Optimal. Leaf size=282 \[ \frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.24, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2743, 2833,
2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 d \left (a^2-b^2\right )^3 \sqrt {a+b \cos (c+d x)}}-\frac {16 a b \sin (c+d x)}{15 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))^{3/2}}-\frac {2 b \sin (c+d x)}{5 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{5/2}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2743
Rule 2831
Rule 2833
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (c+d x))^{7/2}} \, dx &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {5 a}{2}+\frac {3}{2} b \cos (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{5 \left (a^2-b^2\right )}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\frac {3}{4} \left (5 a^2+3 b^2\right )-2 a b \cos (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{15 \left (a^2-b^2\right )^2}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}-\frac {8 \int \frac {-\frac {1}{8} a \left (15 a^2+17 b^2\right )-\frac {1}{8} b \left (23 a^2+9 b^2\right ) \cos (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}-\frac {(8 a) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{15 \left (a^2-b^2\right )^2}+\frac {\left (23 a^2+9 b^2\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{15 \left (a^2-b^2\right )^3}\\ &=-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}+\frac {\left (\left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{15 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (8 a \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}{15 \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {2 \left (23 a^2+9 b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {16 a \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{15 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {2 b \sin (c+d x)}{5 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{5/2}}-\frac {16 a b \sin (c+d x)}{15 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))^{3/2}}-\frac {2 b \left (23 a^2+9 b^2\right ) \sin (c+d x)}{15 \left (a^2-b^2\right )^3 d \sqrt {a+b \cos (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.57, size = 189, normalized size = 0.67 \begin {gather*} \frac {2 \left (\frac {\left (\frac {a+b \cos (c+d x)}{a+b}\right )^{5/2} \left (\left (23 a^2+9 b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )+8 a (-a+b) F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )\right )}{(a-b)^3}+\frac {b \left (34 a^4-5 a^2 b^2+3 b^4+2 a b \left (27 a^2+5 b^2\right ) \cos (c+d x)+b^2 \left (23 a^2+9 b^2\right ) \cos ^2(c+d x)\right ) \sin (c+d x)}{\left (-a^2+b^2\right )^3}\right )}{15 d (a+b \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.62, size = 616, normalized size = 2.18
method | result | size |
default | \(-\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{10 b^{2} \left (a -b \right ) \left (a +b \right ) \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{3}}+\frac {8 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{15 b \left (a -b \right )^{2} \left (a +b \right )^{2} \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+\frac {a -b}{2 b}\right )^{2}}+\frac {4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (23 a^{2}+9 b^{2}\right )}{15 \left (a -b \right )^{3} \left (a +b \right )^{3} \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b -a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {2 \left (15 a^{2}-8 a b +9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )}{\left (15 a^{5}+15 a^{4} b -30 a^{3} b^{2}-30 a^{2} b^{3}+15 a \,b^{4}+15 b^{5}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}-\frac {2 \left (23 a^{2}+9 b^{2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {\frac {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a -b}{a -b}}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {-\frac {2 b}{a -b}}\right )\right )}{15 \left (a -b \right )^{2} \left (a +b \right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +\left (a +b \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}\right )}{\sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a +b}\, d}\) | \(616\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.19, size = 985, normalized size = 3.49 \begin {gather*} -\frac {6 \, {\left (34 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 3 \, b^{6} + {\left (23 \, a^{2} b^{4} + 9 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (27 \, a^{3} b^{3} + 5 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + {\left (\sqrt {2} {\left (-i \, a^{3} b^{3} + 33 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, \sqrt {2} {\left (i \, a^{4} b^{2} - 33 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, \sqrt {2} {\left (i \, a^{5} b - 33 i \, a^{3} b^{3}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-i \, a^{6} + 33 i \, a^{4} b^{2}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) + {\left (\sqrt {2} {\left (i \, a^{3} b^{3} - 33 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3} - 3 \, \sqrt {2} {\left (-i \, a^{4} b^{2} + 33 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 3 \, \sqrt {2} {\left (-i \, a^{5} b + 33 i \, a^{3} b^{3}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (i \, a^{6} - 33 i \, a^{4} b^{2}\right )}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right ) - 3 \, {\left (\sqrt {2} {\left (23 i \, a^{2} b^{4} + 9 i \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (23 i \, a^{3} b^{3} + 9 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (23 i \, a^{4} b^{2} + 9 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (23 i \, a^{5} b + 9 i \, a^{3} b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (-23 i \, a^{2} b^{4} - 9 i \, b^{6}\right )} \cos \left (d x + c\right )^{3} + 3 \, \sqrt {2} {\left (-23 i \, a^{3} b^{3} - 9 i \, a b^{5}\right )} \cos \left (d x + c\right )^{2} + 3 \, \sqrt {2} {\left (-23 i \, a^{4} b^{2} - 9 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-23 i \, a^{5} b - 9 i \, a^{3} b^{3}\right )}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) + 2 \, a}{3 \, b}\right )\right )}{45 \, {\left ({\left (a^{6} b^{4} - 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{3} + 3 \, {\left (a^{7} b^{3} - 3 \, a^{5} b^{5} + 3 \, a^{3} b^{7} - a b^{9}\right )} d \cos \left (d x + c\right )^{2} + 3 \, {\left (a^{8} b^{2} - 3 \, a^{6} b^{4} + 3 \, a^{4} b^{6} - a^{2} b^{8}\right )} d \cos \left (d x + c\right ) + {\left (a^{9} b - 3 \, a^{7} b^{3} + 3 \, a^{5} b^{5} - a^{3} b^{7}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________