Integrand size = 35, antiderivative size = 266 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {(47 A+24 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}} \]
[Out]
Time = 1.06 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3121, 3063, 3064, 2728, 212, 2852} \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=-\frac {(47 A+24 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a \cos (c+d x)+a}}+\frac {(5 A+3 C) \tan (c+d x) \sec ^2(c+d x)}{6 a d \sqrt {a \cos (c+d x)+a}}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}-\frac {(13 A+6 C) \tan (c+d x) \sec (c+d x)}{12 a d \sqrt {a \cos (c+d x)+a}} \]
[In]
[Out]
Rule 212
Rule 2728
Rule 2852
Rule 3063
Rule 3064
Rule 3121
Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (a (5 A+3 C)-\frac {1}{2} a (7 A+3 C) \cos (c+d x)\right ) \sec ^4(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2} \\ & = -\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-a^2 (13 A+6 C)+\frac {5}{2} a^2 (5 A+3 C) \cos (c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{6 a^3} \\ & = -\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (\frac {9}{2} a^3 (7 A+4 C)-\frac {3}{2} a^3 (13 A+6 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{12 a^4} \\ & = \frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-\frac {3}{4} a^4 (47 A+24 C)+\frac {9}{4} a^4 (7 A+4 C) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{12 a^5} \\ & = \frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}+\frac {(17 A+9 C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a}-\frac {(47 A+24 C) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx}{16 a^2} \\ & = \frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}}-\frac {(17 A+9 C) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{2 a d}+\frac {(47 A+24 C) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{8 a d} \\ & = -\frac {(47 A+24 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \text {arctanh}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {3 (7 A+4 C) \tan (c+d x)}{8 a d \sqrt {a+a \cos (c+d x)}}-\frac {(13 A+6 C) \sec (c+d x) \tan (c+d x)}{12 a d \sqrt {a+a \cos (c+d x)}}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(5 A+3 C) \sec ^2(c+d x) \tan (c+d x)}{6 a d \sqrt {a+a \cos (c+d x)}} \\ \end{align*}
Time = 3.01 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.77 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {-48 (17 A+9 C) \text {arctanh}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^5\left (\frac {1}{2} (c+d x)\right )+12 \sqrt {2} (47 A+24 C) \text {arctanh}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^5\left (\frac {1}{2} (c+d x)\right )-\cos ^3\left (\frac {1}{2} (c+d x)\right ) (106 A+48 C+3 (55 A+36 C) \cos (c+d x)+(74 A+48 C) \cos (2 (c+d x))+63 A \cos (3 (c+d x))+36 C \cos (3 (c+d x))) \sec ^3(c+d x) \sin \left (\frac {1}{2} (c+d x)\right )}{48 d (a (1+\cos (c+d x)))^{3/2} \left (-1+\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1611\) vs. \(2(231)=462\).
Time = 11.65 (sec) , antiderivative size = 1612, normalized size of antiderivative = 6.06
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1612\) |
default | \(\text {Expression too large to display}\) | \(2028\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.41 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {12 \, \sqrt {2} {\left ({\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 3 \, {\left ({\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{5} + 2 \, {\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (9 \, {\left (7 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (37 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{2} - 6 \, A \cos \left (d x + c\right ) + 8 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{96 \, {\left (a^{2} d \cos \left (d x + c\right )^{5} + 2 \, a^{2} d \cos \left (d x + c\right )^{4} + a^{2} d \cos \left (d x + c\right )^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}}{\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.56 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.43 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\frac {\frac {6 \, \sqrt {2} {\left (17 \, A \sqrt {a} + 9 \, C \sqrt {a}\right )} \log \left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {6 \, \sqrt {2} {\left (17 \, A \sqrt {a} + 9 \, C \sqrt {a}\right )} \log \left (-\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3 \, {\left (47 \, A + 24 \, C\right )} \log \left ({\left | \frac {1}{2} \, \sqrt {2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} + \frac {3 \, {\left (47 \, A + 24 \, C\right )} \log \left ({\left | -\frac {1}{2} \, \sqrt {2} + \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {12 \, \sqrt {2} {\left (A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {2 \, \sqrt {2} {\left (204 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 96 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 176 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 96 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 45 \, A \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C \sqrt {a} \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (2 \, \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a^{2} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{48 \, d} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{(a+a \cos (c+d x))^{3/2}} \, dx=\int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
[In]
[Out]