Integrand size = 33, antiderivative size = 90 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f} \]
[Out]
Time = 0.18 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3603, 3568, 45} \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=\frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}-\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}} \]
[In]
[Out]
Rule 45
Rule 3568
Rule 3603
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\sec ^6(e+f x)}{(c-i c \tan (e+f x))^{7/2}} \, dx \\ & = \frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {(c-x)^2}{(c+x)^{3/2}} \, dx,x,-i c \tan (e+f x)\right )}{c^2 f} \\ & = \frac {\left (i a^3\right ) \text {Subst}\left (\int \left (\frac {4 c^2}{(c+x)^{3/2}}-\frac {4 c}{\sqrt {c+x}}+\sqrt {c+x}\right ) \, dx,x,-i c \tan (e+f x)\right )}{c^2 f} \\ & = -\frac {8 i a^3}{f \sqrt {c-i c \tan (e+f x)}}-\frac {8 i a^3 \sqrt {c-i c \tan (e+f x)}}{c f}+\frac {2 i a^3 (c-i c \tan (e+f x))^{3/2}}{3 c^2 f} \\ \end{align*}
Time = 2.20 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.54 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {2 i a^3 \left (23-10 i \tan (e+f x)+\tan ^2(e+f x)\right )}{3 f \sqrt {c-i c \tan (e+f x)}} \]
[In]
[Out]
Time = 0.65 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-4 c \sqrt {c -i c \tan \left (f x +e \right )}-\frac {4 c^{2}}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(66\) |
default | \(\frac {2 i a^{3} \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-4 c \sqrt {c -i c \tan \left (f x +e \right )}-\frac {4 c^{2}}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}\) | \(66\) |
parts | \(\frac {2 i a^{3} c \left (\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}-\frac {1}{2 c \sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f}-\frac {2 i a^{3} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+c \sqrt {c -i c \tan \left (f x +e \right )}+\frac {c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4}+\frac {c^{2}}{2 \sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f \,c^{2}}+\frac {3 i a^{3} \left (-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {1}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f}-\frac {6 i a^{3} \left (\sqrt {c -i c \tan \left (f x +e \right )}-\frac {\sqrt {c}\, \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4}+\frac {c}{2 \sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f c}\) | \(290\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.82 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {4 \, \sqrt {2} {\left (3 i \, a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 12 i \, a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 i \, a^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3 \, {\left (c f e^{\left (2 i \, f x + 2 i \, e\right )} + c f\right )}} \]
[In]
[Out]
\[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=- i a^{3} \left (\int \frac {i}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {\tan ^{3}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {3 i \tan ^{2}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.78 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {2 i \, {\left (\frac {12 \, a^{3} c}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} - \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}} a^{3} - 12 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} a^{3} c}{c}\right )}}{3 \, c f} \]
[In]
[Out]
\[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=\int { \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}} \,d x } \]
[In]
[Out]
Time = 5.94 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.26 \[ \int \frac {(a+i a \tan (e+f x))^3}{\sqrt {c-i c \tan (e+f x)}} \, dx=-\frac {2\,a^3\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,23{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}-7\,\sin \left (2\,e+2\,f\,x\right )-3\,\sin \left (4\,e+4\,f\,x\right )+20{}\mathrm {i}\right )}{3\,c\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
[In]
[Out]