Integrand size = 23, antiderivative size = 339 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {\left (12 a^2-54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 (a-b)^{9/2} d}+\frac {\left (12 a^2+54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 (a+b)^{9/2} d}-\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}} \]
[Out]
Time = 0.45 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2747, 755, 837, 843, 841, 1180, 212} \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=-\frac {\left (12 a^2-54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 d (a-b)^{9/2}}+\frac {\left (12 a^2+54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 d (a+b)^{9/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\sec ^2(c+d x) \left (2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)+b \left (3 a^2+11 b^2\right )\right )}{16 d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 d \left (a^2-b^2\right )^4 \sqrt {a+b \sin (c+d x)}}-\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^{3/2}} \]
[In]
[Out]
Rule 212
Rule 755
Rule 837
Rule 841
Rule 843
Rule 1180
Rule 2747
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {1}{(a+x)^{5/2} \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {b^3 \text {Subst}\left (\int \frac {\frac {1}{2} \left (6 a^2-11 b^2\right )+\frac {9 a x}{2}}{(a+x)^{5/2} \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 \left (a^2-b^2\right ) d} \\ & = -\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac {b \text {Subst}\left (\int \frac {\frac {1}{4} \left (-12 a^4+19 a^2 b^2-77 b^4\right )-\frac {5}{2} a \left (3 a^2-10 b^2\right ) x}{(a+x)^{5/2} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \\ & = -\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}+\frac {b \text {Subst}\left (\int \frac {\frac {1}{4} a \left (12 a^4-49 a^2 b^2+177 b^4\right )+\frac {1}{4} \left (18 a^4-81 a^2 b^2-77 b^4\right ) x}{(a+x)^{3/2} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^3 d} \\ & = -\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac {b \text {Subst}\left (\int \frac {\frac {1}{4} \left (-12 a^6+67 a^4 b^2-258 a^2 b^4-77 b^6\right )-\frac {1}{2} a \left (3 a^4-16 a^2 b^2-127 b^4\right ) x}{\sqrt {a+x} \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d} \\ & = -\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac {b \text {Subst}\left (\int \frac {\frac {1}{2} a^2 \left (3 a^4-16 a^2 b^2-127 b^4\right )+\frac {1}{4} \left (-12 a^6+67 a^4 b^2-258 a^2 b^4-77 b^6\right )-\frac {1}{2} a \left (3 a^4-16 a^2 b^2-127 b^4\right ) x^2}{-a^2+b^2+2 a x^2-x^4} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{4 \left (a^2-b^2\right )^4 d} \\ & = -\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}}-\frac {\left (12 a^2-54 a b+77 b^2\right ) \text {Subst}\left (\int \frac {1}{a-b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a-b)^4 d}+\frac {\left (12 a^2+54 a b+77 b^2\right ) \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sin (c+d x)}\right )}{32 (a+b)^4 d} \\ & = -\frac {\left (12 a^2-54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a-b}}\right )}{32 (a-b)^{9/2} d}+\frac {\left (12 a^2+54 a b+77 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b}}\right )}{32 (a+b)^{9/2} d}-\frac {b \left (18 a^4-81 a^2 b^2-77 b^4\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^{3/2}}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {a b \left (3 a^4-16 a^2 b^2-127 b^4\right )}{8 \left (a^2-b^2\right )^4 d \sqrt {a+b \sin (c+d x)}}+\frac {\sec ^2(c+d x) \left (b \left (3 a^2+11 b^2\right )+2 a \left (3 a^2-10 b^2\right ) \sin (c+d x)\right )}{16 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 2.11 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.87 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\frac {\frac {1}{2} \left (18 a^4-81 a^2 b^2-77 b^4\right ) \left ((a+b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \sin (c+d x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \sin (c+d x)}{a+b}\right )\right )-12 (a-b)^2 (a+b)^2 \sec ^4(c+d x) (-b+a \sin (c+d x))-15 a \left (3 a^2-10 b^2\right ) \left ((a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \sin (c+d x)}{a-b}\right )+(-a+b) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \sin (c+d x)}{a+b}\right )\right ) (a+b \sin (c+d x))-3 (a-b) (a+b) \sec ^2(c+d x) \left (3 a^2 b+11 b^3+\left (6 a^3-20 a b^2\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^2 \left (-a^2+b^2\right ) d (a+b \sin (c+d x))^{3/2}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(631\) vs. \(2(307)=614\).
Time = 1.02 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.86
method | result | size |
default | \(\frac {\frac {2 b^{5}}{3 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {12 b^{5} a}{\left (a +b \right )^{4} \left (a -b \right )^{4} \sqrt {a +b \sin \left (d x +c \right )}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 \left (a -b \right )^{4} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {17 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 \left (a -b \right )^{4} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 \left (a -b \right )^{4} \left (b \sin \left (d x +c \right )+b \right )^{2}}-\frac {25 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 \left (a -b \right )^{4} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {19 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 \left (a -b \right )^{4} \left (b \sin \left (d x +c \right )+b \right )^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a^{2}}{8 \left (a -b \right )^{4} \sqrt {-a +b}}-\frac {27 b \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right ) a}{16 \left (a -b \right )^{4} \sqrt {-a +b}}+\frac {77 b^{2} \arctan \left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {-a +b}}\right )}{32 \left (a -b \right )^{4} \sqrt {-a +b}}-\frac {3 b \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} a}{16 \left (a +b \right )^{4} \left (b \sin \left (d x +c \right )-b \right )^{2}}-\frac {17 b^{2} \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}}}{32 \left (a +b \right )^{4} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 b \sqrt {a +b \sin \left (d x +c \right )}\, a^{2}}{16 \left (a +b \right )^{4} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {25 b^{2} \sqrt {a +b \sin \left (d x +c \right )}\, a}{32 \left (a +b \right )^{4} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {19 b^{3} \sqrt {a +b \sin \left (d x +c \right )}}{32 \left (a +b \right )^{4} \left (b \sin \left (d x +c \right )-b \right )^{2}}+\frac {3 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a^{2}}{8 \left (a +b \right )^{\frac {9}{2}}}+\frac {27 b \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right ) a}{16 \left (a +b \right )^{\frac {9}{2}}}+\frac {77 b^{2} \operatorname {arctanh}\left (\frac {\sqrt {a +b \sin \left (d x +c \right )}}{\sqrt {a +b}}\right )}{32 \left (a +b \right )^{\frac {9}{2}}}}{d}\) | \(632\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1191 vs. \(2 (308) = 616\).
Time = 1.36 (sec) , antiderivative size = 5313, normalized size of antiderivative = 15.67 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx=\text {Hanged} \]
[In]
[Out]