}{2 a^2}+\frac {(-d+i c) \sqrt {c+d \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\) |
((4*(-1)^(3/4)*a^(3/2)*d^(3/2)*ArcTanh[((-1)^(3/4)*Sqrt[d]*Sqrt[a + I*a*Ta n[e + f*x]])/(Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])])/f - (I*Sqrt[2]*a^(3/2)*( c - I*d)^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[a]*Sqrt[c + d*Tan[e + f*x]])/(Sqrt[c - I*d]*Sqrt[a + I*a*Tan[e + f*x]])])/f)/(2*a^2) + ((I*c - d)*Sqrt[c + d*Ta n[e + f*x]])/(f*Sqrt[a + I*a*Tan[e + f*x]])
3.12.46.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*c - a*d))*(a + b*Tan[e + f*x])^m* ((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*m)), x] + Simp[1/(2*a^2*m) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[c*(a*c*m + b*d*(n - 1)) - d*(b*c*m + a*d*(n - 1)) - d*(b*d*(m - n + 1) - a*c*(m + n - 1))*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, 0] && GtQ[n, 1] && (In tegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[b*(B/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[ a^2 + b^2, 0] && EqQ[A*b + a*B, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(A*b + a*B)/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n, x], x] - Simp[B/b Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^n*(a - b*Tan[ e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[A*b + a*B, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1168 vs. \(2 (152 ) = 304\).
Time = 0.85 (sec) , antiderivative size = 1169, normalized size of antiderivative = 5.99
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1169\) |
| default | \(\text {Expression too large to display}\) | \(1169\) |
-1/4/f/a*(I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x +e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+ e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c*tan(f*x+e)^2-2*I*(I*a*d)^(1 /2)*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan( f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^ (1/2))/(tan(f*x+e)+I))*d*tan(f*x+e)+(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/ 2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c) )^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*d*tan (f*x+e)^2-I*(I*a*d)^(1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x +e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+ e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x+e)+I))*c+2*(I*a*d)^(1/2)*2^(1/2)*(-a *(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*tan(f*x+e)+2*2^(1/2 )*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2))/(tan(f*x +e)+I))*c*tan(f*x+e)+8*I*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f* x+e))*(c+d*tan(f*x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^2*tan( f*x+e)-4*ln(1/2*(2*I*a*d*tan(f*x+e)+I*a*c+2*(a*(1+I*tan(f*x+e))*(c+d*tan(f *x+e)))^(1/2)*(I*a*d)^(1/2)+a*d)/(I*a*d)^(1/2))*a*d^2*tan(f*x+e)^2-4*c*(I* a*d)^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+e)))^(1/2)*tan(f*x+e)-(I*a*d)^ (1/2)*2^(1/2)*(-a*(I*d-c))^(1/2)*ln((3*a*c+I*a*tan(f*x+e)*c-I*a*d+3*a*d*ta n(f*x+e)+2*2^(1/2)*(-a*(I*d-c))^(1/2)*(a*(1+I*tan(f*x+e))*(c+d*tan(f*x+...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (145) = 290\).
Time = 0.32 (sec) , antiderivative size = 895, normalized size of antiderivative = 4.59 \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Too large to display} \]
-1/4*(sqrt(2)*a*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a*f^2))*e^(I* f*x + I*e)*log((sqrt(2)*a*f*sqrt(-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a*f ^2))*e^(I*f*x + I*e) + sqrt(2)*((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c + d)*s qrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*s qrt(a/(e^(2*I*f*x + 2*I*e) + 1)))/(I*c + d)) - sqrt(2)*a*f*sqrt(-(c^3 - 3* I*c^2*d - 3*c*d^2 + I*d^3)/(a*f^2))*e^(I*f*x + I*e)*log(-(sqrt(2)*a*f*sqrt (-(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)/(a*f^2))*e^(I*f*x + I*e) - sqrt(2)*( (I*c + d)*e^(2*I*f*x + 2*I*e) + I*c + d)*sqrt(((c - I*d)*e^(2*I*f*x + 2*I* e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) )/(I*c + d)) + a*f*sqrt(-4*I*d^3/(a*f^2))*e^(I*f*x + I*e)*log(2*(4*sqrt(2) *(d^3*e^(3*I*f*x + 3*I*e) + d^3*e^(I*f*x + I*e))*sqrt(((c - I*d)*e^(2*I*f* x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(a/(e^(2*I*f*x + 2*I* e) + 1)) - ((I*a*c*d + 3*a*d^2)*f*e^(2*I*f*x + 2*I*e) + (I*a*c*d - a*d^2)* f)*sqrt(-4*I*d^3/(a*f^2)))/(I*c^3 + c^2*d + I*c*d^2 + d^3 + (I*c^3 + c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) - a*f*sqrt(-4*I*d^3/(a*f^2))*e^(I* f*x + I*e)*log(2*(4*sqrt(2)*(d^3*e^(3*I*f*x + 3*I*e) + d^3*e^(I*f*x + I*e) )*sqrt(((c - I*d)*e^(2*I*f*x + 2*I*e) + c + I*d)/(e^(2*I*f*x + 2*I*e) + 1) )*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1)) - ((-I*a*c*d - 3*a*d^2)*f*e^(2*I*f*x + 2*I*e) + (-I*a*c*d + a*d^2)*f)*sqrt(-4*I*d^3/(a*f^2)))/(I*c^3 + c^2*d + I *c*d^2 + d^3 + (I*c^3 + c^2*d + I*c*d^2 + d^3)*e^(2*I*f*x + 2*I*e))) + ...
\[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \]
\[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int { \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{\sqrt {i \, a \tan \left (f x + e\right ) + a}} \,d x } \]
Exception generated. \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Non regular value [0,0] was discard ed and replaced randomly by 0=[77,-80]Warning, replacing 77 by -85, a subs titution
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx=\int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \,d x \]