Optimal. Leaf size=152 \[ -\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}}-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^4-a^4 \sin (c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.15, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2749, 2759,
2715, 2721, 2720} \begin {gather*} \frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac {10 a^4 \sin (c+d x) \sqrt {e \cos (c+d x)}}{d e^3}-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}}+\frac {12 a^8 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^4-a^4 \sin (c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2715
Rule 2720
Rule 2721
Rule 2749
Rule 2759
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^4}{(e \cos (c+d x))^{5/2}} \, dx &=\frac {a^8 \int \frac {(e \cos (c+d x))^{11/2}}{(a-a \sin (c+d x))^4} \, dx}{e^8}\\ &=\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}-\frac {\left (3 a^6\right ) \int \frac {(e \cos (c+d x))^{7/2}}{(a-a \sin (c+d x))^2} \, dx}{e^6}\\ &=\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (15 a^4\right ) \int (e \cos (c+d x))^{3/2} \, dx}{e^4}\\ &=-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (5 a^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{e^2}\\ &=-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}-\frac {\left (5 a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{e^2 \sqrt {e \cos (c+d x)}}\\ &=-\frac {10 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {e \cos (c+d x)}}-\frac {10 a^4 \sqrt {e \cos (c+d x)} \sin (c+d x)}{d e^3}+\frac {4 a^7 (e \cos (c+d x))^{9/2}}{3 d e^7 (a-a \sin (c+d x))^3}+\frac {12 a^6 (e \cos (c+d x))^{5/2}}{d e^5 \left (a^2-a^2 \sin (c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.06, size = 66, normalized size = 0.43 \begin {gather*} \frac {16 \sqrt [4]{2} a^4 \, _2F_1\left (-\frac {9}{4},-\frac {3}{4};\frac {1}{4};\frac {1}{2} (1-\sin (c+d x))\right ) (1+\sin (c+d x))^{3/4}}{3 d e (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 3.56, size = 263, normalized size = 1.73
method | result | size |
default | \(\frac {2 \left (-8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+30 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-48 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-15 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-18 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+48 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-20 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{4}}{3 \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, e^{2} d}\) | \(263\) |
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 = 0.12, size = 146, normalized size = 0.96 \begin {gather*} -\frac {15 \, {\left (-i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) + i \, \sqrt {2} a^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, {\left (i \, \sqrt {2} a^{4} \sin \left (d x + c\right ) - i \, \sqrt {2} a^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 11 \, a^{4} \sin \left (d x + c\right ) + 19 \, a^{4}\right )} \sqrt {\cos \left (d x + c\right )}}{3 \, {\left (d e^{\frac {5}{2}} \sin \left (d x + c\right ) - d e^{\frac {5}{2}}\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 = -0.01 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________