Integrand size = 23, antiderivative size = 455 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))} \]
[Out]
Time = 1.53 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3317, 3930, 4183, 4187, 4191, 3934, 2884, 3872, 3856, 2719, 2720} \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {b^2 \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a \sec (c+d x)+b)^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a \sec (c+d x)+b)}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right )}{4 a^4 d (a-b)^2 (a+b)^3}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{12 a^3 d \left (a^2-b^2\right )^2}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{12 a^3 d \left (a^2-b^2\right )^2} \]
[In]
[Out]
Rule 2719
Rule 2720
Rule 2884
Rule 3317
Rule 3856
Rule 3872
Rule 3930
Rule 3934
Rule 4183
Rule 4187
Rule 4191
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^{\frac {11}{2}}(c+d x)}{(b+a \sec (c+d x))^3} \, dx \\ & = \frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {\int \frac {\sec ^{\frac {5}{2}}(c+d x) \left (\frac {5 b^2}{2}-2 a b \sec (c+d x)+\frac {1}{2} \left (4 a^2-7 b^2\right ) \sec ^2(c+d x)\right )}{(b+a \sec (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )} \\ & = \frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\int \frac {\sec ^{\frac {3}{2}}(c+d x) \left (\frac {3}{4} b^2 \left (13 a^2-7 b^2\right )-a b \left (4 a^2-b^2\right ) \sec (c+d x)+\frac {1}{4} \left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\int \frac {\sqrt {\sec (c+d x)} \left (\frac {1}{8} b \left (8 a^4-61 a^2 b^2+35 b^4\right )+\frac {1}{2} a \left (2 a^4+14 a^2 b^2-7 b^4\right ) \sec (c+d x)-\frac {3}{8} b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{3 a^3 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {2 \int \frac {\frac {3}{16} b^2 \left (24 a^4-65 a^2 b^2+35 b^4\right )+\frac {1}{4} a b \left (20 a^4-64 a^2 b^2+35 b^4\right ) \sec (c+d x)+\frac {1}{16} \left (8 a^6+128 a^4 b^2-223 a^2 b^4+105 b^6\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 a^4 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {2 \int \frac {\frac {3}{16} b^3 \left (24 a^4-65 a^2 b^2+35 b^4\right )-\left (\frac {3}{16} a b^2 \left (24 a^4-65 a^2 b^2+35 b^4\right )-\frac {1}{4} a b^2 \left (20 a^4-64 a^2 b^2+35 b^4\right )\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 a^4 b^2 \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right )\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\left (b \left (24 a^4-65 a^2 b^2+35 b^4\right )\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \int \sqrt {\sec (c+d x)} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac {\left (b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{8 a^4 \left (a^2-b^2\right )^2} \\ & = \frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))}+\frac {\left (b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}+\frac {\left (\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2} \\ & = \frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \left (63 a^4-86 a^2 b^2+35 b^4\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{a+b},\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac {\left (8 a^4-61 a^2 b^2+35 b^4\right ) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac {b^2 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (b+a \sec (c+d x))^2}+\frac {b^2 \left (13 a^2-7 b^2\right ) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (b+a \sec (c+d x))} \\ \end{align*}
Time = 6.60 (sec) , antiderivative size = 747, normalized size of antiderivative = 1.64 \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\frac {\frac {2 \left (16 a^6+328 a^4 b^2-641 a^2 b^4+315 b^6\right ) \cos ^2(c+d x) \left (\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )-\operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right )\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {2 \left (160 a^5 b-512 a^3 b^3+280 a b^5\right ) \cos ^2(c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) (b+a \sec (c+d x)) \sqrt {1-\sec ^2(c+d x)} \sin (c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac {\left (72 a^4 b^2-195 a^2 b^4+105 b^6\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (-4 a b+4 a b \sec ^2(c+d x)-4 a b E\left (\left .\arcsin \left (\sqrt {\sec (c+d x)}\right )\right |-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 (2 a-b) b \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}-4 a^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}+2 b^2 \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\sqrt {\sec (c+d x)}\right ),-1\right ) \sqrt {\sec (c+d x)} \sqrt {1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {\sec (c+d x)} \left (-\frac {b \left (24 a^4-65 a^2 b^2+35 b^4\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2}-\frac {b^3 \sin (c+d x)}{2 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {3 \left (5 a^2 b^3 \sin (c+d x)-3 b^5 \sin (c+d x)\right )}{4 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {2 \tan (c+d x)}{3 a^3}\right )}{d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(2100\) vs. \(2(499)=998\).
Time = 160.23 (sec) , antiderivative size = 2101, normalized size of antiderivative = 4.62
[In]
[Out]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
[In]
[Out]