Integrand size = 26, antiderivative size = 89 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {i a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d}-\frac {i a^3 \sqrt {a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))} \]
[Out]
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3568, 43, 65, 212} \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {i a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d}-\frac {i a^3 \sqrt {a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))} \]
[In]
[Out]
Rule 43
Rule 65
Rule 212
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {\sqrt {a+x}}{(a-x)^2} \, dx,x,i a \tan (c+d x)\right )}{d} \\ & = -\frac {i a^3 \sqrt {a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))}+\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{2 d} \\ & = -\frac {i a^3 \sqrt {a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))}+\frac {\left (i a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d} \\ & = \frac {i a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d}-\frac {i a^3 \sqrt {a+i a \tan (c+d x)}}{d (a-i a \tan (c+d x))} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {i a^{5/2} \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} d}+\frac {a^2 \sqrt {a+i a \tan (c+d x)}}{d (i+\tan (c+d x))} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (73 ) = 146\).
Time = 18.58 (sec) , antiderivative size = 603, normalized size of antiderivative = 6.78
method | result | size |
default | \(\frac {i \left (\tan \left (d x +c \right )-i\right )^{2} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \left (i \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )-i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+i \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )-i \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right )-\operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \left (\cos ^{2}\left (d x +c \right )\right )-\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \cos \left (d x +c \right ) \sin \left (d x +c \right )-i \left (\cos ^{2}\left (d x +c \right )\right )-\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (d x +c \right )}{\left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \cos \left (d x +c \right )-\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \arctan \left (\sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\right ) \sin \left (d x +c \right )-i \cos \left (d x +c \right )+\sin \left (d x +c \right ) \cos \left (d x +c \right )\right )}{d \left (-2 i \left (\cos ^{2}\left (d x +c \right )\right )+2 \sin \left (d x +c \right ) \cos \left (d x +c \right )-i \cos \left (d x +c \right )+\sin \left (d x +c \right )+i\right )}\) | \(603\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (68) = 136\).
Time = 0.25 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.65 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\frac {\sqrt {2} \sqrt {-\frac {a^{5}}{d^{2}}} d \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2}}\right ) - \sqrt {2} \sqrt {-\frac {a^{5}}{d^{2}}} d \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {-\frac {a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{a^{2}}\right ) - 2 \, \sqrt {2} {\left (i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} + i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{4 \, d} \]
[In]
[Out]
Timed out. \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10 \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=-\frac {i \, {\left (\sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) - \frac {8 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{4}}{2 i \, a \tan \left (d x + c\right ) - 2 \, a}\right )}}{4 \, a d} \]
[In]
[Out]
\[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int { {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{2} \,d x } \]
[In]
[Out]
Timed out. \[ \int \cos ^2(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx=\int {\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
[In]
[Out]