Integrand size = 25, antiderivative size = 495 \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \left (6 a^2 b c-a b^2 c-3 b^3 c-3 a^3 d+a^2 b d\right ) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \]
[Out]
Time = 0.89 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4008, 4145, 4143, 4006, 3869, 3917, 4089} \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=-\frac {2 c \sqrt {a+b} \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a f \left (a^2-b^2\right ) (a+b \sec (e+f x))^{3/2}}+\frac {2 \left (-4 a^3 d+7 a^2 b c-3 b^3 c\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{3 a^2 b f (a-b) (a+b)^{3/2}}-\frac {2 \left (-3 a^3 d+6 a^2 b c+a^2 b d-a b^2 c-3 b^3 c\right ) \cot (e+f x) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (\sec (e+f x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{3 a^2 b f (a-b) (a+b)^{3/2}}+\frac {2 b \left (-4 a^3 d+7 a^2 b c-3 b^3 c\right ) \tan (e+f x)}{3 a^2 f \left (a^2-b^2\right )^2 \sqrt {a+b \sec (e+f x)}} \]
[In]
[Out]
Rule 3869
Rule 3917
Rule 4006
Rule 4008
Rule 4089
Rule 4143
Rule 4145
Rubi steps \begin{align*} \text {integral}& = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}-\frac {2 \int \frac {-\frac {3}{2} \left (a^2-b^2\right ) c+\frac {3}{2} a (b c-a d) \sec (e+f x)-\frac {1}{2} b (b c-a d) \sec ^2(e+f x)}{(a+b \sec (e+f x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )} \\ & = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {4 \int \frac {\frac {3}{4} \left (a^2-b^2\right )^2 c-\frac {1}{4} a \left (6 a^2 b c-2 b^3 c-3 a^3 d-a b^2 d\right ) \sec (e+f x)-\frac {1}{4} b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \sec ^2(e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {4 \int \frac {\frac {3}{4} \left (a^2-b^2\right )^2 c+\left (\frac {1}{4} b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right )-\frac {1}{4} a \left (6 a^2 b c-2 b^3 c-3 a^3 d-a b^2 d\right )\right ) \sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}-\frac {\left (b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right )\right ) \int \frac {\sec (e+f x) (1+\sec (e+f x))}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2} \\ & = \frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}}+\frac {c \int \frac {1}{\sqrt {a+b \sec (e+f x)}} \, dx}{a^2}+\frac {\left (a b^2 c+3 b^3 c+3 a^3 d-a^2 b (6 c+d)\right ) \int \frac {\sec (e+f x)}{\sqrt {a+b \sec (e+f x)}} \, dx}{3 a^2 (a-b) (a+b)^2} \\ & = \frac {2 \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \cot (e+f x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}+\frac {2 \left (a b^2 c+3 b^3 c+3 a^3 d-a^2 b (6 c+d)\right ) \cot (e+f x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{3 a^2 (a-b) b (a+b)^{3/2} f}-\frac {2 \sqrt {a+b} c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (e+f x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (e+f x))}{a+b}} \sqrt {-\frac {b (1+\sec (e+f x))}{a-b}}}{a^3 f}+\frac {2 b (b c-a d) \tan (e+f x)}{3 a \left (a^2-b^2\right ) f (a+b \sec (e+f x))^{3/2}}+\frac {2 b \left (7 a^2 b c-3 b^3 c-4 a^3 d\right ) \tan (e+f x)}{3 a^2 \left (a^2-b^2\right )^2 f \sqrt {a+b \sec (e+f x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(1589\) vs. \(2(495)=990\).
Time = 14.93 (sec) , antiderivative size = 1589, normalized size of antiderivative = 3.21 \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\frac {(b+a \cos (e+f x))^3 \sec ^2(e+f x) (c+d \sec (e+f x)) \left (\frac {2 \left (-7 a^2 b c+3 b^3 c+4 a^3 d\right ) \sin (e+f x)}{3 a^2 \left (a^2-b^2\right )^2}-\frac {2 \left (b^3 c \sin (e+f x)-a b^2 d \sin (e+f x)\right )}{3 a^2 \left (a^2-b^2\right ) (b+a \cos (e+f x))^2}-\frac {2 \left (-8 a^2 b^2 c \sin (e+f x)+4 b^4 c \sin (e+f x)+5 a^3 b d \sin (e+f x)-a b^3 d \sin (e+f x)\right )}{3 a^2 \left (a^2-b^2\right )^2 (b+a \cos (e+f x))}\right )}{f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{5/2}}-\frac {2 (b+a \cos (e+f x))^{5/2} \sec ^{\frac {3}{2}}(e+f x) (c+d \sec (e+f x)) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (-7 a^3 b c \tan \left (\frac {1}{2} (e+f x)\right )-7 a^2 b^2 c \tan \left (\frac {1}{2} (e+f x)\right )+3 a b^3 c \tan \left (\frac {1}{2} (e+f x)\right )+3 b^4 c \tan \left (\frac {1}{2} (e+f x)\right )+4 a^4 d \tan \left (\frac {1}{2} (e+f x)\right )+4 a^3 b d \tan \left (\frac {1}{2} (e+f x)\right )+14 a^3 b c \tan ^3\left (\frac {1}{2} (e+f x)\right )-6 a b^3 c \tan ^3\left (\frac {1}{2} (e+f x)\right )-8 a^4 d \tan ^3\left (\frac {1}{2} (e+f x)\right )-7 a^3 b c \tan ^5\left (\frac {1}{2} (e+f x)\right )+7 a^2 b^2 c \tan ^5\left (\frac {1}{2} (e+f x)\right )+3 a b^3 c \tan ^5\left (\frac {1}{2} (e+f x)\right )-3 b^4 c \tan ^5\left (\frac {1}{2} (e+f x)\right )+4 a^4 d \tan ^5\left (\frac {1}{2} (e+f x)\right )-4 a^3 b d \tan ^5\left (\frac {1}{2} (e+f x)\right )-6 a^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+12 a^2 b^2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 b^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 a^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+12 a^2 b^2 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}-6 b^4 c \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+(a+b) \left (-7 a^2 b c+3 b^3 c+4 a^3 d\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}+a (a+b) \left (-2 b^2 c+3 a^2 (c-d)+a b (3 c-d)\right ) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {a-b}{a+b}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {a+b-a \tan ^2\left (\frac {1}{2} (e+f x)\right )+b \tan ^2\left (\frac {1}{2} (e+f x)\right )}{a+b}}\right )}{3 a^2 \left (a^2-b^2\right )^2 f (d+c \cos (e+f x)) (a+b \sec (e+f x))^{5/2} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (a \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )-b \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(7172\) vs. \(2(456)=912\).
Time = 17.34 (sec) , antiderivative size = 7173, normalized size of antiderivative = 14.49
method | result | size |
parts | \(\text {Expression too large to display}\) | \(7173\) |
default | \(\text {Expression too large to display}\) | \(7241\) |
[In]
[Out]
\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int \frac {c + d \sec {\left (e + f x \right )}}{\left (a + b \sec {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int { \frac {d \sec \left (f x + e\right ) + c}{{\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {c+d \sec (e+f x)}{(a+b \sec (e+f x))^{5/2}} \, dx=\int \frac {c+\frac {d}{\cos \left (e+f\,x\right )}}{{\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
[In]
[Out]