Integrand size = 28, antiderivative size = 107 \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\frac {i 2^{\frac {7+m}{2}} a \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-5-m),\frac {m}{2},\frac {2+m}{2},\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^m (1+i \tan (c+d x))^{\frac {1}{2} (-5-m)} (a+i a \tan (c+d x))^{5/2}}{d m} \] Output:
I*2^(7/2+1/2*m)*a*hypergeom([1/2*m, -5/2-1/2*m],[1+1/2*m],1/2-1/2*I*tan(d* x+c))*(e*sec(d*x+c))^m*(1+I*tan(d*x+c))^(-5/2-1/2*m)*(a+I*a*tan(d*x+c))^(5 /2)/d/m
Time = 3.00 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.74 \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=-\frac {i 2^{\frac {7}{2}+m} e^{3 i (c+2 d x)} \sqrt {e^{i d x}} \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{\frac {1}{2}+m} \left (1+e^{2 i (c+d x)}\right )^{\frac {1}{2}+m} \operatorname {Hypergeometric2F1}\left (\frac {7}{2}+m,\frac {7+m}{2},\frac {9+m}{2},-e^{2 i (c+d x)}\right ) \sec ^{-\frac {7}{2}-m}(c+d x) (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2}}{d (7+m) (\cos (d x)+i \sin (d x))^{7/2}} \] Input:
Integrate[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
((-I)*2^(7/2 + m)*E^((3*I)*(c + 2*d*x))*Sqrt[E^(I*d*x)]*(E^(I*(c + d*x))/( 1 + E^((2*I)*(c + d*x))))^(1/2 + m)*(1 + E^((2*I)*(c + d*x)))^(1/2 + m)*Hy pergeometric2F1[7/2 + m, (7 + m)/2, (9 + m)/2, -E^((2*I)*(c + d*x))]*Sec[c + d*x]^(-7/2 - m)*(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2))/(d*(7 + m)*(Cos[d*x] + I*Sin[d*x])^(7/2))
Time = 0.52 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 3986, 3042, 4006, 80, 79}
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.
\(\displaystyle \int (a+i a \tan (c+d x))^{7/2} (e \sec (c+d x))^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (a+i a \tan (c+d x))^{7/2} (e \sec (c+d x))^mdx\) |
\(\Big \downarrow \) 3986 |
\(\displaystyle (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \int (a-i a \tan (c+d x))^{m/2} (i \tan (c+d x) a+a)^{\frac {m+7}{2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \int (a-i a \tan (c+d x))^{m/2} (i \tan (c+d x) a+a)^{\frac {m+7}{2}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a^2 (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{-m/2} (e \sec (c+d x))^m \int (a-i a \tan (c+d x))^{\frac {m-2}{2}} (i \tan (c+d x) a+a)^{\frac {m+5}{2}}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 80 |
\(\displaystyle \frac {a^4 2^{\frac {m+5}{2}} (1+i \tan (c+d x))^{\frac {1}{2} (-m-1)} (a-i a \tan (c+d x))^{-m/2} (a+i a \tan (c+d x))^{\frac {m+1}{2}-\frac {m}{2}} (e \sec (c+d x))^m \int \left (\frac {1}{2} i \tan (c+d x)+\frac {1}{2}\right )^{\frac {m+5}{2}} (a-i a \tan (c+d x))^{\frac {m-2}{2}}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {i a^3 2^{\frac {m+5}{2}+1} (1+i \tan (c+d x))^{\frac {1}{2} (-m-1)} (a+i a \tan (c+d x))^{\frac {m+1}{2}-\frac {m}{2}} (e \sec (c+d x))^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (-m-5),\frac {m}{2},\frac {m+2}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d m}\) |
Input:
Int[(e*Sec[c + d*x])^m*(a + I*a*Tan[c + d*x])^(7/2),x]
Output:
(I*2^(1 + (5 + m)/2)*a^3*Hypergeometric2F1[(-5 - m)/2, m/2, (2 + m)/2, (1 - I*Tan[c + d*x])/2]*(e*Sec[c + d*x])^m*(1 + I*Tan[c + d*x])^((-1 - m)/2)* (a + I*a*Tan[c + d*x])^(-1/2*m + (1 + m)/2))/(d*m)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_.), x_Symbol] :> Simp[(d*Sec[e + f*x])^m/((a + b*Tan[e + f*x])^(m/ 2)*(a - b*Tan[e + f*x])^(m/2)) Int[(a + b*Tan[e + f*x])^(m/2 + n)*(a - b* Tan[e + f*x])^(m/2), x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
\[\int \left (e \sec \left (d x +c \right )\right )^{m} \left (a +i a \tan \left (d x +c \right )\right )^{\frac {7}{2}}d x\]
Input:
int((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x)
Output:
int((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x)
\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \left (e \sec \left (d x + c\right )\right )^{m} \,d x } \] Input:
integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="fricas")
Output:
integral(8*sqrt(2)*a^3*(2*e*e^(I*d*x + I*c)/(e^(2*I*d*x + 2*I*c) + 1))^m*s qrt(a/(e^(2*I*d*x + 2*I*c) + 1))*e^(7*I*d*x + 7*I*c)/(e^(6*I*d*x + 6*I*c) + 3*e^(4*I*d*x + 4*I*c) + 3*e^(2*I*d*x + 2*I*c) + 1), x)
Timed out. \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\text {Timed out} \] Input:
integrate((e*sec(d*x+c))**m*(a+I*a*tan(d*x+c))**(7/2),x)
Output:
Timed out
\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} \left (e \sec \left (d x + c\right )\right )^{m} \,d x } \] Input:
integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="maxima")
Output:
integrate((I*a*tan(d*x + c) + a)^(7/2)*(e*sec(d*x + c))^m, x)
Exception generated. \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Timed out. \[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^m\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \] Input:
int((e/cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^(7/2),x)
Output:
int((e/cos(c + d*x))^m*(a + a*tan(c + d*x)*1i)^(7/2), x)
\[ \int (e \sec (c+d x))^m (a+i a \tan (c+d x))^{7/2} \, dx=\frac {e^{m} \sqrt {a}\, a^{3} \left (-2 \sec \left (d x +c \right )^{m} \sqrt {\tan \left (d x +c \right ) i +1}\, i -\left (\int \sec \left (d x +c \right )^{m} \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{3}d x \right ) d i -3 \left (\int \sec \left (d x +c \right )^{m} \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )^{2}d x \right ) d +2 \left (\int \sec \left (d x +c \right )^{m} \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )d x \right ) d i m +4 \left (\int \sec \left (d x +c \right )^{m} \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right )d x \right ) d i \right )}{d} \] Input:
int((e*sec(d*x+c))^m*(a+I*a*tan(d*x+c))^(7/2),x)
Output:
(e**m*sqrt(a)*a**3*( - 2*sec(c + d*x)**m*sqrt(tan(c + d*x)*i + 1)*i - int( sec(c + d*x)**m*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**3,x)*d*i - 3*int(se c(c + d*x)**m*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)**2,x)*d + 2*int(sec(c + d*x)**m*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x),x)*d*i*m + 4*int(sec(c + d *x)**m*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x),x)*d*i))/d