Integrand size = 33, antiderivative size = 532 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{12 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 a^2 A+35 A b^2-20 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]
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Time = 2.13 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {3079, 3134, 3138, 2734, 2732, 3081, 2742, 2740, 2886, 2884} \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {\left (4 a^2 A-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right ) \sin (c+d x)}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\left (-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {b \left (-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right ) \sin (c+d x)}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {\left (-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3079
Rule 3081
Rule 3134
Rule 3138
Rubi steps \begin{align*} \text {integral}& = \frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{2} (-7 A b+4 a B)+a A \cos (c+d x)+\frac {5}{2} A b \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a} \\ & = -\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{4} \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {5}{2} a A b \cos (c+d x)-\frac {3}{4} b (7 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a^2} \\ & = -\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {3}{8} \left (a^2-b^2\right ) \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {3}{4} a b \left (3 a^2 A-7 A b^2+4 a b B\right ) \cos (c+d x)-\frac {1}{8} b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 a^3 \left (a^2-b^2\right )} \\ & = -\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\left (\frac {3}{16} \left (a^2-b^2\right )^2 \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {1}{8} a b \left (3 a^4 A-54 a^2 A b^2+35 A b^4+36 a^3 b B-20 a b^3 B\right ) \cos (c+d x)+\frac {1}{16} b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\left (-\frac {3}{16} b \left (a^2-b^2\right )^2 \left (4 a^2 A+35 A b^2-20 a b B\right )+\frac {1}{16} a b \left (a^2-b^2\right ) \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{24 a^4 \left (a^2-b^2\right )^2} \\ & = -\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (4 a^2 A+35 A b^2-20 a b B\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{8 a^4}-\frac {\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )}+\frac {\left (\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{24 a^4 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \\ & = \frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (\left (4 a^2 A+35 A b^2-20 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{8 a^4 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{24 a^3 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}} \\ & = \frac {\left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{12 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (4 a^2 A+35 A b^2-20 a b B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (27 a^2 A b-35 A b^3-12 a^3 B+20 a b^2 B\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (33 a^4 A b-170 a^2 A b^3+105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.19 (sec) , antiderivative size = 820, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\frac {\frac {2 \left (12 a^5 A b-216 a^3 A b^3+140 a A b^5+144 a^4 b^2 B-80 a^2 b^4 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (24 a^6 A+195 a^4 A b^2-566 a^2 A b^4+315 A b^6-132 a^5 b B+344 a^3 b^3 B-180 a b^5 B\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (33 a^4 A b^2-170 a^2 A b^4+105 A b^6-12 a^5 b B+104 a^3 b^3 B-60 a b^5 B\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b+b \cos (c+d x)}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-b^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-b^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) (-11 A b \sin (c+d x)+4 a B \sin (c+d x))}{4 a^4}-\frac {2 \left (-A b^4 \sin (c+d x)+a b^3 B \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {2 \left (-13 a^2 A b^4 \sin (c+d x)+9 A b^6 \sin (c+d x)+10 a^3 b^3 B \sin (c+d x)-6 a b^5 B \sin (c+d x)\right )}{3 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^3}\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(2003\) vs. \(2(585)=1170\).
Time = 40.59 (sec) , antiderivative size = 2004, normalized size of antiderivative = 3.77
method | result | size |
default | \(\text {Expression too large to display}\) | \(2004\) |
parts | \(\text {Expression too large to display}\) | \(3298\) |
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Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {\left (A + B \cos {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx=\int \frac {A+B\,\cos \left (c+d\,x\right )}{{\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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