Integrand size = 27, antiderivative size = 272 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) (c-i d)^{5/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) (c+i d)^{5/2} f}-\frac {2 b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{5/2} f}+\frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \]
[Out]
Time = 2.05 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {3650, 3730, 3734, 3620, 3618, 65, 214, 3715} \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=-\frac {2 b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) (b c-a d)^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a) (c-i d)^{5/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a) (c+i d)^{5/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{f \left (c^2+d^2\right )^2 (b c-a d)^2 \sqrt {c+d \tan (e+f x)}}+\frac {2 d^2}{3 f \left (c^2+d^2\right ) (b c-a d) (c+d \tan (e+f x))^{3/2}} \]
[In]
[Out]
Rule 65
Rule 214
Rule 3618
Rule 3620
Rule 3650
Rule 3715
Rule 3730
Rule 3734
Rubi steps \begin{align*} \text {integral}& = \frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}+\frac {2 \int \frac {-\frac {3}{2} \left (a c d-b \left (c^2+d^2\right )\right )-\frac {3}{2} d (b c-a d) \tan (e+f x)+\frac {3}{2} b d^2 \tan ^2(e+f x)}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{3/2}} \, dx}{3 (b c-a d) \left (c^2+d^2\right )} \\ & = \frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {4 \int \frac {-\frac {3}{4} \left (2 a b c^3 d-a^2 d^2 \left (c^2-d^2\right )-b^2 \left (c^2+d^2\right )^2\right )-\frac {3}{2} c d (b c-a d)^2 \tan (e+f x)-\frac {3}{4} b d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{3 (b c-a d)^2 \left (c^2+d^2\right )^2} \\ & = \frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {b^4 \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{\left (a^2+b^2\right ) (b c-a d)^2}+\frac {4 \int \frac {-\frac {3}{4} (b c-a d)^2 \left (2 b c d-a \left (c^2-d^2\right )\right )-\frac {3}{4} (b c-a d)^2 \left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{3 \left (a^2+b^2\right ) (b c-a d)^2 \left (c^2+d^2\right )^2} \\ & = \frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}+\frac {\int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b) (c-i d)^2}+\frac {\int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b) (c+i d)^2}+\frac {b^4 \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) (b c-a d)^2 f} \\ & = \frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c-i d x}} \, dx,x,i \tan (e+f x)\right )}{2 (i a+b) (c-i d)^2 f}+\frac {\text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {c+i d x}} \, dx,x,-i \tan (e+f x)\right )}{2 (i a-b) (c+i d)^2 f}+\frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{\left (a^2+b^2\right ) d (b c-a d)^2 f} \\ & = -\frac {2 b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{5/2} f}+\frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}}-\frac {\text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a-i b) (c-i d)^2 d f}-\frac {\text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{(a+i b) (c+i d)^2 d f} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) (c-i d)^{5/2} f}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) (c+i d)^{5/2} f}-\frac {2 b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (b c-a d)^{5/2} f}+\frac {2 d^2}{3 (b c-a d) \left (c^2+d^2\right ) f (c+d \tan (e+f x))^{3/2}}-\frac {2 d^2 \left (2 a c d-b \left (3 c^2+d^2\right )\right )}{(b c-a d)^2 \left (c^2+d^2\right )^2 f \sqrt {c+d \tan (e+f x)}} \\ \end{align*}
Time = 5.48 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\frac {\frac {3 \left (\frac {(-i a+b) (c+i d)^2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {(i a+b) (c-i d)^2 (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {2 b^{7/2} \left (c^2+d^2\right )^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) (-b c+a d) \left (c^2+d^2\right )}-\frac {2 d^2}{(c+d \tan (e+f x))^{3/2}}-\frac {6 d^2 \left (-2 a c d+b \left (3 c^2+d^2\right )\right )}{(b c-a d) \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{3 (-b c+a d) \left (c^2+d^2\right ) f} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(13966\) vs. \(2(238)=476\).
Time = 0.95 (sec) , antiderivative size = 13967, normalized size of antiderivative = 51.35
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(13967\) |
default | \(\text {Expression too large to display}\) | \(13967\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 15564 vs. \(2 (232) = 464\).
Time = 190.87 (sec) , antiderivative size = 31143, normalized size of antiderivative = 114.50 \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right ) \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(a+b \tan (e+f x)) (c+d \tan (e+f x))^{5/2}} \, dx=\text {Hanged} \]
[In]
[Out]